Chapter 5: Deportation Ghettoes in Mali: Expelled Migrants Between State Exclusion and Self-Organization

Clara Lecadet

in Gaibazzi, P., Bellagamba, A., Dünnwald, S. (eds.), EurAfrican Borders and Migration Management, Palgrave Macmillan, 2017, 109-125

From expulsion to the ghetto

The expulsion of foreigners is a powerful, symbolic means of tracing the shape and limits of citizenship and nation. As Daniel [START_REF] Kanstroom | Deportation nation: outsiders in American history[END_REF] and Mae [START_REF] Ngai | Impossible subjects: illegal aliens and the making of modern America[END_REF] have shown in relation to American history, the expulsion of foreigners has been a central element in defining the limits of citizenship and in the construction and reinforcement of the prerogatives of the federal government. The evolution of the European project seems to follow the same pattern, since recent developments in European politics have promoted the notion of sending back illegal immigrants as part of a shared framework for a common migration policy [START_REF] Favell | L'européanisation ou l'émergence d'un nouveau « champ politique » : le cas de la politique d'immigration[END_REF][START_REF] Guiraudon | Les effets de l'européanisation des politiques d'immigration et d'asile[END_REF]. The main meaning and purpose of expulsion is thus to be found in the delimitation and differentiation of political space and entities in a national and/or federal sense. [START_REF] Kanstroom | Deportation nation: outsiders in American history[END_REF] uses the expression 'extend border control' to refer to this wide and diffuse political function, since the use of expulsion goes beyond basic border control. Expulsion is in fact rooted in a series of distinctions that make up the nation: defining what is inside and outside state sovereignty and differentiating foreigners from nationals.

As [START_REF] Kanstroom | Aftermath: deportation law and the new American diaspora[END_REF] remarks, however, hand in hand with these political distinctions go social rejection and separation. In the United States, expulsion policy, with its stated aim of maintaining state control of territorial borders, in reality hides a process of social selection aimed in particular at foreigners, residing legally or illegally in the country, who have committed an offence and who are thus the immediate target of expulsion measures. The fundamental legitimacy of the American expulsion system is to be found in the criminalization of immigrants, which reinforces the social and racial inequalities already at work in American society.

However the process of social differentiation caused by deportation is not only at work within the countries where deportations are taking place. It appears as a major element when trying to consider the general consequences of deportation [START_REF] Peutz | Embarking on an Anthropology of Removal[END_REF]. The deportation aftermath [START_REF] Kanstroom | Aftermath: deportation law and the new American diaspora[END_REF] has legal, political, and social dimensions [START_REF] Drotbohm | On the durability and the decomposition of citizenship: the social logics of forced return migration in Cape Verde[END_REF]2012) that are mostly unexplored. The effects produced by expulsion in terms of both political exclusion and social marginalization are incalculable. Liz [START_REF] Fekete | The deportation machine: Europe, asylum and human rights[END_REF] shows the risk and also the danger of some expulsions which expose migrants to reprisals by the state or repercussions in terms of social and family relationships in their country of origin: being expelled can lead to death, she explains, using a series of examples which show sharply how the individual is crushed by a dehumanizing system. Among the diverse effects of expulsion across the globe (de Genova and [START_REF] Peutz | Criminal alien' deportees in Somaliland[END_REF], are family separation, bad treatment to which expelled migrants are exposed during and after their expulsion, and also the establishment in certain countries, such as the Azores, Guatemala and Ecuador, of agencies supposed to facilitate reintegration [START_REF] Kanstroom | Aftermath: deportation law and the new American diaspora[END_REF]. As initiatives in favor of integration for expelled migrants remain scarce, exclusion and marginalization increasingly mark the post-deportation stage.

The setting up of ghettoes in Mali by expelled migrants, which this chapter addresses,1 is an example of the collective social and political reconstruction which characterizes the post-expulsion stage. The settlement of ghettoes corresponds to the locations where deportations, either by air or by road, take place. In the capital city Bamako, expelled migrants flewn back mostly from Europe, America or Saudi Arabia but also returned overland after being expelled from neighboring countries such as Mauritania, Libya and Algeria. At the border zone between Algeria and Mali, in the area of Tinzawaten, as well as in the cities of Kidal and Gao in Northern Mali, migrants expelled from Algeria were either in transit or settled for long periods. Expellees organized themselves on the basis of nationality within areas which they called ghettoes and which served not only as refuge and shelter but also as a base for arranging their onward travel. The emergence of ghettoes seems to have arisen out of two demands, social and political, following the expulsion of foreigners by individual countries.

Ghettoes are the social translation of the banishment inherent in expulsion, the social form of its political dimension (Lecadet, 2014b). They testify to fellowship and to the internalization, by the group and by the individuals within it, of their excluded status. At the same time, they bear the stamp of an underlying criticism of state powers as regards this status, because the historical and symbolic resonance of the term 'ghetto' includes a ferment of claims and collective protest [START_REF] Becker-Ho | Le premier ghetto ou l'exemplarité vénitienne[END_REF][START_REF] Lecadet | Tinzawaten, c'est le grand danger pour nous les immigrés ![END_REF]Lecadet, , 2014a)). It speaks, in simple terms, almost a universal language, of the experience of rejection, of the exclusion inherent in expulsion. It describes marginalization at the border or in the towns which line the postexpulsion routes in Kidal, Gao and Bamako and where migrants stop temporarily or are stuck after their expulsion. It tells also of the possibility of a collective life after expulsion, of a form of autonomy in exclusion and separation. It speaks of a form of self-help and minimal solidarity in expulsion and testifies to the fact that after expulsion something remains, persists and resists, something which cannot simply be reduced to the prospective return of the migrants to their point of departure envisioned by deportation policies.

Expelled migrants's choice of the term 'ghetto' to describe their living areas consequently points to socio-political understandings of expulsion. My aim in this chapter is to examine, on the basis of three ghettoes set up by migrants in respectively Tinzawaten, Gao and Bamako after their expulsion from Algeria, three essential meanings and functions of ghettoes in relation to expulsion. As expulsion from Algeria largely reflects European pressures on Northern African states to prevent African migrants from reaching southern Europe, the chapter also contributes to a more nuanced understanding of the spatial reconfigurations induced, though not entirely determined, by the externalization of European borders into Africa. It equally highlights the role of African contexts and migrants in forging particular political cultures in response to such border regimes. At Tinzawaten on the northern border of Mali, where the Algerian police abandoned the expelled migrants, the ghettoes were places of both refuge and survival, emblematic of state selection and exclusion, and places where migrants reaffirmed national affiliations. In Gao, the most important town in Northern Mali, 600 km from the Algerian border and 1200 km from the capital Bamako, the ghettoeswhich were home to migrants en route to Algeria and to others who had been expelled from that country once or even several timesseemed to be intermediary places within a largely mobile structure. Finally, Bamako ghetto corresponded to the more classically accepted meaning of the term as a place of urban marginalization, housing migrants who have no way of moving on after their expulsion. The configuration and function of these three ghettoes are partially overlapping: taken all together as they spread out from the borders of Mali and Algeria, they show ghettoes as part of a paradoxical dynamic of abandonment and survival, of isolation and fellowship, of stagnation and mobility.

Tinzawaten: the border ghetto

Deportation is an ancient phenomenon inherent in the social and political structure of the African continent, but it seems to have taken on a new momentum with the externalization of European borders, which since 2000 has been increasing the pressure on the Maghreb countries to limit the flow of migrants trying to reach Europe. The Tamanrasset-Tinzawaten axis has been functioning as a gathering place for undocumented migrants of various nationalities arrested in Algeria, and to a lesser extent in Morocco and Libya, and who are soon to be deported. The Algerian state put in place measures for expelling foreigners overland from Tamanrasset, in the south of Algeria, to Tinzawaten, a village located on the border between Algeria and Mali, from 1990 up to the outbreak of war in Mali in 2012, which marked a decrease in expulsions to this area. The result was the establishment of new places, corresponding to the geo-political areas defined by the expulsion process but which were also self-organizing centers where destitute and abandoned migrants set up the conditions necessary for survival.

Migrants described Tinzawaten as an infernal, unbearable place in almost proverbial pronouncements. Many said 'Tinzawaten is 7 kilometers of hell', which presents it as, paradoxically, a place of death, a place of inhumanity and struggle, yet where life goes on and is organized. This has resonances of the description of the border given by Etienne Balibar, for whom this is a lifeless area, but one which by necessity ends up being inhabited by the people who cross it:

For a poor man from a poor country, the border tends to be something else: not only is it an obstacle that is very difficult to cross, it is also a place which one continually comes up against, one which, at the mercy of expulsions and family gatherings, one crosses and re-crosses, and where finally one ends up. It is an extraordinarily glutinous spacio-temporal zone, a place for 'almost living'a waiting-room for life, a non-life. The psychoanalyst André Green wrote somewhere that it is difficult enough to live on a border, but that this is nothing to actually being oneself a border. He meant this in the sense of the pull of multiple identities, migrant identities, but one must also look at the material basis of this (Balibar, 2007, p. 529).

The geographical location of Tinzawaten in the middle of the Sahara, the lack of resources and the absence of infrastructure, could be the most eloquent of metaphors for a world partially removed from the rules that govern normal social life, removed from the habits, the duration and the stability of established communities. In an unknown, inhospitable place without resources and as a matter of urgency, migrants are forced to organize some kind of 'community'. Every element of this singular situation seems to lead to a withdrawal from the politics that normally bases the life of an individual within a permanent community: the absence of any legal dimension in the expulsion process from Tamanrasset, the fact that the expelled migrants are left 'in the middle of nowhere', in a a no man's land between the two borders, the frequent confiscation of the legal documents proving identity. Yet, despite this, something is in fact immediately re-formed. Tinzawaten is where immediate re-socialization takes place, however dramatic the circumstances, because national identity is a decisive criterion in regrouping and an essential factor in the solidarity to be found in this little island of abandoned houses, formed into ghettoes, with each one allocated to a separate nationality.

From the end of the 1990s Tamanrasset-Tinzawaten was the principal route of regular expulsions in the Saharan region, in terms of both its use by the Algerian police for convoys of expelled migrants, and the number of convoys that made the trip. A convoy was made up of between three and seven lorries, each transporting about 40 people, mainly men. When the Tuareg rebellion in the 1990s led to increased control by the central Malian state over the region between Gao and the border with Algeria, the former police commissioner in Kidal, a small town about 350 kilometers to the south of the Algerian-Malian border, took the initiative of registering, from 2000 onwards, all foreigners passing through the town, thereby listing more than 2,000 expelled migrants arriving in this zone each year. In the context of a political crisis such as the war in Libya in 2011, the return of massive numbers of foreign Africans fleeing Libya meant that this figure increased considerably. With the outbreak of the war in Mali in 2012, the expulsions from Algeria seem to have decreased, but this made the situation of foreigners on Algerian soil even more difficult because of overpopulation in detention centers there. Mali has never objected to the arrival on its territory of migrants expelled overland from other countries such as Algeria, Mauritania, Morocco or Libya, whatever their nationality. Amongst these migrants were Malians, of course, but also people from Ghana, Gambia, Liberia, Cameroon, Nigeria, Senegal, Guinea, Burkina Faso and Niger amongst other nationalities.

