domain: shs sdv

text:
Evelyn Rivera-Toapanta

Zein Kallas

Meta Čandek-Potokar

Joel Gonzalez

Marta Gil

Elsa Varela

Justine Faure

Marija Cerjak

Tomažin Urška

Chiara

Chiara Aquilani

Bénédicte Lebret

Danijel Karolyi

Carolina Pugliese

José Maria Gil

Marketing strategies to self

Keywords: Analytical hierarchy process, autochthonous swine breeds, local meat products, marketing, PDO, PGI products, World Café Focus Group

Introduction

The sustainable conservation of autochthonous swine breeds and local chains could be partly achieved by highlighting the high-quality intrinsic and extrinsic properties of the products derived from them and by promoting their consumption by specific market niches [START_REF] Vitale | Consumers' expectations and liking of traditional and innovative pork products from European autochthonous pig breeds[END_REF]. Moreover, traditional food products constitute an important element of European culture, identity and heritage [START_REF] Ilbery | Niche markets and regional speciality food products in Europe: towards a research agenda[END_REF][START_REF] Lebret | Consumers' study on traditional pork products from local breeds: expectations and hedonic evaluation[END_REF], and are increasingly being proposed as a way to raise societal awareness of rural landscapes [START_REF] Soy-Massoni | Landscape eaters: supporting rural development and ecosystem services delivery by eating[END_REF]. However, innovation in traditional products may face some challenges related to the likely contested understanding of these concepts [START_REF] Guerrero | Cross-cultural conceptualization of the words traditional and innovation in a food context by means of sorting task and hedonic evaluation[END_REF], making it particularly difficult to develop innovations acceptable to consumers. Improving breed profitability through niche markets and increased added-value products [i.e., healthier nutritional properties [START_REF] Romanzin | Effect of rearing system (mountain pasture vs. indoor) of Simmental cows on milk composition and Montasio cheese characteristics[END_REF]] , ensuring a return to breed populations high enough to maintain adequate genetic diversity and hence safeguard breed survival [START_REF] Bozzi | Conservational issues in local breeds-state of the art[END_REF]. [START_REF] Varga | Changing year-round habitat use of extensively grazing cattle, sheep and pigs in East-Central Europe between 1940 and 2014: consequences for conservation and policy[END_REF] reported that swine grazing had almost disappeared in Europe by the 1970s. In this context, research on market opportunities and analysis of consumer preferences and the factors affecting sustainability will be a key strategy. This will focus on the renewability and resilience, thereby aiming to regenerate the social, environmental and economic impacts [START_REF] Swisher | Value chains in renewable and sustainable food systems[END_REF].

Some studies [START_REF] Brown | Consumers' preferences for locally produced food: a study in southeast Missouri on JSTOR[END_REF][START_REF] Conner | Consumer demand for local produce at extended season farmers' markets: guiding farmer marketing strategies[END_REF] have concluded that local foods are related to spatial proximity, freshness and higher sensorial quality, but for the authors, the most important 'local' promotion message may be 'Grown in' and the name of the region or city rather than the distance (from food production to consumers). [START_REF] Howard | Central coast consumers want more food-related information, from safety to ethics[END_REF] also found that consumers prefer to obtain information that includes the word 'local' on labels or in brochures rather than through direct interaction with the seller. Unlike intensive production, the swine in the five local systems studied are raised outdoors/extensively and so depend on the resources of their natural surroundings, defined as grazing habitats in the review carried out by [START_REF] Varga | Changing year-round habitat use of extensively grazing cattle, sheep and pigs in East-Central Europe between 1940 and 2014: consequences for conservation and policy[END_REF]. Moreover, consumers perceive pasture-raised products as healthy, natural and environmentally friendly and some of them are willing to pay a premium price for pasture-raised meat-based products (Font-i-Furnols and [START_REF] Guerrero | Consumer preference, behavior and perception about meat and meat products: an overview[END_REF][START_REF] Stampa | Consumer perceptions, preferences, and behavior regarding pasture-raised livestock products: a review[END_REF]. These Localized Agro-Food Systems also open new perspectives for territorial and regional development projects [START_REF] Muchnik | Systèmes agroalimentaires localisés: état des recherches et perspectives[END_REF].

In this context, research on market opportunities, analysis of consumer preferences and the factors affecting their acceptance of different types of product is a crucial strategy to support traditional autochthonous swine breeds and promote their development. Effective promotional and marketing strategies have been defined for potential markets through developing the role of location in promoting more advantageous sales methods for farmers in nontraditional distribution channels (short distribution channels, direct sales). The present study has a twofold objective: first, to carry out a Strengths, Weaknesses, Opportunities and Threats (SWOT) analysis on five European autochthonous swine breed production systems to identify the critical points of the adopted marketing channels; and secondly, to identify appropriate marketing strategies for each study case to provide stakeholders along the supply chain with guidelines for enhancing added-value products and for developing direct marketing alternatives to analyze viability and feasibility.

Material and methods

The aim of the TREASURE H2020 project (Horizon 2020Programme, 2015) was to improve the conservation status of 20 European autochthonous swine breeds through innovative strategies that included untapped new market options for these breeds (Čandek-Potokar and Nieto Linan, 2019). Five of them and their related value chains with contrasting properties in terms of level of development and organization and representing different EU regions were considered for a study of their marketing strategies [START_REF] Karolyi | Turopolje Pig (Turopoljska svinja)[END_REF][START_REF] Lukač | Krškopoljski prašič (Krškopolje Pig)[END_REF][START_REF] Mercat | Gascon Pig[END_REF][START_REF] Pugliese | Cinta Senese Pig[END_REF][START_REF] Tibau | Negre Mallorquí (Majorcan Black) Pig[END_REF]. The reason for including the untapped breed is that it was mandatory in the UE call for proposals (Research and Innovation Action, Grant agreement N°: 634476).

Experimental design

We followed a mixed-method research methodology and applied a joint qualitative and quantitative approach [START_REF] Weible | Citizens' perception of modern pig production in Germany: a mixed-method research approach[END_REF], carrying out focus group (FG) sessions using the World Café (WC) method and the Analytical Hierarchical Process (AHP) because of their specific strengths. Five FG sessions, one for each of the case studies, were performed between May and July 2018 in Louey (Hautes-Pyrénées-France), Florence (Tuscany-Italy), Palma de Mallorca (Spain), Ljubljana (Slovenia) and Zagreb (Turopolje region-Croatia).