Once expelled, these groups formed national micro-communities in a little island of ruined houses separated from Algeria by a dried up wadi. The terminology used by the migrants to describe this zone may seem surprising, as there were 'presidents', 'ministers' and also 'coups' and 'civil wars'; in short, all the functions and tensions to be found in traditional political societies. In the various ghettoes -Cameroonian, Nigerian, Liberian, Malianorganizations linked to their nationality supported the migrant groups. On this autonomous site, migrants reappropriated the regulations, functions and protocol of their various countries.

Rapidly and in simplified form, each national community established regulations loaded with specific political, historical and cultural references.

The term 'ghetto' suggests isolation, exclusion and destitution, but also the reconstruction of a collective life based on nationality. In the ghettoes, conflicts are resolved collectively within the national micro-communities led by chiefs, generals, ministers and presidents. Each ghetto mimics a power structure, whether it is chiefdom, government or the army. This mimicry carries with it an equal degree of subversion and irony in relation to the situation of extreme abandonment in which the migrants have been left. These national groupings thus have a social dimension and political significance. Nationality is not here the prerogative of a state, or a simple marker for the individual, rather it is the process of recomposition itself and the reorganization of individuals at the time of their expulsion. This moment, when the migrants have often lost everything, is also the time for spontaneously reappropriating political norms and rebuilding a form of civilian life. The strong national link makes up for the absence of institutional support. Moreover, this phenomenon operates beyond the border zone, as will be seen in the organization of the Cameroonians in Gao and Bamako.

In ghettoes as in other organized groups, a whole terminology is in fact set up, redirecting and subverting the names, norms and functions of institutional politics. For instance, the term chosen by Liberians to describe their organization following their sporadic expulsion from Algeria, seems emblematic of the invention and use by expelled migrants of a frontier vocabulary (see Introduction). The International Brothers Association Network or Interban, was created in 2006 by Liberian migrants with the aim of facing up to the most dramatic consequences of their expulsion from Algeria. Their organization was born out of this banishment and was defined by it, to the extent that the contraction of the organization's name, which the Liberians used amongst themselves in the desert or during their expulsion, played on the double meaning of banishment and solidarity. Interban is the family created in banishment which emerged from expulsion and which affirms its unity in segregation by restating their national link. This terminological creativity is thus emblematic of the fellowship that allows migrants to face up to the highly dramatic consequences of the limitations set on their mobility.

In sum, ghettoes became a symbol of the rejection and selection operated at the border, but also of the self-organization of the migrants in the face of expulsion. Left entirely alone, exhausted, some of them ill, these men who were abandoned in the desert were very likely to die. Lifeless corpses were regularly found in the desert; digging a grave was too much for the others who would have exhausted their remaining strength in the effort. The consolidation of the national link became the only means of facing up to the most dramatic consequences of expulsion in this zone. From the year 2000 onwards the ghettoes took shape. The Nigerians in particular invested a great deal in their ghetto, which was inhabited and organized as a destination for those expelled from Algeria, as a sort of base or platform for those passing through on their way there, and an opportunity for those already expelled to set off again. In this way the ghetto aimed to overcome the limitations of expulsion by strengthening the conditions for group organization and promoting mobility. These national micro-communities were ephemeral, filled up and emptied, formed and broken up, subject to arrivals and departures. When a ghetto emptied, the register was entrusted to the leader of another ghetto who would hand it over to the migrants of the relevant nationality when the next convoy arrived. These micro-communities were indicative of the migrants' self-organization in the very place of their rejection and their most tragic relegation. At this time and in this place following expulsion, networks were organized on a regional scale. The migrants themselves occupied a central position in the provision of aid and of the means of survival for those arriving and for those returning home, but also, more discretely, for those preparing to set off again to Algeria.

From the moment of expulsion a micro-economy of mobility was thus put in place. By organizing the convoys of those who, despite their expulsion, wanted to return to Algeria, the leaders of the ghettoes and their assistants usually earned a free trip for themselves. Those wanting to return to Algeria could thus settle in the Tinzawaten ghettoes for periods of up to several months at a time. Despite the almost constant renewal of its members through the various convoys arriving from Algeria, the organization of each ghetto remained the same. Individuals chose to stay or to set off again according to the material, physical and moral resources they had left when they were expelled. There were other elements influencing this choice: a fortnightly mission was set up by the Malian Red Cross between 2009 and 2011, collecting around fifty migrants each time and taking them to the migrant center in Gao where they were helped to return to their country of origin; the migrants also knew on their arrival at Tinzawaten that in Kidal, Gao and Bamako there were places where they could be temporarily housed [START_REF] Lecadet | Itinéraires de la faim' des migrants subsahariens expulsés d'Algérie au Mali[END_REF]. The permanence of the ghettoes in Tinzawaten and Gao shows, however, that human movement outstrips to a certain extent political constraints and tries to bypass them.

The political structure of the Tinzawaten ghettoes therefore supported their strategies for survival and for facilitating new movement, new migrations denied to them by the expulsion process. The Gao ghettoes followed this same pattern. Secret, discrete places, not readily accessible, the ghettoes set up by migrants seemed in fact to have a function that ran counter to the aid initiatives such as the Red Cross' that began to be set up for migrants in Northern Mali and in Bamako from 2009 onwards, with the aim of assisting their return to their country of origin. The attempt to manage mobility through institutionalized expulsion policies, as well as by the various mechanisms promoting the return of foreigners to their country of origin, is constantly tested on the ground by the ineradicable desire for autonomy that is evident in the journeys undertaken by migrants and in the places in which they choose to live, individually and collectively. The organization of border ghettoes is partly aimed at thwarting these attempts at expulsion and marginalization by re-creating forms of autonomy and potential mobility.

Gao: the intermediate ghetto

Because they were established at the very point of expulsion from Algeria, the Tinzawaten ghettoes were an extreme kind of ghetto, places that felt the effects of mass expulsion and autonomous reorganization. It was in this preliminary situation that the ghetto took on its most pronounced political expression. Whereas the migrant ghettoes near the border reproduced and redirected the vocabulary and practices of politics, the other kinds of ghetto, which lined the route of migrants after their expulsion into Mali, took on more of a domestic flavor as regards the choice of place and their internal organization. Mahamet [START_REF] Timera | Aventuriers ou orphelins de la migration internationale: nouveaux et anciens migrants subsahariens au Maroc[END_REF] uses the description of a migrant house in Morocco to show their importance as an intermediary space where different personalities and different strata of society meet, each with their separate motives for migration. In this respect, the organization of the Gao ghettoes in the phase following expulsion showed many similarities with the temporary migrant houses in Morocco.

Like all the places where migrants regroup and which are marginal to establishments offering institutional support, the ghettoes played a central but relatively unnoticed role in the overall movement of people and the attempts to set off again after expulsion. They were places for living and also negotiating places in which post-expulsion departures were set up and where people stayed briefly before setting off again. As in Tinzawaten, national identity continued to be a determining criterion in the reorganization of expelled migrants in the various parts of Northern Mali and in Bamako. Even so, the ghettoes in Gao or Bamako took on neither the symbols of state nor titles nor political and/or military functions in their organization. The Gao ghettoes were closed, private places, in privately owned houses rented or lent to migrants by their Malian owners who had links with people 'passers'.

From this point of view, the ghetto seems more like a stage on a journey, but it is also an intermediary place in an overall mobility economy linking the migrants, the people 'passers' and the house owners. A 'fixer' would often pay rent on a house to its Malian owner. In exchange, the migrants would have to organize their transport through him. He received a commission on the trips made by Tamashek drivers. This mobility economy ran counter to the aims of institutionalized reception centers, which promoted the return of migrants to their countries of origin. International NGOs such as Secours Catholique or the Catholic Committee against Hunger and for Development (CCFD) gave financial support via migrant associations in Gao and Bamako to destitute expelled migrants, but only for travel to their country of origin. Inside the ghettoes, by contrast, the migrants were free to choose their destination. The migrant center which operated in Gao between 2009 and 2011 was an institutional space where migrants were welcomed, but the rigorous protocol, the very limited time they were allowed to stay and the financing of their return home all amounted to a very limited offer in relation to the aspirations of the migrants themselves. The mobility economy, which operated underground in the ghettoes, ran counter to institutional, humanitarian policies which saw expulsion and returning home as synonymous and focused on financing the return of migrants to their country of origin.

In Gao, the limits placed on the movement of expelled migrants by the Malian Red Cross and the migrant center caused bad feelings in the Cameroonian ghetto towards the NGOs and migration policies. Situated near the Tizi-Mizi Hotel, where American soldiers from the military base in Gao were billeted, the Cameroonian ghetto was in a house separated from the road by a low wall and with a large interior courtyard like all traditional Malian houses. The rules of the house were posted on the front door. Rules of this kind and this need for organizing the group were to be found in all the places where migrants gathered after expulsion. In spite of the fact that it presented itself as an autonomous community, the ghetto was also a place where transactions took place between the migrants and a 'fixer' who paid all of the rent: the residents of the ghetto were tied to the Malian owner by an agreement covering the transport needs of the group. Housing was closely linked to transport. The Cameroonian ghetto in Gao, like others in the town, was built on this exchange of transport and housing: the expelled migrants did not pay for the places they stayed in, the profits being made by both the people 'passers' and the owners of the houses via commission on the price of transport. On the other hand, the owner seemed to have no rights over the organization of the migrants inside the house. The organization and sharing of domestic tasks and the principles of communal life were the business of the migrants alone.

Bamako: the urban ghetto

The reappropriation of the terminology of segregation through the use of the word 'ghetto' extends expulsion well beyond the event itself. Informal post-expulsion communities can be seen as a by-product of expulsion methods, in the sense that these methods encourage the formation of new groups and sort individuals into the political categories which led to their expulsion. These people, weighed down by the failure of their adventure and set apart by what is often the impossibility of return, use the ghettoes as a way of reformulating their situation, which continues therefore to have a political significance.

The spaces occupied by migrants after their expulsion extended well beyond the border zone and its surrounding area, and recreated small islands of exclusion and ghettoization far beyond what might be considered the end point of their expulsion. Once they had left Tinzawaten, many migrants hoped to get to Bamako, a crossroads town where various forms of solidarity are in operation. Some benefited from the help that is offered there by local communities of their own nationality through the tontine system 2 . Thus, since the year 2000 Bamako has gained various sites where expelled migrants from different Central African countries take refuge and regroup. While many migrants lived in a state of complete abandonment around the Sogoniko bus station, forced to sleep in the street when the shame of returning home was too great, others had grouped together in a relatively permanent way, creating a place which for a long time was known as the Magnambougou ghetto. This ghetto in fact moved on through various houses each time that its inhabitants were no longer able to pay the rent. The ghetto was therefore as much a physical placea locationas a network of cooperating people, a space of social interaction that rooted itself according to circumstances. Around 2003 the migrants finally set themselves up in a rented property until they were removed from it in October 2009. This event coincided with the beginning of a complete institutionalization of the reception process, contributing to a change in the kind of places where they could take shelter after their expulsion. More than a place specifically given over to expelled migrants, the Magnambougou ghetto was actually a crossroads for the mainly Cameroonian travelers who were heading north and for those who were forced to return from Algeria. Although it was essentially set up as a staging post, some migrants remained there for several months if not years. As an emblematic site, on the one hand testifying to the capacity for self-organization of non-Malian migrants expelled from Algeria, and on the other becoming a place of abandonment and poverty, this ghetto demonstrated the various social situations experienced after their expulsion by a number of migrants in transit, mainly from Cameroon or Central Africa.

The role played by migrant centers in the Maghreb and in Europe is well known. Abdelmalek [START_REF] Sayad | L'immigration et les paradoxes de l'altérité 1. L'illusion du provisoire[END_REF] showed how, in the 1950s, they provided good accommodation for poor immigrants in France. Claiming and reappropriating the term 'ghetto' for places which allowed migrants to gather together and organize their daily lives after expulsion, seemed to suggest the opposite: a sort of negative version from the point of view of language and image. Nonetheless, the function of the ghetto partly overlaps that of the migrant center, by offering at the very least shelter and a meeting place. But these ghettoes are also places where migrants become stalled, blocked and destitute. The use of the term symbolizes the break which expulsion represents in the migratory journey. It is the opposite of 'adventure', the word used in African stories and songs to refer to departures. It is certainly the case that expulsion does not entail the same consequence for each individual or group, but even so the choice of the word 'ghetto' seems to testify to the failure and relegation brought about by expulsion.

Expulsion redraws for migrants the horizon of the ghetto, a term which expresses both the harshness of the living conditions that follow expulsion and the difficulty of returning when there has been no departure in any real sense. It also redraws the reality of a ghetto which here corresponds more to a place where people gather together to cope with desperation and survival.

At the heart of the Magnambougou ghetto there was a very high level of collective awareness which, in February 2008, had led the migrants to decide as a group to close the area to all visitors from outside the ghetto: on various occasions many of the inhabitants of the ghetto attended discussion days on migration policies organized after 2007 by the Malian Expelled Migrants Association 3 , to listen to speeches by Roméo Ntamag, president of the Association of Central African Migrants Expelled in Mali 4 , and to see the sketch written and performed by members of the ghetto specifically for such events. Their common experience of expulsion seemed to be a determining factor in the formation of this group, illustrating failure and the dead-end that returning home represented. In this context the reappropriation of the term 'ghetto' testified to the fact that expulsion meant social segregation, but in the context of demonstrations organized by the various associations and of public discussion days, the term also became a strong symbol of the post-expulsion experience and helped to bring this experience out into the open. From this point of view the term 'ghetto' moved on from the fellowship of a self-organizing community to receive recognition by locally based associations and by the international NGOs involved in aid to expelled migrants.

Initially conceived as transit camps for migrants before they returned home, ghettoes such as Magnambougou owed their paradoxical existence to their being permanently temporary. The inhabitants mostly remained blocked there, with the distinctive feature of having been expelled from a country that was not their own: at the heart of the precariousness and marginalization that expulsion creates they found themselves in a situation of even greater social rejection and relegation than that experienced by Malian 'nationals'. Unlike the Tinzawaten ghettoes, the Magnambougou ghetto was not one that had been created by the migrants themselves, in the sense that it existed in a classic urban context and operated through rental agreements. Nor was it, like the Cameroonian ghetto in Gao, a place which in its very organization presupposed close links with people 'passers'.

It functioned, however, with the same respect for rules and the same protoadministrative habits which prevailed in the other ghettoes further to the north. The inhabitants paid a monthly rent of 10,000 CFA francs per room, with some of them occasionally sleeping in corridors or on terraces. The dilapidated building had 14 rooms. The (adult) inhabitants of each room which could run to as many as 17, shared the cost. The ghetto mainly lived on credit, and with rent arrears about which the manager, described as a drunkard, was very understanding. Every new arrival paid a ghetto fee which varied between 2,500 and 5,000 CFA francs and which was an entry payment allowing the group to buy cleaning products and other daily needs: mops, buckets, matting, saucepans and so on. Those who had no money on arrival were allowed a little more time to pay. The payment of the ghetto fee implied acceptance of the admission regulations, recorded and posted in a written set of rules: these stipulated that every adult should take his or her turn on the cleaning rota, and also set out disciplinary rules that had to be observed in the communal house, and which included in particular a ban on fighting and stealing, and called for respect for elders and so on.

The rules which this micro-society set up seemed to conform to the paradoxical definition of a ghetto (Bauman, 2006, pp. 148-49) as a place of shelter aimed at providing for the daily needs of its inhabitants, while also being a place that was set apart, segregated. On a very small scale this place reproduced the social conditions of poverty and social relegation of the ghettoized areas in large cities, with the people within it being confronted daily by the difficulty of feeding themselves and paying the rent. Promiscuity and other problems presented by different standards of hygiene were also a source of tension. The ghetto thus appeared to be a pocket of extreme insecurity and privation within the town, noted for the political and social situation of its members. Relations with neighbors were characterized by mistrust, but also by exchanges and demonstrations of solidarity, such as the friendly shopkeeper who regularly donated sacks of rice, or the locals who shared their meals with the migrants. But apart from problems over food and rent, it was destitution and the difficulties inherent in the situation of an expelled migrant in a foreign land which often led to a crisis when a migrant died. In this sort of critical moment, the Association of Central African Migrants (ARACEM) would try to mobilize the association's network in order to find the necessary solution to the problem of repatriating or burying in Mali the person who had died; and when twenty or so migrants from the ghetto were imprisoned for theft in 2008, AME and ARACEM led a joint delegation to the Bamako prison to visit the prisoners and bring them sacks of rice bought thanks to financial support from the Swiss NGO Helvetas.

The ghetto got its identity and memory from these shared tragedies and experiences. The people whom I met in February 2009 had ambivalent feelings towards the ghetto which they wanted to leave -'We really need to get out of the situation that we're in', one of them said to me -but which at the same time they did not want to forget: 'We have to be able to tell others about what we've experienced […] When you get to the end of the tunnel you forget a lot of things. You can't remember on your own.' Some wrote their story down, kept a personal diary and their words were stamped with a mixture of pain and pride. In contrast to the misery of their living conditions in the ghetto, those whom I met spoke of hoping for change, for an improvement in their own situation and that of the group as a whole. Together the inhabitants of the ghetto and the members of ARACEM set up a hair-dressing salon called 'Hope'.

The ghetto, an enclave in a complex geography of expulsion and mobility

In the Euro-African border zone, expulsion traces a specific geography of the routes and places where migrants gather and pass through, where groups form and then scatter. If in this chapter I have laid particular stress on the ghetto as a major socio-political reality in order to understand the effects of the expulsions in Mali, the coexistence of centers for expelled migrants, the centers for those en route to Algeria and the ghettoes, reveals the complexity of these patterns of mobility and shelter. On the border between Cameroon and Equatorial Guinea, in the Cameroonian town of Kye-Ossi, or in the town of Kidal in Northern Mali, expelled migrants stay in centers which are a step nearer to institutionalization and recognition by public authorities than the ghettoes, and are generally managed by a third party, typically a migrant living in the town to which he has been expelled and where he organizes housing and subsistence for newly arrived expelled migrants. In 2009, various associations based in Bamako (Association Malienne des Expulsés, Association Retour Travail et Dignité, Association des Refoulés d'Afrique Centrale au Mali, Association Tounkan Tè Danbé Don) had set up reception centers for expelled migrants in Bamako and other regions.

Whereas expulsion policies mean that being sent back to countries of origin is the unavoidable result for immigrants, the diversity of these places seems by contrast to demonstrate the existence of different kinds of mobility after expulsion; if some, exhausted and having lost everything, try to go back home, others extend their stay in the area to which they were expelled, in the hope of setting off once again. From this point of view, the existence of ghettoes reveals that a fellowship is born within migration which in addition to its role in facing up to the most dramatic consequences of expulsion, sets up the collective as a response to exclusion by the state. The ghettoes are part of these migratory journeys and are formed, in different places and modalities, by the pressure of expulsion. They are to be found in Tinzawaten, Gao and Bamako, but also in Arlit and Agadez in Niger [START_REF] Brachet | Migrations transsahariennes: vers un désert cosmopolite et morcelé (Niger)[END_REF] and certainly in countless other places in Africa and elsewhere along migration routes, and they show the capacity for social and political organization in the context of uncertainty, destitution and abandonment.

The ghettoes and centers set up in the post-expulsion period are part of a wider structure of places for those on the move. They must be seen alongside other ghettoes, other centers in which migrants organize themselves in Algeria, for example, or in Morocco, and also in the numerous towns which line the routes of this traffic, intra-African and/or heading for Europe [START_REF] Daniel | Les routes clandestines: l'Afrique des immigrés et des passeurs[END_REF][START_REF] Pian | Aux portes de Ceuta et Melilla: regard sociologique sur les campements informels de Bel Younes et de Gourougou[END_REF]. They find an echo in the jungles and ghettoes set up by migrants in Europe at key border crossings (Calais, Patras) [START_REF] Agier | Je me suis réfugié là!» Bords de route en exil[END_REF]. A homology of form, based on grouping by nationality, the appointment of a leader with responsibility for daily life and for transport, a sense of hierarchy, and collective conflict resolution, all link these different places in Europe and Africaghettoes, centers, jungles. The proliferation of ghettoes and other places of shelter created along their way by immigrants and expelled migrants illustrates the fellowship, the forms of self-help and self-organization that arise out of the political pressures placed on the movement of people by individual states, and the attempt to overcome these.

Ghettoes symbolize the state rejection of immigrants at the border, but behind their ruined walls, the poverty and the desolation, they also become places of bare-fisted 'resistance' to discriminatory state laws and systems [START_REF] Lecadet | Tinzawaten, c'est le grand danger pour nous les immigrés ![END_REF]. The intensification of migration control and the regularity of expulsions give the term 'ghetto' a new context, which updates its meaning, and its organizational methods in the light of these border issues.They show how social and political life is recreated in the experience of crossing borders. Even in a minimal and mostly lowly form, these places show that the strengthening of borders induces strategies of subversion and avoidance. Borders and the sites along them make visible the naked confrontation between state politics and the collective desire for mobility.

This fieldwork was carried out for my Ph.D thesis entitled The moving tide of expelled migrants. Centres of displacement, collective mobilisation and the risks experienced by expelled migrants in Mali (EHESS,

2011). Its aim was to offer a general description of places, networks and associations formed by expelled migrants after deportation.

The tontine is a system for raising capital, and in Sub-Saharan Africa it is generally supported by associations or groups of people who pay contributions that can be used to provide assistance to deported compatriots.

This association, founded in Bamako in 1996 by Ousmane Diarra, played a pioneer role in the self-organization of expelled migrants in Mali and in the emergence of protest movements against, and political criticism of, the policies and practices of the expulsion of illegal foreigners from western countries and within Africa.

This association, set up in Bamako in 2006 by Roméo Ntamag and Patrice Boukar, had the specific aim of representing non-Malian migrants expelled into Mali. In particular it allowed the attention of various associations to be focused on the inhabitants of the Magnambougou ghetto and it was responsible for the opening of an institutionalized reception center for expelled migrants in Bamako in 2009, a center that was supported by the NGOs Medico International and Médecins du Monde.