Each FG included at least 12 stakeholders from each local system, with a balanced representation of the five stages in the value chain (Fig. 1). The suggested shares of stakeholder profiles in the FG were three farmers, two processors, a civil servant/policy maker working in public administration (either regional or local), a representative related to the governance of the quality seal [protected designation of origin (PDO)/protected geographical indication (PGI)], if applicable, a traditional store retailer, a supermarket retailer, a food service representative (restaurants) and two people working in consumer associations (n = 60 in total). The final number of participants in each profile was adapted slightly in each case study, depending on the characteristics of the corresponding supply chain. The WC was applied to gain insights into two sets of attributes in each case study: (a) the SWOT matrix and (b) the most suitable marketing strategies based on the 4Ps of marketing (Product, Price, Place and Promotion) for the breed production system. The WC is a structured conversational process intended to facilitate open and intimate discussions, and share ideas within a larger group to obtain collective consensus decisions.

All the participants were balanced (in terms of number and also the type of stakeholder in the chain) and separated into four thematic tables (groups for the discussion) with one moderator per table. Each table was designed to gather information about one of the four components of the SWOT analysis. For 10-12 min, the participants at each table expressed their thoughts about the SWOT regarding the pertinent swine production systems and their meat products. Further, the groups agreed among themselves on a list of the most relevant items, along with the moderator. Afterwards, they moved to another table, ensuring that no two participants coincided at two consecutive tables. This rotation was repeated until all the stakeholders had participated at all the four tables. The most relevant items raised in terms of SWOT at each discussion table were then voted in a plenary session. Each participant chose three prominent items depending on their choice, and further ranked them to assign three, two or one point based on the rank. In a second step, participants addressed the marketing strategies (4Ps). The same procedure was then followed to obtain the preference values for the 4Ps, conducting four rounds of discussion (one P per table: Product, Price, Place and Promotion) and voting in a plenary session to determine the three most important items for each marketing strategy. At the end of the WC sessions, the information collected was used to design an AHP survey, which was emailed to the WC participants with a briefing on how to fill out the questionnaire, to be returned within the next 48 h after reception.

The relative importance of four marketing strategies: the AHP The AHP was carried out to measure the relative importance of the main elements of the marketing mix strategies involved in the product, price, place and promotion policies extracted from the WC discussion sessions. The AHP methodology has been used previously to evaluate marketing mix elements [START_REF] Abedi | Prioritizing of marketing mix elements effects on patients' tendency to the hospital using analytic hierarchy process[END_REF], but to the best of our knowledge, very few studies on this have been directed at the food sector [START_REF] Borrisser-Pairó | Towards entire male pigs in Europe: a perspective from the Spanish supply chain[END_REF]. Prioritizing marketing mix elements can help members of the supply chain make strategic decisions to attract customers and meet their needs. The AHP is a multicriteria decision-supporting method in discrete environments, which aims to breakdown a complex decision problem into a hierarchy of smaller constituent sub-problems [START_REF] Saaty | A scaling method for priorities in hierarchical structures[END_REF][START_REF] Saaty | The Analytic Hierarchy Process[END_REF]. Determining the most preferred policy from a set of policies to help the untapped breeds to thrive is a decision problem for which the top level of the hierarchy represents the marketing mix strategy. It is broken down into a predefined number of characteristics (policy decisions) on the second level and their policy types on the third level (Fig. 2).

The AHP estimates elicit weights (w) for each policy and policy level to explain individual behavior in choosing the preferred marketing mix strategy. The relative importance of weight (w) for each policy (P n ) and policy type (L n.p ) [where n (1…N) is the number of policies and p (1…P) is the number of policy levels] is obtained from pairwise comparisons. To implement the AHP, participants in the WC discussion were asked to make two types of pairwise comparisons: (a) pairwise comparisons of the policy level; and (b) pairwise comparisons between the policies (product, price, place and promotion). The respondent first had to indicate the relative importance of the two elements compared. A nine-point scale was then used to measure the strength of this relative importance using verbal judgments [START_REF] Saaty | The Analytic Hierarchy Process[END_REF].

An example of the pairwise comparison can be seen in Table 1. Using the result elicited from this, the AHP estimated the weights (w) for each policy and policy level to explain individual behavior in choosing the preferred marketing mix strategy.

Results and discussion

Main results and discussion of the SWOT analysis of local swine systems

The most relevant SWOT for Majorcan Black Pig (MBP), Cinta Senese (CS), Noir de Bigorre (NB), Krškopolje Pig (KRS) and Turopolje pig (TP) are detailed below in Table 2. It provides a summary of the stakeholders' opinions gathered from the SWOT analysis of each local swine breed system in which differences and common aspects can be appreciated.

The strengths of these swine breeds are based on their being differentiated native breeds adapted to the land and different agro-climatic conditions, reared within traditional extensive systems, and fed using natural resources (e.g., pasture grass, cereals, legume seeds, figs, almond, acorns and several Mediterranean shrubs). Another strength is that the meat and its products have the PGI or PDO seal. Furthermore, cultural heritage and terroir products were highlighted by the stakeholders of the NB and KRS local systems.

One of the main strengths identified in the SWOT analysis was the high quality of the products and their link to the «heritage patrimony». This is a very important aspect of these local production systems underlined by Pensado-Leglise and Sanz Cañada (2018). However, their research also shows that these characteristics are insufficient to generate the incomes needed by farmers, which will put the sustainability of these chains at risk in the near future. In relation to this, Čandek-Potokar et al. (2018) recommended that the majority of local swine breed producers still need to be supported by subsidies to ensure their conservation. However, these authors also reinforce the idea that the best strategy is to breed systems aiming to reach self-sustainability. This may be possible through an effective marketing strategy of the meat products obtained from local swine breeds [START_REF] Čandek-Potokar | Improving sustainability of local pig breeds using quality labels-case review and trademark development in project TREASURE[END_REF].