Cyril Grunspan

Ricardo Pérez-Marco

Ricardo P Érez-Marco

PING-PONG SWAPS PEER-TO-PEER CROSSCHAIN SWAPS WITHOUT ESCROW NOR TRUSTED THIRD PARTY

Keywords: 2010 Mathematics Subject Classification. 68M01, 91A05, 91G80 Bitcoin, blockchain, peer-to-peer, payment channel, swap. 1

Introduction

Satoshi Nakamoto's foundational article [START_REF] Nakamoto | Bitcoin: a peer-to-peer electronic cash system[END_REF] describes Bitcoin protocol as a peer-topeer payment protocol. After Bitcoin's invention, many other cryptocurrencies where developed following the Bitcoin code or ideas. Bitcoin is also the first programmable form of money and smartcontracts of different sorts can be implemented using its scripting language. Some years later, smartchains were build, specifically dedicated to smartcontracts, and it was easy to create tokens for various purposes using smartcontracts (see also [START_REF] Friedenbach | Freimarkets[END_REF] for an earlier proposal of extension of the Bitcoin protocol), like for example ERC-20 tokens in the Ethereum network. These tokens are useful for ICO's but also some of them represent stable coins, like for example Tether whose value is tied to the USD. Most of the trading occuring in cryptocurrency exchanges involves trading pairs A/B of two cryptocurrencies or tokens. These trades in exchanges are centralized are processed in the internal balances of the exchanges and require the trust on the exchange, and thus have a counterparty risk (for example the risk of fractional reserve and insolvency of the exchange). Our goal is to extend Nakamoto's peer-to-peer payment philosophy to the global cryptocurrency market.

When both tokens A and B are in the same smartchain, one can devise smartcontracts that allow a decentralized trade of the pair A/B through the interaction with the smartcontract. This is the basic principle of Decentraliced Exchanges (DEX's), like Uniswap in the ETH network, or Cake in the BSC network. These are fundamental pieces of the DeFi technology that is being build.

When both tokens or cryptocurrencies A and B do not belong to the same blockchain, the situation is more complex. Some cryptograhic solutions of "Atomic Swaps" ( [START_REF]Atomic swaps[END_REF]) have been proposed in certain blockchains with specific properties (see for example the Decred project [START_REF]Decred atomic swaps[END_REF]). Atomic swaps required more cryptographic functionalities than the solution of ping-pong swaps proposed here. More precisely, they need also a hashlock mechanism which is not needed for the solution we present here.

Other peer-to-peer protocol exchanges work in the base of an escrow system and penalties in case of dishonest behavior, as for example Bisq ([5]) or Hodlhodl ([6]). Also a "bridge industry" is being developed for crosschain swaps (Ren, Anyswap, Axelar, Thorchain, etc). Many of these projects aim to be decentralized but so far they do not achieve the decentralization quality that one expects from Bitcoin standards.

The goal of this article is to propose a very simple procedure that allows peer-topeer decentralized swaps without escrow nor trusted third party involved (and, of course, no mutual trust required between both parties). The only important (but mild) technical requirement is that in both blockchains we can open simple unidirectional payment channels. An unidirectional payment channel can be open if we can create 2-2 multisignature transactions and we dispose of timelock scripts.

Payment channels

2.1. General payment channels. Payment channels allow off-chain transactions that are instant and secure. They also allow micropayments and have no commission cost, except for the opening or closing of the channel on-chain. They require smartcontracts with a 2-2 multisignature and a timelock script. They can be unidirectional, i.e. allowing the exchange of the cryptocurrency in only one direction, or bidirectional. Unidirectional payment channels are much simpler than bidirectional ones that require a constant monitorization of the blockchain. This is not necessary in the case of unidirectional payment channels. Payment channels were first proposed by Satoshi Nakamoto himself in some private emails, and J. Spillman proposed a simple unidirectional payment channel that could be implemented after Segwit. Later, other sorts of more complex payment channels were proposed (see [START_REF]Payment channels[END_REF]). The Lightning Network that is currently being implemented, is build using the composition property of bidirectional Poon-Dryja payment channels, as described in the Lightning Network article [START_REF] Poon | The Bitcoin Lightning Network: Scalable off-chain instant payments[END_REF]. For our purposes we only need the simpler Spillman's unidirectional channels, that do not need a hashlock scrypt. But of course, the same protocol will work with bidrectional payment channels or their composition, and in particular we can use in the Bitcoin blockchain the Lightning Network currently implemented. We describe in the next section how to build the simpler unidirectional payment channels in the Bitcoin network. In other smartchain with a rich scripting programming languages, one can build smartcontracts that simulate multisignatures, timelocks and unidirectional payment channels. We will not discuss the specifics for each smartchain or token type. With a rich scripting language the task is in general much easier than in the Bitcoin network. For example, in the Ethereum network, the goal of the Raiden project [START_REF][END_REF] is to create a Lightning Network for tokens in the ETH blockchain and they have developped payment channels.

Unidirectional payment channel in Bitcoin.

We describe how to build an unidirectional payment channel from Alice to Bob in the Bitcoin protocol (or similar). Alice wants to pay Bob. Alice and Bob agree on the total amount of bitcoins that will be locked in a 2-2 multisignature address with two outputs to addresses controlled by each one of them. Bob provides a signed transaction with the total amount and a timelock, so that Alice can recover the totality of her funds if the channel is not closed before some predetermined time t 0 (in blockchain time). Then Alice sends the total amount of bitcoins to this 2-2 multisignature address. This on-chain transaction opens the payment channel. Now, each time Alice wants to pay Bob, she only needs to sign a 1-2 signature new transaction which transferts the new balance to the output addresses. Bob can cash-out anytime by providing the second signature and broadcasting the transaction. This action will close the payment channel. But Bob waits for more payments from Alice. In any case, Bob must close the channel before t 0 or Alice will be able to recover all the funds. With this procedure, Alice can make instant payments or micro-payments with no fee involved that are done off-chain by simple data communication from Alice to Bob. The computational cost is only formatting the transaction with 1 of 2 signatures. The ECDSA (or Snorr) signature requires only a small fraction of a second.

Ping-pong swaps.

We explain in this section ping-pong swaps with a very specific example using the Bitcoin and Litecoin networks, from which the general procedure is clear. The code of Litecoin is a fork of the code of Bitcoin and has incorporated most of the script functionalities from Bitcoin, thus we can also open in the Litecoin network Spillman unidirectional payment channels.

3.1. Preliminary setup and opening channels. Imagine now the situation where Alice wants to buy from Bob a cryptocurrency. For example, Alice has BTC and wants to buy LTC from Bob. In both of these two blockchains we can create unidirectional payment channels as described in the previous section. First they agree on an amount and a price for the trade of the pair BTC/LTC. Suppose that Alice wants to spend 1 BTC and they agree on the price 1 LTC = 0.003521 BTC (current exchange rate at the moment). This means that Bob has to provide 284 LTC to Alice.

In the BTC network, Alice opens an unidirectional channel with Bob for the total amount of 1 BTC. In the LTC network, Bob opens an unidirectional channel with Alice in the amount of 284 LTC. They both agree on timelocks (1 hour for example, enough time to avoid double spends with the timelocked recovery transactions).

They also agree on the granularity of the swap, which can be, for example, of 0.1% of the total amount. This is some small fraction of the total amount that is not worth cheating from any of the parties involved in the trade.

3.2.

Ping-pong micropayments. After this setup, Alice and Bob engage in a "ping-pong" sequence of micropayments. First Alice pays 0.1% of the total amount (that is, 100.000 satoshis) to Bob through the Bitcoin payment channel. Once Bob has received the 1-2 signed transaction from Alice, he pays 0.2% of the total amount of LTC (0.568 LTC to be precise) to Alice through the LTC payment channel. Then, once Alice has received Bob's payment, she sends 0.2% of the total amount of BTC, then Bob sends again 0.2% of LTC, and so on. In the last transaction Alice will send only 0.1%. In this example, Alice will perform a total of 501 payments and Bob 500 payments. Once they both transfer the total amount, they can close the channels and recover the funds in an address of their own. Obviously, all these micropayments are done by a software wallet designed for this swap purpose. In the example, the total transaction is completed after 1001 signatures, thus in less than a second if the channel payments (signature) take under a millisecond. In general, the processing time is inversely proportional to the granularity of the swap.

3.3.

Cheating. The important observation, is that the only way that Alice or Bob can cheat is by interrupting the "ping-pong" transactions and closing their receiving channel, pocketing the granularity of the payment (0.1% in our example). The other party will then close the channel to secure the payment already received. The granularity can be adjusted to have a small enough value to desincentivize any cheating.

If the granularity is smaller than the fees for opening and closing the channels, there is no profit for the cheater (although he saves one granularity).

Implementation.

In principle, this can be easily implemented as a mobile app having some functionalities of a software wallet in a computer or mobile phone. Once Alice and Bob agree on their trade, the app provides the necessary information to the other party to open the channels, and each one opens its unidirectional channel. Both apps communicate directly for the "ping-pong" of transactions, and finalizing the swap by closing the channels. This should be almost instantaneous, except for the waiting time of onchain transaction to open and close the channels. If both Alice and Bob have open payment channels in the Lightning Network in BTC, they can use them as one of the channels.

Non-custodial exchanges (NCEX).

In order to find counterparts, the app can connect to a server where people can post their trade offers. The matching can be done automatically or manually. The advertisement of the offers involves then a centralized server, but only for posting and reputation purposes. In this way we can build a centralized urely cryptocurrency Non-Custodial EXchange (NCEX), where the users have control of their funds all the time. This type of exchange limits the liability of the exchange operator since the exchange cannot be hacked in order to access its wallets. Also it is interesting to note that if the exchange charges a minimal fee higher than the granularity and the fees for opening the payment channels, there is no point in cheating on the granularity of the ping-pong swap.

In the implementation of non-custodial exchanges, there is the possibility of doing the ping-pong swaps through the exchange. The added value of the exchange could be, for example, in having a reputation system and database of trusted dealers. Alice and Bob can be users of the exchange and open an important payment channel with the exchange for the total amount of BTC that Alice wants to trade and the total amount of LTC that Bob wants to trade. Then, once the orders are matched, two ping-pong swaps can be performed through the exchange by the composition of the BTC payment channel from Alice to the exchange and from the exchange to Bob, and the LTC channels from Bob to the exchange and the exchange to Alice. The dynamics of the composed ping-pong swaps is easy to figure out. First Alice send his initial 0.1% payment to the exchange, then the exchange forwards it to Bob. Once Bob receives the first micropayment it sends his first 0.2% LTC micropayment to the exchange that forwards it to Alice, etc. If Alice has a large sell order of BTC that can only be absorbed by several buyers, in this way she only needs to open two channels with the exchange (thus only two on-chain fees). Thus this type of procedure saves important on-chain fees although is not P2P having the exchange as middle-man, but it is still a trustless procedure.

A decentralized global market.

One can imagine a better decentralized exchange system using a decentralized communication protocol similar to the one used in the Bitcoin network. In this setup, offers can be propagated flooding the network, as is done in the broadcasting of transactions in the Bitcoin network. Each app is a communication node, connects to 8 or more neighbours, and relays the "mempool" of offers for each trading pair A/B. This creates an international decentralized bid-ask order book for each trading pair. Another possibility is to build a federation of Nostr relays where the offers are posted and matched by the clients ( [START_REF]Nostr project[END_REF]).

In this way we can build a global purely decentralized exchange and worldwide global market, which is uncensorable and with no centralized structure. Properly implemented, this glogal market offers anonymous trading, with almost no fees. The classical exchange and broker industry will still be useful for crypto/fiat pairs, or for those pairs of cryptocurrencies that involve one whose blockchains does not allow payment channels.

Application: Wrapping cryptocurrencies into other chains.

Non-native cryptocurrencies can be used in other smartchains if they are wrapped into native tokens. For example, bitcoins are wrapped into the Ethereum blockchain as tokens WBTC, in the Binance Smart Chain as BTCB, in RSK network ( [START_REF][END_REF]) as R-BTC, or in Liquid network ( [12], [START_REF] Back | Wuille Enabling Blockchain Innovations with Pegged Sidechains[END_REF]) as L-BTC. The procedure of wrapping can be more or less decentralized or elaborate depending on the chain. In Ethereum and the Binance smartchains, the tokenization is done by a trusted third party, a company or an exchange. In most of the situations, the tokenization can be done through a centralized exchange. In Liquid and RSK the procedure can be more decentralized, but requires an important number of confirmations of the transaction in the Bitcoin network (over 100 for RSK and Liquid, thus more than 16 hours). The wrapping procedure may also involve extra fees other than the transaction fees.

With ping-pong swaps we can simplify the procedure, taking advantage of anyone willing to unwrap tokens, with a ping-pong swap between the native and wrapped tokens. For example, Alice having native BTC, can initiate a ping-pong swap with Bob having wrapped BTC. Alice opens a payment channel with Bob in the BTC network, and Bob one to Alice in the smartchain. If the smartchain has fast confirmation times, the procedure takes the time of the confirmations for opening and closing the BTC channel. For example, we could wrap BTC and obtain R-BTC in RSK with a ping-pong swap in 4 confirmations (2 confirmations for opening and closing the channel is enough for security purposes, see [START_REF] Grunspan | On profitability of Nakamoto double spend[END_REF]) instead of 100 confirmations required by RSK's powpeg procedure.

8. Application: Rebalancing Lightning Network channels.

One of the most common problem of the LN as is being implemented is the rebalancing of payment channels. The channels used are all bidirectional. Unbalanced channels become unidirectional and their routing capability is impaired. Several solutions have been proposed to balance LN payment channels. By circulary, if we can find a convenient circular composition of channels that allows the rebalancing. Since the general flow of payments is from customers to vendors, this is not a realistic general solution. Other solutions like splines involve on-chain transaction, thus on-chain fees and waiting time. Here we propose a solution using ping-pong swaps using a secondary chain that has fast confirmations and cheap fees.

In case Alice and Bob have an open LN payment channel that they want to rebalance by sending funds from Alice to Bob, they can run a simple ping-pong swap. They will use the existing bidirectional LN channel as Alice payment channel, and Bob will open an unidirectional channel to Alice in the Bitcoin network or another blockchain (in this last case, they need to agree first on the conversion rate). They will proceed exactly as before. This procedure only needs two on-chain transactions for opening and closing the channel from Bob to Alice.

Notice that this allows to rebalance your Bitcoin LN channel using another cryptocurrency or token. If we use one that has fast confirmation time in its blockchain (as the Ethereum network), the whole procedure of rebalancing can be performed in seconds. If we use one that has cheap transaction fees, the cost will be small.

Fiat swaps

Ping-pong swaps can be performed with cryptocurrencies/fiat pairs if commercial banks incorporate the payment channel technology. In their centralized setting, this is very simple. Banks need to allow high frequency micropayments between client accounts. They need to have API's that allow the interaction of the users cryptowallet with their bank account in order to check the dynamics of the ping-pong micropayments. Then you can setup for example a BTC/USD ping-pong swap. Both Alice and Bob need to be customers of the bank, and Alice will open a payment channel to Bob's account internal to the bank. The app will perform the ping-pong swap checking with the APIs the bank account balance and ordering the micropayments.

Incorporating this technology, banks can become cryptoexchanges. They will only be custodial of the fiat funds. The users keep at all times control of their cryptocurencies with their private keys.

Remark. Without doubt, the core idea of making first small trades in order to build confidence with an unknown dealer goes back to the origins of trade. Some traders use this procedure when buying from early miners new cryptocurrencies not listed in exchanges. The contribution of ping-pong swaps is the combination of this old folklore procedure with micropayments through payment channels. Not surprinsingly, this simple idea appeared before somewhere else. After writing the article, it was pointed out to the authors the blog post by Micah Zoltu [START_REF] Zoltu | :\\micahzoltu.medium.com\how-to-build-a-censorship-resistant-cross[END_REF] where a ping-pong swap of the pair ETH/ZEC is described.

Author's Bitcoin Beer Address (ABBA) 1 : 1KrqVxqQFyUY9WuWcR5EHGVvhCS841LPLn

1 If you find this article useful, you can send some anonymous satoshis to support our research at the pub.


Ricardo P Érez-Marco

DIFFERENCE EQUATIONS AND OMEGA FUNCTIONS

We introduce Omega functions that generalize Euler Gamma functions and study the functional difference equation they satisfy. Under a natural exponential growth condition, the vector space of meromorphic solutions of the functional equation is finite dimensional. We construct a basis of the space of solutions composed by Omega functions. Omega functions are defined as exponential periods. They have a meromorphic extension to the complex plane of order 1 with simple poles at negative integers. They are characterized by their growth property on vertical strips and their functional equation. This generalizes Wielandt's characterization of Euler Gamma function. We also introduce Incomplete Omega functions that play an important role in the proofs.

These equations are linear and we have a vector space of meromorphic solutions. A natural motivation for studying these functional equations comes from the study of subspaces generated by natural linear operators. For instance, we can consider, in the space of meromorphic functions, the shift (or integer translation) linear operator

T (f (s)) = f (s + 1)
and the multiplication by s linear operator S(f (s)) = sf (s) 2010 Mathematics Subject Classification. Primary: 30D10, 39A05. Secondary: 30D15, 30B50. Key words and phrases. Euler Gamma function, exponential periods, Omega functions, difference equation.

Observe that S has no eigenvectors and the minimal invariant subspace invariant by S containing the constant functions is the space of polynomials C[s]. The space generated by S and the function f is the vector space C[s]f f, S(f ), S 2 (f ), . . . = C[s]f

It is natural to find the functions f such that the space C[s]f is generated by f and T . This happens if and only if f is solution of the difference equation [START_REF] Aycock | Euler and the multiplication formula for the Γ-function[END_REF].

Already, in the simplest case of the functional equation of Euler Gamma function, the space of solutions is infinite dimensional since any function of the form e 2πins Γ(s) for an integer n ∈ Z is also a solution. It is classical to add conditions to characterize Euler Gamma function as the only normalized solution to this functional equation. One can mention Weierstrass characterization imposing some asymptotic behavior when s → +∞ (1856, [START_REF] Weierstrass | Über die Theorie der analytischen Fakultäten[END_REF]), or Wielandt's characterization (1939, [23], see also [START_REF] Remmert | Wielandt's theorem about the Γ-function[END_REF], [START_REF] Remmert | Classical topics in complex function theory[END_REF]) requiring boundedness on vertical strips of width larger than 1, or, more recently, requiring finite order of the solutions and a right half plane free of zeros nor poles (2022, [START_REF] Érez-Marco | On the definition of Euler Gamma function[END_REF]). Wielandt's boundedness condition has been weakened by Fuglede to a moderate growth in the vertical strip (2008, [START_REF] Fuglede | A sharpening of Wielandt's characterization of the Gamma function[END_REF]).

In the spirit of Wielandt, we search for solutions with some growth control on vertical strips. Under a suitable growth condition, we prove that the space of solutions is finite dimensional: for some constant C > 0 and 0 ≤ c < 2π, is finite dimensional of dimension d.

Moreover, we build an explicit basis of the vector space of solutions with Special Functions, that we call Omega functions, that generalize Euler Gamma function. There is a large classical literature on linear difference equations with polynomial coefficients by Poincaré [START_REF] Poincar É | Sur les intégrales irrégulières des équations linéaires[END_REF], Birkhoff [START_REF] Birkhoff | General theory of linear difference equations[END_REF], Carmichael [START_REF] Carlson | Sur une classe de séries de Taylor[END_REF], Nörlund [START_REF] Örlund | Leçons sur les équations linéaires aux différences finies[END_REF], and, more recently, solutions with vertical exponential growth have been studied by Barkatou [2] and Duval [START_REF] Duval | Étude asymptotique d'une intégrale analogue à la fonction "Γ modifiée[END_REF] following work of Ramis. The analysis of the equation of this article is self-contained and independent of the classical theory. 1.2. Omega functions. Historically, Euler Gamma function appears for the first time in a letter from Euler to Goldbach, dated January 8th 1730 ( [START_REF] Euler | Letter to Goldbach[END_REF]). Euler defines the Gamma function for real values s > 0, by the integral formula

Γ(s) = +∞ 0 t s-1 e -t dt .
which is also convergent for complex values of s with Re s > 0. In this integral formula, the value Γ(s) appears as an exponential period.

Algebraic periods are integrals of algebraic differential forms over cycles of an algebraic variety. In the special case of an algebraic curve, when we represent the curve as a Riemann domain over the complex plane or the Riemann sphere, algebraic periods are also the integrals of algebraic differential forms on paths joining two ramification points where we have singularities of the differential form. From the transalgebraic point of view, it is natural to consider exponential periods, where integrals involve exponential expressions, and the singularities can be exponential singularities. More general periods can be envisioned where the differential form has transcendental singularities with monodromy as t s in a local variable (geometrically these corespond to differential forms living in a branched Riemann domain with an infinite ramification). There is a vast literature on classical algebraic periods, but almost none on the transalgebraic periods. We refer to [START_REF] Konsevich | Periods, Mathematics unlimited-2001 and beyond[END_REF] for a survey about classical periods, and to [START_REF] Biswas | Log-Riemann surfaces[END_REF] and [START_REF] Biswas | The Ramificant Determinant, Symmetry, Integrability and Geometry: Methods and Applications[END_REF] for exponential periods and their relation with log-Riemann surfaces. Also we refer the reader to [START_REF] Érez-Marco | On the definition of Euler Gamma function[END_REF] for an historical survey of different definitions of Euler Gamma function and their generalizations, and to [START_REF] Whittaker | A course in modern analysis[END_REF] for its classical properties.