The weaknesses included the high costs of farming derived from a low prolificacy (reproduction rate), the consequent high price of the products and the small size of the farms. Aging farmers, product heterogeneity in terms of quality, and a lack of professionalism, collaboration and marketing strategies are challenging for these breeds to adapt to the current market demands. These weaknesses should be overcome to face current consumer demand. Moreover, these local swine breed production systems should be in line with the Local Agro-Food System (SYAL), because this would imply a better mutual cooperation network among stakeholders along the entire chain and new perspectives for public policies and for territorial and regional development projects [START_REF] Muchnik | Systèmes agroalimentaires localisés: état des recherches et perspectives To cite this version[END_REF]. Compare the following policies in a pairwise way to identify the most important one as the best marketing strategies for the added-value meat products from the untapped swine breeds. The opportunities were based on strengthening consumer loyalty within the local market, including the tourist market, and the promotion of traditional products with a PGI or a PDO label, highlighting their international expansion. Growing market demand for different cuts of fresh meat in the retail market and for high-quality fresh meat and ready-to-eat formats in the HORECA sector (HOtels, REstaurants and CAtering) has made the strategy of launching innovative products an attractive business opportunity, with positive consumer perceptions of local products and production systems.

All the main threats center on competition with other local swine breeds, mainly in the MBP and NB chain, and with other low-priced commercial meat products made from quality cuts. Also mentioned were the misconceptions surrounding the word black among the NB and Iberian products. Furthermore, consumers' negative perceptions regarding the high fat content of meat products and product frauds increase mistrust among consumers, and their health and animal welfare concerns may determine the swine breeds' survival. Another concern was regarding the aging farmers who adopt inefficient management practices.

To sum up, one of the main strengths identified from the SWOT analysis was the high quality of the products and their link to cultural patrimony. This is a very important aspect of these local production systems, underlined by Pensado-Leglise and Sanz Cañada (2018). However, these authors also explain that the identified strengths are insufficient to justify the continued payments to farmers, risking the sustainability of these chains in the near future. They also point out that there may not be sufficient market support for these local systems, biodiversity and cultural patrimony. Along these lines, Čandek-Potokar et al. (2018) recommend that most local swine breed production still need to be supported by subsidies to ensure their conservation. However, these authors also reinforce the idea that the best approach is making these breed systems self-sustainable, which may be possible with an efficient marketing strategy for their products. Another approach is the work of Sanz-Cañada and [START_REF] Sanz-Cañada | Geographies of origin and proximity: approaches to local agro-food systems[END_REF], who emphasized that these Local Agro-Food System (SYAL), built on identity-based food products, could be crucial for a more sustainable rural development. But, of utmost importance is the cooperation between farms, firms and institutions.

Main marketing strategies of the local systems based on the 4P analysis: implications for the potential to introduce the products into the market and for conserving the swine breeds No scientific or systematized studies have previously been undertaken or published on marketing strategies for the self-sustainability of the five autochthonous swine breeds. There may only be a few publications containing descriptive characteristics.

Majorcan Black Pig

The most relevant product policy identified by stakeholders was the diversification of meat products to increase the position of the MBP brand in the market (37.43%) by including innovations focused on the nutritional properties of the products, process innovation and format size, thereby meeting consumer demands [START_REF] Vitale | Consumers' expectations and liking of traditional and innovative pork products from European autochthonous pig breeds[END_REF], and strengthening the PGI brand (26.75%) (Fig. 3). Also relevant was improving the sensory quality of the products (35.82%), including extending their shelf life and highlighting their nutritional value. These strategies are similar to the ones mentioned by [START_REF] Guerrero | Marketing PDO (products with denominations of origin) and PGI (products with geographical identities)[END_REF].

Regarding the place strategy, increasing the presence of MBP processed food, including sobrassada (seasoned, dried and fermented sausage with a PGI certification), porcella (suckling piglet) and fresh meat cuts, in local restaurants, was identified as one of the most important strategies (42.98%), together with enhancing the availability of sobrassada in local shops (33.58%). Besides, stakeholders stated that the PGI Regulatory Council and public administrations must be encouraged to support commercial actions that ensure the presence of MBP processed food in large retailers, thereby strengthening the MBP image (21.44%). These findings support the conclusions from [START_REF] Giraud | Consumer perception of typical food products in Europe[END_REF], who suggested that marketing plans of typical foods should mainly focus on local consumers.

The most relevant action (42.29%) for the promotion strategy was to increase the presence of MBP products at sectoral and gastronomic fairs via public relations, followed by promotion on TV, online and in the press (31.19%), and the promotion of MBP products in primary schools with public administration support (26.52%). Furthermore, the proposed branding strategy using the TREASURE projects as support was considered as a 'marketing tool that could promote the MBP as a differentiated and autochthonous swine production system'.

The stakeholders' price strategy (40.32%) was to maintain a high price based on a premium category. Some stakeholders also advocated for a different distribution of prices along the production chain, which would favor higher revenues for farmers, indirectly promoting the development of the primary sector. This aligns with some policy initiatives such as those launched by the French and Spanish governments to develop value chain legislation aimed at ensuring a minimum price for farmers.

Cinta Senese

In this system, product heterogeneity is considered a hurdle for building a loyal customer base. Thus, the most relevant product strategies were to improve the standardization of the food processing among producers (38.15%) (Fig. 4) and to promote the healthier nutritional properties of the processed food (35.75%). Modifying some of these characteristics, including reducing the salt and additive content, would help to better match customer needs. Increasing sales of raw meat should also eventually be promoted (26.10%); indeed, this type of product can be sold directly by the farmer, representing an immediate source of income and an interesting tool for rural development [START_REF] Chambers | Local, national and imported foods: a qualitative study[END_REF][START_REF] Sanjuán | Consumers' willingness to pay for beef direct sales. A regional comparison across the Pyrenees[END_REF]. Moreover, producers believe that the sensory traits associated with CS are more appreciated in fresh meat than in processed food. There was a certain disagreement among stakeholders regarding the prioritization of place strategies, with two strategies of almost equal importance emerging. The two most relevant strategies for the sale and distribution of CS processed food were high-quality restaurants (37.15%) followed by stores selling top-quality products (36.33%). The latter category also includes GDO (large-scale distribution) gourmet lines, even though most of the producers were against GDO sales policies. Despite the increasing difficulties faced by local stores such as butchers' shops to compete with GDO, they remain an important selling channel, making this placing strategy still worthwhile (26.53%). Moreover, short supply chains and local retailers are preferred over GDO for high added-value products such as PDO and/or PGI-labeled products, since they can provide consumers with direct information about meat origin, processing methods and animal rearing systems [START_REF] Marsden | Food supply chain approaches: exploring their role in rural development[END_REF][START_REF] Ilbery | Forecasting food supply chain developments in lagging rural regions: evidence from the UK[END_REF].