We introduce (resp. Incomplete) Omega functions which are a natural generalization of the (resp. Incomplete) Gamma function. They are defined as exponential periods of the form

Ω k (s) = +∞.ω k 0 t s-1 e P 0 (t) dt , Ω k (s, z) = z 0 t s-1 e P 0 (t) dt
where P 0 (t) ∈ C[t] and ω k is a root of unity pointing to a direction where the polynomial P 0 diverges to -∞. Some critical computations in the proof of the main Theorem generalize computations carried out for exponential periods appearing in [START_REF] Biswas | The Ramificant Determinant, Symmetry, Integrability and Geometry: Methods and Applications[END_REF]. The generalization of the Ramificant Determinant is the key result for the proof of the linear independence of the Omega functions (Ω k ). In this magical calculation, we compute a determinant of a matrix of exponential periods which are individually not computable. Omega functions appeared before in the literature under the name of "modified Gamma functions" and their asymptotic behavior at infinite was studied by N. G. De Bruijn, [START_REF] De Bruijn | Asymptotic methods in analysis[END_REF] p.119, and A. Duval [START_REF] Duval | Étude asymptotique d'une intégrale analogue à la fonction "Γ modifiée[END_REF]. We know no earlier references for Incomplete Gamma functions.

Definition.

Let P 0 (t) ∈ C[t] be a degree d ≥ 1 polynomial such that P 0 (0) = 0 and lim t→+∞ Re P 0 (t) = -∞, normalized by

P 0 (t) = - 1 d t d + d-1 k=1 a k t k
We also denote a d = -1/d and a 0 = 0. Let ω be the primitive d-th root of unity given by

ω = e 2πi d
and write

ω k = ω k . Definition 2.1. Let d ≥ 1.
For k = 0, 1, . . . , d -1, the Omega functions, or Ωfunctions, associated to P 0 , are defined for Re s > 0, by

Ω k (s) = +∞.ω k 0 t s-1 e P 0 (t) dt
The roots ω k point to the directions where the polynomial P 0 diverges exponentially to -∞, lim t→+∞.ω k Re P 0 (t) = -∞ so the integral is converging and we have a sound definition. Usually, we spare the reference to P 0 , but for some results it will be crucial to keep track of the dependence on parameters and we will write Ω(s) = Ω(s|P 0 ) = Ω(s|a 1 , . . . , a d-1 ) .

For d = 1, we have P 0 (t) = -t and Ω 1 = Γ is Euler Gamma function.

If P 0 ∈ R[t],
then Ω 0 is real analytic, and Ω k (s|P 0 ) = Ω d-k (s| P0 ), where P0 is the conjugate polynomial

P0 (t) = - 1 d t d + d-1 k=1 āk t k .
Sometimes we will be interested in the case where α k ∈ K where K ⊂ C is a number field. In that case we say that the Omega functions are defined over K.

Meromorphic extension, poles and residues.

Theorem 3.1. The Omega functions (Ω k ) 0≤k≤d-1 extend to the complex plane into meromorphic functions of order 1 satisfying the fundamental functional equation

(2) Ω k (s + d) + α d-1 Ω k (s + d -1) + . . . + α 1 Ω k (s + 1) = s Ω k (s)
where α l = -la l .

Moreover, the function Ω k is holomorphic in C -N, and has simple poles at the negative integers. The residue at s = 0 is

Res s=0 Ω k = 1 .
Observe that for d = 1, Ω 0 = Γ and the functional equation ( 2) is the classical functional equation Γ(s + 1) = sΓ(s).

Proof. We have for Re s > 0 and by integration by parts,

Ω k (s + d) + d-1 l=1 α l Ω k (s + l) = +∞.ω k 0 t s (-P ′ 0 (t)).e P 0 (t) dt = -t s e P 0 (t) +∞.ω k 0 + s +∞.ω k 0 t s-1 e P 0 (t) dt = s Ω k (s)
and we get the functional equation ( 2). Now, using once the functional equation we extend meromorphically Ω k to {Re s > -1}, and by induction to {Re s > -n}, for n = 1, 2, . . ., hence to all of C. The only poles that can be introduced by this extension procedure using the functional equation are those created from the pole at s = 0 at the negative integers. The functional equation shows that sΩ k (s) is holomorphic at s = 0, hence the pole at s = 0 is simple. It follows from the functional equation and the extension procedure that the other poles are also simple. We compute the residue at s = 0 using the functional equation,

Res s=0 Ω k = lim s→0 sΩ k (s) = d l=1 α l Ω k (l) = +∞.ω k 0 (-P ′ 0 (t)).e P 0 (t) dt = -e P 0 (t) +∞.ω k 0 = 1
More generally, we can compute the residues at the negative integers.

Theorem 3.2. Let (λ n ) n≥0 be the coefficients of the power series expansion of e P 0 (t) ,

e P 0 (t) = +∞ n=0 λ n t n .
Then the residue of

Ω k at s = -n is λ n , Res s=-n Ω k = λ n .
Proof. For n ≥ 0 let r n ∈ C be the residue of Ω k at s = -n, with r n = 0 if there is no pole, and r n = 0 for n < 0. The functional equation [START_REF] Barkatou | Contribution à l'étude des équations différentielles et aux différences dans le chanp complexe[END_REF] gives

r n = lim h→0 hΩ k (h -n) = lim h→0 1 h -n d l=1 α l hΩ k (h -n + l) = - 1 n d l=1 α l r n-l
hence the recurrence relation

(3) nr n = - d l=1 α l r n-l
Now, consider the generating power series

F (t) = +∞ n=0 r n t n
The recurrence relation [START_REF] Birkhoff | General theory of linear difference equations[END_REF] gives

F ′ (t) = +∞ n=0 nr n t n-1 = - d l=1 α l +∞ n=0 r n-l t n-1 = - d l=1 α l t l-1 +∞ n=l r n-l t n-l = - d l=1 α l t l-1 F (t) = P ′ 0 (t)F (t) Since we have F (0) = r 0 = 1 from Theorem 3.1, we get F (t) = e P 0 (t) thus r n = λ n as claimed.

Example.

For d = 1, the generating power series is

F (t) = e -t = +∞ n=0 (-1) n n! t n
and we recover the classical result that Res s=-n Γ = (-1) n n! .

Note that the residues at the simple poles at the negative integers are the same for all the functions Ω 0 , . . . , Ω d-1 . Indeed, for k = l, we can check directly that Ω k -Ω l is an entire function because of the convergence for all s ∈ C of the integral

Ω k (s) -Ω l (s) = +∞.ω k +∞.ω l t s-1 e P 0 (t) dt = γ lk t s-1 e P 0 (t) dt
where the integral can be taken over any path γ lk assymptotic to +∞.ω l and +∞.ω k in the proper direction and with 0 winding number around 0. This integral depends holomorphically on the parameter s ∈ C.

Observe also that if the coefficients of P 0 belong to a number field K, P 0 (t) ∈ K[t], then the residues of Ω k belong also to K. Another arithmetical observation is the following: Corollary 3.3. We assume that the only non-zero coefficients of P 0 are for powers divisible by an integer n 0 ≥ 2, that is, if a k = 0 then n 0 |k.

Then, if n 0 does not divide n, we have r n = 0.

Proof. From the previous Theorem we have

e P 0 (t) = d k=1 e a k t k = d k=1 m≥0 a m k m! t mk
and when we expand the last product we get the result.

We have a more precise result than just the computation of the residues. We can determine the Mittag-Leffler decomposition of Ω k .This is an analytic result that requires some estimates.

Ω k (s) = +∞ n=0 λ n s + n + +∞.ω k 1 t s-1 e P 0 (t) dt
where the integral is an entire function of order 1.

Observe that this Theorem shows that the Omega function Ω k is a meromorphic function of order 1.

Corollary 3.5. The Omega functions Ω k are meromorphic functions of order 1.

Proof. We write

Ω k (s) = 1 0 t s-1 e P 0 (t) dt + +∞.ω k 1 t s-1 e P 0 (t) dt
and we compute the first integral expanding the exponential in power series (uniformly convergent in [0, 1]),

1 0 t s-1 e P 0 (t) dt = +∞ n=0 λ n 1 0 t s+n-1 dt = +∞ n=0 λ n t s+n s + n 1 0 = +∞ n=0 λ n s + n
The second integral can be bounded by the next Lemma that shows that it is an entire function of order 1 (using that Euler Gamma function is of order 1). Lemma 3.6. We have the estimate

+∞.ω k 1 t s-1 e P 0 (t) dt ≤ e -2π k d Im s C 0 + C 1 d Re s/d Γ Re s d Proof.
We make the change of variables t = ω k u

+∞.ω k 1 t s-1 e P 0 (t) dt = ω s k +∞ ω -1 k u s-1 e -1 d u d (1+O(u -1 )) du = e 2πi k d s +∞ ω -1 k u s-1 e -1 d u d (1+O(u -1 )) du
This gives the bound

+∞.ω k 1 t s-1 e P 0 (t) dt ≤ e -2π k d Im s +∞ ω -1 k u s-1 e -1 d u d (1+O(u -1 )) du
Now, taking an integration path of finite length from 1 to ω k and bounded away from 0, we get (using the same letter C to denote several universal constants C > 0)

+∞.ω k 1 t s-1 e P 0 (t) dt ≤ e -2π k d Im s C + +∞ 1 u s-1 e -1 d u d (1+O(u -1 )) du
The last integral can be estimated by

+∞ 1 u s-1 e -1 d u d (1+O(u -1 )) du ≤ (1 + C) +∞ 1 u s-1 e -1 d u d du ≤ (1 + C) +∞ 0 u s-1 e -1 d u d du ≤ (1 + C)d Re s/d Γ Re s d
(for the computation of the last integral we use the change of variable v = u d /d).

Incomplete Omega functions.

We define the Incomplete Omega functions that generalize the Incomplete Gamma function.

Definition 4.1. For z, s ∈ C, Re s > 0, the Incomplete Omega function Ω(s, z) is defined by

Ω(s, z) = z 0 t s-1 e P 0 (t) dt
For P 0 (t) = -t this is the classical Incomplete Gamma function. Observe that we recover all the (Ω k ) functions by taking the appropriate limit of Ω(s, z) as z → ∞,

Ω k (s) = lim z→+∞.ω k Ω(s, z)
For the particular values of s = 1, 2, . . . , d -1 these are the transcendental entire functions of the variable z ∈ C studied in [START_REF] Biswas | The Ramificant Determinant, Symmetry, Integrability and Geometry: Methods and Applications[END_REF] which form a basis of the fundamental for the transcendental vector space of functions on a simply connected log-Riemann surface with exactly d infinite ramification points. Some of the results proved here generalize some results from [START_REF] Biswas | The Ramificant Determinant, Symmetry, Integrability and Geometry: Methods and Applications[END_REF]. Following the same Abel's philosophy that inspires [START_REF] Biswas | The Ramificant Determinant, Symmetry, Integrability and Geometry: Methods and Applications[END_REF], we prove that we only need to use a finite number of transcendentals (Ω(s + k, z)) 0≤k≤d-1 to compute integrals of the form z 0 Q(t, t s )e P 0 (t) dt where Q(x, y) ∈ C[x, y] is a polynomial. We start by studying the simpler case when

Q(t) ∈ C[t]. Proposition 4.2. Let Q(t) ∈ C[t]. For d ≥ 2, the integral z 0 t s Q(t)e P 0 (t) dt is of the form z 0 t s Q(t)e P 0 (t) dt = z s A(s, z) e P 0 (z) + d-1 k=0 c k (s) Ω(s + k, z)
where A ∈ C[s, z], and the polynomial coefficients c k (s) ∈ C[s] have coefficients depending polynomially on the coefficients of P 0 , (a 1 , . . . , a d-1 ).

Proof. First we consider the case d = 1. We prove the result for Q(t) = t n integrating by parts n + 1 times z 0 t s+n e -t dt = -t s+n e -t z 0 + (s + n -1)

z 0 t s+n-1 e -t dt = -z s+n e -z + (s + n) z 0 t s+n-1 e -t dt = -(z s+n + (s + n)z s+n-1 )e -z + (s + n)(s + n -1) z 0 t s+n-2 e -t dt . . . = -z s (z n + (s + n)z n-1 + . . .)e -z + (s + n)(s + n -1) . . . s Ω(s, z)
For a general polynomial Q(t) we have the the result by linear decomposition of the integral.

In the rest of the proof we assume d ≥ 2. If q = deg Q ≤ d -2, by linearity, the integral is a linear combination (with the coefficients of tQ(t)) of (Ω(s + k, z)) 0≤k≤d-1 and the result follows.

If deg Q ≥ d -1, then we consider the Euclidean division of Q(t) by P ′ 0 (t), Q(t) = A 1 (t)P ′ 0 (t) + B 1 (t) with A 1 , B 1 ∈ C[t], deg B 1 ≤ d -2 and deg A 1 = deg Q -(d -1) = q -(d -1) ≤ q -1.
We proceed splitting the integral:

z 0 t s Q(t)e P 0 (t) dt = z 0 t s A 1 (t)P ′ 0 (t)e P 0 (t) dt + z 0 t s B 1 (t)e P 0 (t) dt Since deg B 1 ≤ d -2
, the second integral is a linear combination of Ω(s, z), Ω(s + 1, z), . . . , Ω(s + d -1, z), thus of the desired form, and we can forget about it. We work on the first integral integrating by parts,

z 0 t s A 1 (t)P ′ 0 (t)e P 0 (t) dt = t s A 1 (t)e P 0 (t) z 0 - z 0 (t s A 1 (t)) ′ e P 0 (t) dt
Then we get:

z 0 t s A 1 (t)P ′ 0 (t)e P 0 (t) dt = z s A 1 (z)e P 0 (z) - z 0 t s A ′ 1 (t)e P 0 (t) dt -s z 0 t s-1 A 1 (t)e P 0 (t) dt
The first integral in the right hand side is of the same form as the initial one with

Q(t) but with deg A ′ 1 = deg A 1 -1 ≤ deg Q -(d -1) -1 = q -d ≤ q - 2
(using here d ≥ 2), hence by descending induction we can forget about it. For the second integral, we can write

s z 0 t s-1 A 1 (t)e P 0 (t) dt = A 1 (0)s Ω(s, z) + s z 0 t s A 1 (t) -A 1 (0) t e P 0 (t) dt and t -1 (A 1 (t) -A 1 (0)) is a polynomial of degree deg A 1 -1 ≤ q -2.
Then the descending induction gives the expression for the integral as announced.

The coefficients of P 0 appear first linearly in the Euclidean divisions by P ′ 0 then, by repeated Euclidean divisions the dependence of the c k (s) is polynomial on the coefficients of P 0 . Now, we can get easily the general result for Q(t, t s ).

Corollary 4.3. Let Q(t, t s ) ∈ C[t, t s ]. For d ≥ 2, the integral z 0 Q(t, t s )e P 0 (t) dt is of the form z 0 t s-1 Q(t, t s )e P 0 (t) dt = A(s, z, z s )e P 0 (z) + d-1 k=0 c k (s)Ω(s + k, z)
where A ∈ C[s, z, z z ] and the coefficients c k (s) ∈ C[s] have coefficients that are polynomials on the coefficients of P 0 (a 1 , . . . , a d-1 ).

Proof. We write Q(t, t s ) = Q(t)(t s ) as a polynomial on the variable t s with coefficients polynomials in t, and we split linearly the integral, observing that the part corresponding to each monomial is like the integral when Q is a polynomial in t as before, but with s shifted into s + l by some positive integer l. The result follows from the previous Proposition. Now, we can prove that the Omega functions (Ω k (s)) 0≤k≤d-1 generate a large class of exponential periods:

Corollary 4.4. Let Q(t, t s ) ∈ C[t, t s ] and 0 ≤ n ≤ d -1. The exponential period +∞.ωn 0 Q(t, t s ) e P 0 (t) dt is a linear combination of the exponential periods (Ω k (s)) 0≤k≤d-1 +∞.ωn 0 Q(t, t s ) e P 0 (t) dt = d-1 k=0 c k (s) Ω k (s)
where the coefficients c k (s) are polynomials on s and on the coefficients (a 1 , . . . , a d-1 ).

Proof. For d ≥ 2, we have from the previous Proposition that z 0 Q(t, t s )e P 0 (t) dt = A(s, z, z s )e P 0 (z)

+ d-1 k=0 c k (s)Ω(s + k, z)
When z → +∞.ω n the terms in the first sum vanish, since A(s, z, z s )e P 0 (z) → 0 for a polynomial A(s, z, z s ) ∈ C[z] (the exponential decay of e P 0 (z) takes over the polynomial divergence of A(z)), and we get

+∞.ωn 0 Q(t, t s )e P 0 (t) dt = d-1 k=0 c k (s)Ω n (s + k)
5. Linear independence.

The row vector build with Omega functions Ω(s) = (Ω k (s)) 0≤k≤d-1 has the following important linear independence property: where

Ω kl = Ω k-1 (s + l).
More precisely, we can compute ∆(s|a 1 , . . . , a d-1 ) = ∆(s|0, . . . , 0) exp (Π d (s, a 1 , . . . , a d-1 ))

where Π d (s, a 1 , . . . , a d-1 ) is a universal polynomial with rational coefficients.

In view of the last formula, the result follows from ∆(s|0, . . . , 0) = 0. We will prove the last formula and compute explicitly the determinant ∆(s|0, . . . , 0). These computations are similar to the ones for the Ramificant Determinant (see [START_REF] Biswas | The Ramificant Determinant, Symmetry, Integrability and Geometry: Methods and Applications[END_REF]) that corresponds to the special case s = 0.

We can compute Ω k (s + l|0, . . . , 0) using Euler Gamma function.

Lemma 5.2. We have

Ω k (s + l|0, . . . , 0) = ω k(s+l) d s+l d -1 Γ s + l d
Proof. We first make the change of variables t = ω k u, and then v = u d /d,

Ω k (s + l|0, . . . , 0) = +∞.ω k 0 t s+l-1 e -1 d t d dt = ω k(s+l) +∞ 0 u s+l-1 e -1 d u d du = ω k(s+l) d s+l d -1 +∞ 0 v s+l d -1 e -v dv = ω k(s+l) d s+l d -1 Γ s + l d
Now we recall the following well known elementary Vandermonde Lemma:

Lemma 5.3. If ξ 1 , . . . , ξ d are the d roots of a monic polynomial Q(X), then we can compute the Vandermonde determinant V (ξ 1 , . . . , ξ d ) of the (ξ 1 , . . . , ξ d ) as

V (ξ 1 , . . . , ξ d ) = 1 ξ 1 ξ 2 1 . . . ξ d-1 1 1 ξ 2 ξ 2 2 . . . ξ d-1 2 . . . . . . . . . . . . . . . 1 ξ d ξ 2 d . . . ξ d-1 d = i =j (ξ i -ξ j ) = d i=1 Q ′ (ξ i ) .
Using this Lemma with Q(X) = X d -1 we compute the Vandermonde determinant:

V d = 1 ω 1 ω 2 1 . . . ω d-1 1 1 ω 2 ω 2 2 . . . ω d-1 2 . . . . . . . . . . . . . . . 1 ω d ω 2 d . . . ω d-1 d = i =j (ω i -ω j ) = i (dω d-1 i ) = d d i ω i d-1 = (-1) d-1 d d .
We use this result to compute ∆(s|0, . . . , 0).

Lemma 5.4. We have

∆(s|0, . . . , 0) = (2πd) d 2 √ 2π ω d(d-1) 2 s s Γ(s)
and in particular ∆(s|0, . . . , 0) = 0 for s = -1, -2, . . .. Taking the limit s → 0 we recover the formula from Lemma 3.5 from [START_REF] Biswas | The Ramificant Determinant, Symmetry, Integrability and Geometry: Methods and Applications[END_REF].

Proof. Using Lemma 5.2 we have

∆(s|0, . . . , 0) = = ω 0.(s+1) d s+1 d -1 Γ s+1 d ω 0.(s+2) d s+2 d -1 Γ s+2 d . . . ω 0.(s+d) d s+d d -1 Γ s+d d ω 1.(s+1) d s+1 d -1 Γ s+1 d ω 1.(s+2) d s+2 d -1 Γ s+d d . . . ω 1.(s+d) d s+d d -1 Γ s+d d . . . . . . . . . . . . ω (d-1).(s+1) d s+1 d -1 Γ s+1 d ω (d-1).(s+2) d s+2 d -1 Γ s+2 d . . . ω (d-1).(s+d) d s+d d -1 Γ s+d d = ω d(d-1) 2 s d s d d-1 2 -d Γ s d + 1 d Γ s d + 2 d . . . Γ s d + d d ω 1 0 ω 2 0 . . . ω d 0 ω 1 1 ω 2 1 . . . ω d 1 . . . . . . . . . . . . ω 1 d-1 ω 2 d-1 . . . ω d d-1 = ω d(d-1) 2 s d -d 2 s(2π) d-1 2 Γ(s)d d = d d 2 (2π) d-1 2 sω d(d-1) 2 s Γ(s) = (2πd) d 2 √ 2π ω d(d-1) 2 s s Γ(s)
where we have used Gauss multiplication formula in the second line (that is in fact due to Euler and not to Gauss, see [START_REF] Aycock | Euler and the multiplication formula for the Γ-function[END_REF]) with z = s/d,

Γ(z).Γ z + 1 d . . . Γ z + d -1 d = (2π) d-1 2 d 1 2 -dz Γ(dz) .
and that the determinant in the fourth line is equal to (-1) d-1 V d where V d is the Vandermonde determinant computed previously.

Proof of Theorem 5.1. Consider the entire function of several complex variables ∆(s|a 1 , a 2 , . . . , a d-1 ) on the variables (a 1 , a 2 , . . . , a d-1 ). Observe that Corollary 4.4 proves that each integral

+∞.ω k 0 t s+n-1 e P 0 (t) dt ,
is a linear combination with coefficients that are polynomial on s and the (a k ) of the integrals Ω k (s) for k = 0, 1, . . . , d -1, Therefore, differentiating column by column, we observe that for each k = 0, 1, . . . , d -1, we have

∂ a k ∆ = Q k ∆
where Q k is a polynomial on s and the (a k ). We conclude that the logarithmic derivative of ∆ with respect to each variable a k is a universal polynomial on the variables s and (a k ). This gives the existence of the universal polynomial Υ d such that ∆(s|a 1 , a 2 , . . . , a d-1 ) = c(s).e Υ d (s;a 1 ,a 2 ,...,a d-1 ) , with c(s).e Υ d (s;0,0,...,0) = ∆(s|0, . . . , 0) ∈ C. Then if we define Π d (s, a 1 , . . . , a d-1 ) = Υ d (s; a 1 , a 2 , . . . , a d-1 ) -Υ d (s; 0, 0, . . . , 0) we get the result ∆(s|a 1 , a 2 , . . . , a d-1 ) = ∆(s|0, . . . , 0)e Π d (s,a 1 ,...,a d-1 )

Corollary 5.5. The functions Ω 0 , . . . , Ω d-1 do not have a common zero in C -N -.