Within this context, the promotion policy using social networks was considered the most important one for the CS system (38.18%), followed by collaboration with renowned chefs (34.88%) and cooking schools (26.94%), which might help raise the profile of CS products outside Tuscany. Limited consumer knowledge of the breed and GDO competition for price polices were considered to be strong hurdles for price strategies in the CS chain. However, stakeholders firmly agreed that a further price increase could be an attractive strategy (49.70%) to differentiate CS production systems and acknowledge their value, but that it should go hand in hand with an enhanced marketing strategy, possibly encouraging tasting experiences at purchasing points.

Noir de Bigorre

The product strategy is a priority for the Noir de Bigorre chain-Gascon breed and is related to the development of products and their image. To strengthen product traceability for consumers, it was considered that an 'identity card' for hams and loins would enable a certain product to be associated with a particular farmer and the pig feeding method or practices (45.12%) (Fig. 5). The implementation of open-air drying/ripening rooms (33.28%) and the (increased) possibility of marketing ham slices cut with a knife (rather than a slicer) are possible innovations in the chain (21.60%).

The place strategy is ensured by the coverage of local stores in the PDO area, serving as ambassadors for the products (41.25%) and improving local product visibility. Working with a high-end specialized network (31.10%), possibly outside the geographical area of production, would be another opportunity to promote the distribution and sale of NB processed food.

Regarding the promotion strategy, this is mainly linked to its heritage dimension. This result is in line with the work of Sanz-Cañada and [START_REF] Sanz-Cañada | Geographies of origin and proximity: approaches to local agro-food systems[END_REF], who showed that identitybased food is related to territory. To support these products, it is important to focus on their showcasing and cooking lessons highlighting their sensory characteristics (44.68%). Different promotional strategies have been identified based on the relationship between the food and its story. Training in the product history for all the stakeholders in the sale of the product is a key element (37.49%). A two-pronged product promotion strategy was considered, one for the public, based around the PDO and the Last, the price strategy is based on a modification of the carcass payment grid to include meat quality indicators (48.61%) in an attempt to improve the quality of the meat and fat and to manage its variability.

Krškopolje Pig

The stakeholders described dry-cured products, fresh meat and innovative culinary products to be the three most important product strategies. Following the AHP analysis, the highest relative importance (Fig. 6) was assigned to dry-cured products (49.46%).

Regarding the place strategy, the AHP analysis revealed that stakeholders considered the most important channel for the products from the breed (42.99%) to be direct sales, with business gifts mentioned. This type of distribution is further supported by the significant importance given to state protocol (use as gifts and in gala dinners) (23.29%) as the third most important distribution channel. Good high-cuisine restaurants were recognized as the second most important distribution channel (33.82%).

Focusing on the promotion strategy, having a prestigious seal was recognized by stakeholders as the most important (49.05%), and given twice as much weight as the next two key promotion strategies identified, namely guided tours with tasting (25.92%) and promotion via educational institutions (25.03%). The importance of the use of these products as state protocol gifts and the promotion of the breed and its products in education centers and schools were highlighted. Last, the key price policy strategies established with the AHP methodology were premium price, differentiated price and price adjusted to the sales channel. The premium price strategy was given the greatest weight (42.69%). In summary, the recognized assets were mainly non-intensive traditional rearing, good quality and flavor, and cultural heritage, while the main shortcomings were a disorganized value chain and no processed food standardization. The localized agro-food systems concept (SYAL) could be an interesting tool to improve cooperation in this chain [START_REF] Muchnik | Systèmes agroalimentaires localisés: état des recherches et perspectives[END_REF].

Turopolje Pig

The product strategy for TP processed food should rely on high product quality and be used for product positioning. This is in line with previous research [START_REF] Cerjak | What motivates consumers to buy traditional food products? Evidence from Croatia and Austria using word association and laddering interviews[END_REF] showing that Croatian consumers perceive a better quality of traditional products as one of the major motives for their purchase.

One of the most important elements of product policy is the standardization of processed food (40.58%) (Fig. 7). Next to it is the use of quality labels, with priority given to the PDO quality sign (38.27%) and the organic label, customer perception associated with higher product positioning, purchase frequency and willingness to pay, as shown by [START_REF] Bryła | The perception of EU quality signs for origin and organic food products among Polish consumers[END_REF]. Due to market saturation in terms of exclusive products, TP processed food should be sold in smaller packages and using innovative strategies such as sales in gift baskets. Images of the Turopolje region and/or TP should be used on the packages (21.15%) to highlight the product origin.

Regarding the place strategy, TP products could only be purchased directly from producers and were occasionally served in local restaurants, which was recognized as a major problem. Therefore, while representing an important percentage (29.09%), direct sales were evaluated as less important than sales in specialized shops (40.69%) and restaurants (30.22%). These findings are in line with a previous study [START_REF] Jensen | Heterogeneity in consumers' perceptions and demand for local (organic) food products[END_REF] which showed that Danish local food consumers support short food chains. Only a few respondents agreed with an online sales strategy.

Focusing on the promotion policy, most of the stakeholders stated that the main message they would convey to consumers is that TP processed foods are healthy, environmentally friendly and of high quality, representing the tradition and indigenous nature of the breed and the raw materials involved (47.23%). However, a few respondents stated that they would segment consumers and use messages adapted to each type of target. Storytelling, which involved sharing stories about the history and traditions of the Turopolje region (30.57%), was considered an important promotion strategy. Respondents agreed on the usefulness of direct marketing, including promotion at fairs and local events, word of mouth and others (22.19%). Promotion through social networks (Facebook, Instagram, Twitter, etc.) was likewise considered a prospective promotional strategy. Last, all stakeholders agreed that TP products should cost a premium price due to their expensive production and product exclusivity. However, some stakeholders held a different opinion on the pricing strategy within the supply chain, believing that value to a product (e.g., exclusive packaging) should also influence the selling price (25.21%). Some respondents emphasized the need for common price negotiation if processed foods were to be standardized.

Strategies of the local systems

The extensive or semi-extensive local grazing systems characterizing the five swine breeds addressed in this study (Fig. 8) are essential toward the conservation and enhancement of high natural value farming systems, as described by [START_REF] Varga | Changing year-round habitat use of extensively grazing cattle, sheep and pigs in East-Central Europe between 1940 and 2014: consequences for conservation and policy[END_REF]. Autochthonous breeds play a role in supporting agroecosystem resilience [START_REF] Hajjar | The utility of crop genetic diversity in maintaining ecosystem services[END_REF], maintaining socio-cultural traditions, local identities, traditional knowledge [START_REF] Nautiyal | The sociological importance of social movements: in the perspective of Uttarakhand separate state movement[END_REF] and cultural landscapes [START_REF] Tisdell | Socioeconomic of loss of animal genetic diversity: analysis and assessment[END_REF], all of which are public goods and attributes identified in our study.

Recent research has demonstrated that the best chance of conserving the genetic diversity of autochthonous swine breeds comes from a mix of looking for niche markets for the processed food and increasing awareness among consumers who appreciate the high quality of traditional meat products [START_REF] Kallas | Can innovations in traditional pork products help thriving EU untapped pig breeds? A nonhypothetical discrete choice experiment with hedonic evaluation[END_REF]. To this effect, the stakeholders in the present study identified the following actions as the best product strategies: enhancing and highlighting the sensory quality of the meat products, boosting the geographical indications and traditional specialties of the breed, where appropriate (PGI and PDO), improving the standardization of the food processing among producers to ensure homogeneous products, and increasing product diversity and traceability. This is in line with the emergent interest in local foods, where consumers perceive to have higher quality standards and to be tools for preserving tradition and local know-how [START_REF] Gilg | Quality farm food in Europe: a possible alternative to the industrialised food market and to current agri-environmental policies: lessons from France[END_REF][START_REF] Weatherell | In search of the concerned consumer: UK public perceptions of food, farming and buying local[END_REF][START_REF] Vecchio | Italian and United States farmers' markets: similarities, differences and potential developments[END_REF]. In addition, the PDO and PGI logos are commonly considered the main purchasing motivation for a shopper with excellent knowledge of the EU certification label, according to [START_REF] Vecchio | The role of PDO/PGI labelling in Italian consumers' food choices[END_REF]. However, a better understanding of the relationships between people, products, and territory will be crucial to the survival of these local systems. Our study hints at organizational differences among these systems. Hence, in some of the studied chains (Cinta Senese and Krškopolje Pig), better cooperation among the stakeholders and major collective action is needed. This cooperation would improve the governance and increase negotiation ability with the other stakeholders related to the marketing strategy, as described by [START_REF] Cañada | Valorización de una Indicación Geográfica Protegida. El caso de la carne de la Sierra de Guadarrama[END_REF].

Regarding the price strategy, maintaining a high price based on a premium category while letting consumers taste the product in situ (show-cooking, direct marketing) and establishing price differentiation depending on feeding (extensive vs concentrate) were identified as key elements in our study, which is also in agreement with [START_REF] Stampa | Consumer perceptions, preferences, and behavior regarding pasture-raised livestock products: a review[END_REF].

With regards to the place strategy, the stakeholders identified the need to increase the presence of traditional products in the HORECA sector and to improve and intensify their presence in specialized local food shops selling top-quality products and gourmet foods. This strategy, which was identified in most of the chains studied, is in line with [START_REF] Jensen | Heterogeneity in consumers' perceptions and demand for local (organic) food products[END_REF]. Similarly, [START_REF] Conner | Demand for pasture-raised livestock products in Michigan: results of consumer surveys and experimental auctions[END_REF] showed that natural food stores were the most promising distribution channel.

Several common strategies identified to promote the processed food from local systems were based on public relations, including food and gastronomic events, tourist events, workshops, guided tours, direct sales to companies for business gifts, web pages and collaboration with catering schools, popular chefs and athletes. They highlighted that any promotional message should include information about the history of the product and its healthy properties. However, the concept of 'growing in', referring to the information about the product origin, as discussed in the study from the USA by [START_REF] Conner | Consumer demand for local produce at extended season farmers' markets: guiding farmer marketing strategies[END_REF], must be enhanced. This aspect was not given enough consideration by the stakeholders of these European local systems. Securing public administration support is recommended to develop these aspects.

Concluding remarks

The participative bottom-up mixed-methods approach used in this study involving the key stakeholders in each untapped breed value chain is novel and enabled us to co-construct the identified strategies. This methodology increases the likelihood of successfully implementing these strategies since they are not seen as impositions from external agents.

The main parameters of this study of untapped swine breeds were based on non-intensive traditional rearing, good sensory quality and cultural heritage. The extensive production systems that characterize these native breeds are fully aligned with sustainability since they are an essential element in the conservation and enhancement of high nature value local farming systems.

The stakeholders identified having PGI and PDO seals as being of utmost importance to enhance the sensory quality of the products, improve the standardization and traceability of the food processing to ensure homogeneous products, and to enhance their high price based on the premium category and direct sales. Iberian meat products could be a strong competition mainly to the MBP and NB breeds and hence effective product identification and traceability is highly important. The stakeholders of these autochthonous breeds must place greater emphasis on highlighting grazing as a feeding strategy (extensive or semi-extensive), the meat products derived from the swine and their origin.

Fig. 1 .

 1 Fig. 1. Stakeholders' type included in the focus groups. *: Hotels, Restaurants, and Catering.

Fig. 2 .

 2 Fig. 2. Hierarchical structure used to value the best policy for the untapped swine breeds.

Fig. 3 .