Proof. Otherwise, if s 0 ∈ C -N -is a common zero, then s 0 + 1 ∈ C -N * -and the functional equation shows that the non-zero vector (1, α d-1 , . . . , α 1 ) is in the kernel of the matrix [Ω kl (s 0 + 1)], which contradicts that it has non-zero determinant by Theorem 5.1.

Observe that this simultaneously non-vanishing result relies on the fact that Euler Gamma function has no zeros. This is something that was explained to be a "mini-Riemann hypothesis" in [START_REF] Érez-Marco | On the definition of Euler Gamma function[END_REF], and was the subject of correspondence between Hermite and Stieltjes [START_REF] Hermite | Correspondance d'Hermite et Stieltjes, publiée par Baillaud et Bouguet[END_REF]. Although used in the proof, the non-vanishing of Euler Gamma function is a particular case of this general result for Omega functions.

Corollary 5.6. The functions Ω 0 , . . . , Ω d-1 are C-linearly independent.

Proof. Otherwise there will be a non-trivial null linear combination of the rows of the matrix [Ω kl ] and the determinant will be zero.

Solutions of the functional equation.

Observe that the functional equation ( 1) reduces to the functional equation ( 2) by dividing the equation by α d that is assumed to be non-zero. We can make a first observation that the space of solutions of the functional equation ( 2) is an infinite dimensional vector space. Proposition 6.1. The space of meromorphic solutions f of the functional equation

(4) f (s + d) + α d-1 f (s + d -1) + . . . + α 1 f (s + 1) = s f (s)
is an infinite dimensional vector space.

Proof. The functional equation is linear and there are non-zero solutions (the Ω k functions). Given a non-zero meromorphic solution f (s), we can construct an infinite number of linear independent solutions

g(s) = e 2πins f (s)
where n ∈ Z is any integer.

If we restrict to solutions with a controlled growth, the situation the space of solutions is finite dimensional. Definition 6.2. We consider the C-vector space V of meromorphic functions f satisfying the functional equation ( 2 for some constant 0 ≤ c < 2π.

It is clear that the space V is a subspace of the vector space of general solutions (without a prescribed growth condition). We prove first that V is non-empty by proving the estimates for the functions Ω k for k = 0, 1, . . . d -1. Proof. We make the change of variables t

= ω k u Ω k (s) = +∞.ω k 0 t s-1 e P 0 (t) dt = ω s k +∞ 0 u s-1 e -1 d u d (1+O(u -1 )) du = e 2πi k d s +∞ 0 u s-1 e -1 d u d (1+O(u -1 )) du so, we get for 0 < a ≤ Re s ≤ b |Ω k (s)| ≤ e -2π k d Im s (1 + C 1 ) +∞ 0 u Re s-1 e -1 d u d du ≤ e -2π k d Im s (1 + C 1 )d Re s/d Γ Re s d ≤ Ce -2πk d Im s
where C, C 1 > 0 are constants depending only on a, b > 0 and P 0 .

The growth condition on the strip S(1, d) and the functional equation implies a control of f in the halph plane {Re s ≥ 1}. More precisely, we have Proposition 6.4. Let f ∈ V. Then f there exists constants C 0 , τ > 0 such that for Re s ≥ 1

|f (s)| ≤ C 0 e τ |s| Γ Re s d Proof.
More precisely, we prove that for Re s ≥ 1

|f (s)| ≤ C 0 d 2 s d Γ(s/d) ≤ C 0 d 2 Re s d Γ Re s d
The functional equation gives, for Re s ≥ s 0 , for s 0 > 0 large enough,

|f (s + d)| ≤ d |sf (s)|
We can take the constant C 0 > 0 large enough to have, for Re s ≤ s 0 ,

f (s)| ≤ C 0 d 2 s d Γ(s/d) then |f (s + d)| ≤ d|s|C 0 d 2 s d Γ(s/d) = C 0 d 2 s+d d s d Γ(s/d) ≤ C 0 d 2 s+d d Γ s + d d
Thus the estimate holds for Re s ≥ s 0 + d and by induction for Re s ≥ s o + kd for all k ≥ 1.

Now we prove the main Theorem:

Theorem 6.5. The space of solutions V is a finite dimensional vector space generated by the basis (Ω k ) 0≤k≤d-1 .

We recall Carlson's Theorem [START_REF] Carlson | Sur une classe de séries de Taylor[END_REF]: Theorem 6.6 [START_REF] Carlson | Sur une classe de séries de Taylor[END_REF]. Let C + = {s ∈ C; Re s > 0} and f : C + → C be a holomorphic function extending continuously to C + . We assume that f is of exponential type, that is, there is C, τ > 0 such that for all s ∈ C + ,

|f (s)| ≤ Ce τ |s|
We assume that on the imaginary axes we have a more precise control, for y ∈ R,

|f (iy)| ≤ Ce c|y|
for some constant c < π.

If f (n) = 0 for all n ∈ N, then f is identically 0.

We use Carlson's Theorem in the half plane {Re s > 1} to prove the main Theorem.

Proof. We consider a meromorphic solution f (s) of the functional equation and satisfying the estimate (5). The matrix [Ω kl [START_REF] Aycock | Euler and the multiplication formula for the Γ-function[END_REF]] being invertible, we have a linear combination g(s) = c 0 Ω 0 (s) + . . . + c d-1 Ω d-1 (s) with c 0 , . . . , c d-1 ∈ C such that g(l) = f (l) for l = 1, 2, . . . , d. Since g satisfies also the functional equation, we get by induction using the functional equation that f and g take the same values at all the positive integers s ∈ N * . So the function f -g vanish at all positive integers and satisfies the estimate [START_REF] De Bruijn | Asymptotic methods in analysis[END_REF]. Therefore, the function h(s) = e -iπs (f (s) -g(s))/Γ(s/d) satisfies on Re s = 1 (recall that Γ is bounded on vertical strips), with 0 ≤ c ′ < π.

Also using Proposition 6.4 the function h has exponential growth in the right half plane {Re s ≥ 1}. Therefore using Carlson's Theorem we conclude that h is identically 0, thus f (s) = g(s) for all values s in this half plane, hence in C.

We have proved that the vector space generated by Omega functions can be characterized by the functional equation ( 2) and the growth property [START_REF] De Bruijn | Asymptotic methods in analysis[END_REF]. This generalizes to Omega functions Wielandt's characterization for Euler Gamma function (1939, [23], [START_REF] Remmert | Wielandt's theorem about the Γ-function[END_REF], [START_REF] Remmert | Classical topics in complex function theory[END_REF]).

We also observe that Omega functions provide the general solutions of the functional equation (2) with estimates (5) since given such a functional equations with coefficients (α l ) we can build the coefficients a l = -l -1 α l , then the polynomial P 0 and the Omega functions (Ω k ) 0≤k≤d-1 that form a basis for the space of solutions.

It is also easy to see that we can replace the (5) by an estimate of the form, for s ∈ S(1, b), |f (s)| ≤ Ce -c Im s with 2πn ≤ c < 2π(n + 1) for an integer n ∈ Z. Then the space of solutions is also finite dimensional as the map f (s) → e -2πins f (s) provides an isomorphism of the space of solutions with V.

The structure of the space of solutions is interesting. The space of holomorphic solutions is a subspace of dimension d -1.

Proposition 6.7. The subspace of holomorphic solutions in V is a subspace of dimension d -1 generated by the entire functions Ω l (s) -Ω 0 (s) = γ 0l t s-1 e P 0 dt Proof. As observed before, the functions Ω l (s) -Ω 0 (s) are entire functions and are linearly independent. Some of the results in [START_REF] Biswas | The Ramificant Determinant, Symmetry, Integrability and Geometry: Methods and Applications[END_REF] can be generalized. In particular the Integrability criterion and Abel-like Theorem (Theorems 4.2 and 4.3). This will be studied in a separate article. K. Biswas has extended results from [START_REF] Biswas | The Ramificant Determinant, Symmetry, Integrability and Geometry: Methods and Applications[END_REF] to curves of higher genus [START_REF] Biswas | Algebraic de Rham cohomology of log-Riemann surfaces of finite type[END_REF]. It is interesting to speculate on the extension of the results for Omega functions in higher genus.

1 . Introduction 1 . 1 .

 111 Difference equations. We study in this article difference equations of the form (1) sf (s) = d k=1 α k f (s + k) where α 1 , . . . , α d ∈ C and α d = 0. The simplest case is the functional equation satisfied by Euler Gamma function sΓ(s) = Γ(s + 1)

Theorem 1 . 1 .

 11 The space of meromorphic solutions f of the functional equation sf (s) = d k=1 α k f (s + k) where α 1 , . . . , α n ∈ C, α d = 0, and f satisfies a growth condition, for 1 ≤ Re s ≤ d, |f (s)| ≤ Ce -c Im s

Theorem 3 . 4 .

 34 The Omega function Ω k has the Mittag-Leffler decomposition:

Theorem 5 . 1 .Ω 11 Ω

 5111 For any s ∈ C -N * -, the vectors Ω(s + 1), Ω(s + 2), . . . , Ω(s + d) are linearly independent, ∆(s) = 0 where ∆(s|a 1 , . . . , a d-1 ) = det 12 . . . Ω 1d Ω 21 Ω 22 . . . Ω 2d .

  ) and the estimate in the vertical stripS 1,d = {1 ≤ Re s ≤ d}, for s ∈ S 1,d , (5) |f (s)| ≤ Ce -c Im s

Proposition 6 . 3 .

 63 For k = 0, 1, . . . , d -1, for any strip S a,b = {a ≤ Re s ≤ b} with 0 < a < b, there exists a constant C = C(a, b, P 0 ) > 0, depending only on a, b > 0 and the polynomial P 0 , such that for s ∈ S a,b , we have|Ω k (s)| ≤ Ce -2πk d Im sObviously we can take a = 1 and b = d and since 0 ≤ c = 2πk d < 2π we get that Ω k satisfies the estimate[START_REF] De Bruijn | Asymptotic methods in analysis[END_REF].

  |h(s)| ≤ Ce -(c-π) Im s with 0 ≤ c < 2π. Therefore we have on Re s = 1 |h(s)| ≤ Ce c ′ | Im s|

Acknowledgements.

We thank J.-P. Ramis for pointing out the earlier appearance of Omega fonctions under the name of "modified Gamma functions" in the theory of asymptotic approximations, and to the classical litterature on difference equations, in particular for references on the study of solutions with controlled vertical growth.