 3 Fig. 3. Hierarchical structure used to value the best policy for the MBP breed. Product (Pd): Pd 1 : Boost PGI brand, Pd 2 : Enhance the organoleptic quality of the product, Pd 3 : Diversify the product and adapt to the needs of the young consumer. Place (Pl): Pl 1 : Selling MBP products in local restaurants, Pl 2 : Distribution of sobrassada in local shops, Pl 3 : (MBP sector) with access to different distribution channels. Promotion (Pm): Pm 1 : Marketing strategies based on public relations (food and gastronomic events), Pm 2 : Promotion in primary schools with the support of the Public Administration, Pm 3 : TV promotion, and Price (Pc): Pc 1 : To maintain a high price strategy based on a premium category, Pc 2 : To increase the transparency of prices along the production chain, and Pc 3 : To apply a premium strategy along all the production chain.

Fig. 4 .

 4 Fig. 4. Hierarchical structure used to value the best policy for the CS breed. Product (Pd): Pd 1 : Increase the diffusion of fresh meat, Pd 2 : Healthier products (less salt, less additives), Pd 3 : Improve the standardization of manufacturing process among producers to ensure homogeneous products. Place (Pl): Pl 1 : Stores of topquality products (also GDO with gourmet lines), Pl 2 : Local shops, Pl 3 : High quality restaurants. Promotion (Pm): Pm 1 : Collaboration with chefs, Pm 2 : Collaboration with cooking schools, Pm 3 : Promotion on social networks. Price (Pc): Pc 1 : Different price between organic and conventional products, Pc 2 : Different prices depending on feeding strategy (i.e. finishing with acorn/chestnut or concentrate), and Pc 3 : Higher price, but let the consumers try the products in stores.

Fig. 5 .

 5 Fig. 5. Hierarchical structure used to value the best policy for the Noir de Bigorre chain-Gascon breed. Product (Pd): Pd 1 : To obtain traceability/identification card for ham and loin, Pd 2 : To set up ageing room in the open-air, Pd 3 : To bring to market knife sliced products. Place (Pl): Pl 1 : To extend the coverage of local stores in DPO area, Pl 2 : To develop their export sales. Pl 3 : To work with specialized high-end network. Promotion (Pm): Pm 1 : To support products from processing to tasting (showcase of products, cooking lesson), Pm 2 : To provide training in products history for all actors participating in product's sale, Pm 3 : To build two promotions: one for general public, another for professionals. Price (Pc): Pc 1 : To create carcass payment table at SICA to encourage meat quality, Pc 2 : To raise prices of rare products (dry-cured ham, dry sausage), and Pc 3 : To raise pig price at SICA.

Fig. 6 .

 6 Fig. 6. Hierarchical structure used to value the best policy for the KRS breed. Product (Pd): Pd 1 : Dry-cured products, Pd 2 : Fresh Meat, Pd 3 : Innovative culinary products. Place (Pl): Pl 1 : Good, high-cuisine restaurants, Pl 2 : Direct sales (on farms, and to companies for business gifts), Pl 3 : State protocol. Promotion (Pm): Pm 1 : Prestigious brand, Pm 2 : Guided tours, tasting, Pm 3 : Through educational institutions. Price (Pc): Pc 1 : Premium price, Pc 2 : Differentiated price and Pc 3 : Price adjusted to sales channel.

Fig. 7 .

 7 Fig. 7. Hierarchical structure used to value the best policy for the TP breed. Product (Pd): Pd 1 : To obtain "Protected Designation of Origin" (PDO) status, Pd 2 : Standardization of production (recipe), Pd 3 : Exclusive packaging. Place (Pl): Pl 1 : In specialized shops, Pl 2 : Direct sale Pl 3 : In restaurants. Promotion (Pm): Pm 1 : Promotion of healthy, ecological, indigenous, traditional, high-quality products, Pm 2 : Direct marketing (fairs, local events, oral submission …), Pm 3 : Use Turopolje heritage in promotion. Price (Pc): Pc 1 : Premium price for premium products, Pc 2 : All Turopolje pig products should be in the same premium price segment and Pc 3 : Aligned price strategy along the entire value chain.

Fig. 8 .

 8 Fig. 8. Growing finishing pig from Mallorca island (a), Tuscany (b), Hautes-Pyrénées (c), Slovenia (d) and Turopolje region (e).

Table 1 .

 1 Example of the AHP pairwise comparison across policies

	Policy 1

Table 2 .

 2 Summary of stakeholders' points of view from SWOT analysis of each local swine breed system: differences and common aspects

	Breed	N°Strengths	Weaknesses	Opportunities	Threats
	MBP	1	Differentiated native breed	Elevated farming costs	Expansion to international	Competition with Iberian pig
			adapted to the territory and		market	products
			agro-climatic conditions
		2	PGI of the sobrassada	Small size of the farms and	Building customer loyalty	Low price of meat products
				the heterogeneity of the	within the local Majorcan	from other commercial swine
				products' quality	market
		3	High intrinsic sensory quality of	Lack of professionalism	Promotion of PGI meat	Fat content
			meat and meat products		products
	CS	1	Traditional rearing system and the	CS fresh meat is scarcely	Working on a better	Products' frauds increasing
			know-how of farmers ensure meat	available outside Tuscany	communication strategy	distrust of consumers and
			and meat products with excellent		for the breed and its	their concerns for health and
			sensory features		products	animal welfare
		2	Meat has earned the PDO to give	CS system is fragmented	Collaboration with	CS systems are dealing with
			a strong identity to CS	between farms and	renowned chefs to enlarge	consumers' increasing
				processing industries	the consumer knowledge	disinformation about food
						and agriculture systems
		3	Creating a rewarding niche	Lack of communication	The PDO label	Elder farmers who keep
			market for environmentally	and collaboration and lack		adopting inefficient
			sensitive consumers	of shared marketing		management and market
				strategies		strategies
	NB	1	High sensorial quality (flavor) of	Managing variability at	Heritage of the chain as a	Sanitary crises (epizootic
			the products	each stage of production	factor of attractiveness for	epidemic, African swine fever)
					tourism and institutions	at national, European or
						international levels
		2	PDO label (breed, know-how and	Variability between farms	Use of local resources	Confusion with the adjective
			terroir) as protector of NB chain	in the sanitary risk	developed by the	'black' used by other swine
					NB chain meet the social	chains
					expectations of
					consumers
		3	Feeding natural resources,	No control of use breed	Intrinsic quality (sensory,	Declining meat consumption
			outdoor production system	outside the PDO area	nutritional…)
	KRS	1	Non-intensive traditional way of	The value of breed is not	Need for branding	The inability of breeders to
			rearing	sufficiently distinguished	strategy 'back to nature'	achieve the expected price
						levels
		2	High meat quality (sensory	The quality of meat	Differentiate sales	Intensification of the rearing
			properties)	products is not adequately	channels through tourism,	mode, lack of the quality
				standardized	HoReCa sector, gifts	rules
		3	Cultural heritage related to the	Lack of standardized	Consumers' perceptions	Missing intermediate factors
			KRS systems	production process	of local products and	in the value chain
					extensive production
					systems
	TP	1	The long-standing tradition in its	Long and expensive	Increasing purchasing	Higher exposure to animal
			breeding	production cycle	power of some consumer	diseases and
					segments	unfavorable weather
						conditions
		2	Swine are raised outdoor and fed	Low prolificacy (small	Use of celebrity marketing	Competition with other local
			with naturally available feedstuffs	number of piglets)		swine breeds
		3	Meat products with PGI	High mortality rate and	Support of local	Low interest and
				consequently the high	authorities in application	unfamiliarity of wider range
				price of TP products	for PGI	of consumers with products
						of TP

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Evelyn Rivera-Toapanta et al. Downloaded from https://www.cambridge.org/core. 06 Sep 2021 at 11:30:36, subject to the Cambridge Core terms of use.

Acknowledgements. This study has received funding from the European Union's Horizon 2020 Research and Innovation program under grant agreement No 634476 (project acronym TREASURE). IRTA thanks CERTA funding. Authors thank Jaume Jaume (Semilla-Caib), who supported the organization of the study in Mallorca Island; Florence Garcia-Launay and Ludovic Brossard (PEGASE, INRAE, 35590 Saint-Gilles, France) and Alexandre Fonseca (Consortium Noir de Bigorre) who participated in the organization of the French case study and realization of the focus-groups. E. A. Rivera-Toapanta is a recipient of a doctoral fellowship awarded by the INIA (Instituto Nacional de Investigación y Tecnología Agraria y Alimentaria). This work has been carried out within the framework of the Doctorate Program in Food Sciences of the Autonomous University of Barcelona. The content of this paper reflects only the authors' view and the European Union Agency is not responsible for any use that may be made of the information it contains.

Author contributions. Evelyn Rivera-Toapanta: investigation, formal analysis, writing original draft, supervision, visualization. Zein Kallas: project administration, methodology, conceptualization, investigation and writing review and editing, supervision. Meta Čandek-Potokar: conceptualization, project administration, writing-review and editing, funding acquisition. Joel Gonzalez: conceptualization, investigation and writing original draft. Marta Gil: supervision, project administration, investigation, writing-review and editing. Elsa Varela: conceptualization, investigation, writing-review and editing, methodology. Justine Faure: conceptualization, investigation and writing original draft. Marija Cerjak: investigation, writing-review and editing. Tomažin Urška: investigation, writing-review and editing. Chiara Aquilani: investigation, writing original draft and writing-review and editing. Bénédicte Lebret: conceptualization, project administration, investigation, writing-review and editing, funding acquisition. Danijel Karolyi: conceptualization, project administration, investigation, writing-review and editing, funding acquisition. Carolina Pugliese: conceptualization, project administration, investigation, writing-review and editing, funding acquisition. José Maria Gil: project administration, methodology, conceptualization, funding acquisition.


M Hage-Hassan

The two-dimensional hydrogen atom in The momentum representation

The analytic expression of the momentum representation in terms of the associated Legendre function is determined by a direct integration of Fourier transform of the wave function of coordinates using the Levi-Civita transformation and 2 2 R R → the generating function method. A new generating function for the associated Legendre functions are obtained.

Introduction

The problem of the hydrogen atom in momentum space has been reformulated by Fock [1] which led to an integral form of the Schrödinger equation. This equation is solved by projecting the three-dimensional momentum space onto the surface of a four-dimensional sphere and the eigenfunctions are then expanded in terms of spherical harmonics. The applications of Fock's method to various problems of quantum mechanics are very extensive [2][START_REF] Wulfman | Group Theory and its Applications[END_REF][START_REF] Bethe | Quantum Mechanics of One-And Two-Electron Atoms[END_REF]. Recently, Fock's stereographic projection method had been applied to the theory of anisotropic excites and the two dimensional hydrogen atom. It had physical realization [START_REF] Parfitt | [END_REF][START_REF] Parfitt | [END_REF][START_REF] Aquilanti | The d-dimensional hydrogen atom: Hyperspherical harmonics as momentum space orbital and alternative sturmian basis sets[END_REF] and many papers treated this case [START_REF] Parfitt | [END_REF][START_REF] Parfitt | [END_REF][START_REF] Aquilanti | The d-dimensional hydrogen atom: Hyperspherical harmonics as momentum space orbital and alternative sturmian basis sets[END_REF][START_REF] Torres Del Castillo | On the dynamical symmetry of quantum Kepler problem[END_REF].

In a previous paper [START_REF] Hage-Hassan | On the hydrogen wave function in Momentum-space, Clifford algebra and the Generating function of Gegenbauer polynomials[END_REF] we presented a new and elementary method for the determination of the analytic expression for the wave function in momentum space in the third dimension space by a direct integration of the Fourier transform of the wave function of coordinates. We construct first, the generating function of the basis of coordinates assuming that the energy is a constant. Second, we perform the Fourier transform using Hurwitz transformations [START_REF] Hage-Hassan | Inertia tensor and cross product in n-dimensions space[END_REF][START_REF] Hage-Hassan | Hurwitz's matrices, Cayley transformation and the Cartan-Weyl basis for the orthogonal groups[END_REF], or quadratic transformations to obtain the generating function for the momentum space. Finally, the wave function in the momentum space is derived from the development of the last function and the energy is replaced by its value.

The presented paper is part two of our previous work [START_REF] Hage-Hassan | On the hydrogen wave function in Momentum-space, Clifford algebra and the Generating function of Gegenbauer polynomials[END_REF], we deal with the same method for the two dimensional hydrogen atom and we derive the wave function in the momentum representation. It is important to note that the calculation in the case of 2-dimensions is particular and differs from the treatment of 3-dimensions and from Fock's stereographic projection method. This paper is organized as follows. In section two, we derive the wave function of the two dimensional hydrogen atom. In section three, we construct the generating function for the basis of the hydrogen atom and are also devoted to the presentation of the transformation . In sections four and five, we derive the wave functions of the hydrogen atom in momentum space. In section six, we derive the last wave functions in terms of spherical harmonics. Finally in section seven, we acquire the new generating function for the associated Legendre functions.

R R →

The wave function of two dimensions hydrogen atom

The Schrödinger equation of the two dimensional hydrogen atom is:

) , ( 2 ) , ( 2 2 ϕ ρ ρ μ ϕ ρ Ψ - Δ - = Ψ e H h (2.1)
To solve the equation (2.1) we apply the method of separation of variables

) ( ) ( ) , ( ϕ ρ ϕ ρ Φ = Ψ R (2.2)
) , ( ϕ ρ are the polar coordinates.

Introducing a separation constant, , we can obtain the angular equation

2 m 0 2 2 2 = + ϕ ϕ m d d (2.3)
The solution is:

ϕ π ϕ im e 2 1 ) ( = Φ (2.4)
The corresponding radial equation is:

ER R e m = - + ∂ ∂ ∂ ∂ - - ] ] 1 ) ( 1 [ 2 2 2 2 2 ρ ρ ρ ρ ρ ρ μ h (2.5)
For simplicity, we choose the Gaussian units

q E - = We find the equation [12] 0 ) 2 ( 1 2 2 2 0 2 2 = - - + + R m q d dR d R d ρ ρ ρ ρ ρ (2.6)
We make the substitution

) ( ) ( ) ( 2 / v F e v R v m - = ρ and ρ 0 2q v =
This leads us to the equation:

0 ) 2 1 2 1 ( ) 1 2 ( 0 2 2 = - - + - + + F m q dv dF v m dv F d v (2.7) This is the confluent hypergeometric equation: 0 ' ) 1 2 ( ' ' = + - + + ny y v m vy
This has two linearly independent solutions. If we choose the solution which is regular at the origin, then this becomes a polynomial of finite degree for [START_REF] Gradshteyn | table of integrals series and products[END_REF] .We can now write the real-space wave function in the form:

ϕ ϕ ρ im m m n v m nm nm e v L e v N ) ( ) ( ) , ( 2 2 / - - = Ψ (2.8)
Using the orthogonal relations of Laguerre polynomials and the functional relation [START_REF] Gradshteyn | table of integrals series and products[END_REF] )

( ) ( ) ( ) 1 ( ) ( ) 1 2 ( ) ( 1 1 v L n v L n v L n v vL n n n n α α α α α α - + + - + - + + =
We obtain the normalized wave functions:

ϕ ϕ ρ im m m n v m nm nm e v L e v N ) ( ) ( ) , ( 2 2 / - - = Ψ (2.9) With )! ( )! ( 3 0 m n m n q N nm + - = π

Generating function for the basis of two dimensions hydrogen atom

and the quadratic transformation

2 2

R R →

We construct the generating function for the basis of the hydrogen atom and we define the transformation 2 2 R R →

The generating function of Laguerre polynomial

) ( 2 v L m m n-
The generating function of Laguerre polynomial is:

∑ ∝ = - - + - = 0 1 1 ) ( ) 1 ( 1 ) ( n v z z r r n n e z v L z (3.1) From the property ) ( ) ( ) 1 ( 1
) ( v L v L dv d n n + - - = α α We deduce ) ) 1 ( exp( ) 1 ( ) ( ) ( 1 2 0 2 z zv z z v L z m m n m m n n - - - = + ∝ = - ∑ (3.2)

The generating function

We assume that is a constant. We write

0 q = Ψ = ∑ - ) , ( ! ) , , ( 1 ϕ ρ nm nm m n nm m t z N p t z G r = ∑ - - ϕ ρ ρ ρ im m nm m m m n n q e q m t q L z e ) 2 ( ! ) 2 ( 0 0 2 0 2 0 0 ) 1 ( ) ( 2 ) 1 ( ) 2 ( exp( ) 1 ( 1 0 z iy x tzq z q z e z q - ± + - - - - ρ ρ (3.3) 3.3 The quadratic transformation 2 2 R R → The well known Levi-Civita, transformation is 2 2 R R → (3.4) 2 1 2 2 2 1 2 , u u y u u x = - =

The volume element

We consider the transformation ) , ( ) , ( 2

1 ϕ ρ → u u With . 2 , 1 , , 0 , 2 0 = ∝ + ≤ ∝≤ - ≤∝ ≤ ≤ ≤ i u i ρ π ϕ
The calculation of the Jacobian gives Therefore

u d u u y u x f r d y x f r r ∫ ∫ = 2 2 2 )) ( ), ( ( 4 ) , ( (3.5)

The generating function of the hydrogen atom in momentum space

We first write the Fourier transform in the representation (u) then we perform the integration thus, we obtain the generating function in the momentum representation.

The generating function in {u} representation

The wave function of the hydrogen atom in momentum space is:

2 u u d z iy x tzq z q z e z p t z G u q r p i r r r r 2 2 0 0 . ) 1 ( ) ( ) 1 ( ) 2 ( exp( ) 1 ( 2 ) , , , ( 2 0 -
± + - - - = ∫ - - - ρ π β β ρ (4.3)
We assume that 0 ≥ β as to eliminate the problem of convergence.

We then write ) , , ( )] , , , ( [

0 p t z G p t z G r r = ∂ ∂ - = β β β (4.4)
We chose m ≥0 for the following procedures because the calculations are not affected by this choice.

The generating function of momentum-space

We can do the integration of (4.3) by a direct calculation with the variables (u) using the integral with

∫ - = E n P dx dx dx dx e I ... 3 2 1 ∑ = = n ij j i ij x x a P 1
The well known solution is

) det( , / ) ( 2 ij n a a a I = = π
We have

u d z iy x tzq z q z q e r p i r r r 2 2 0 0 0 . ) 1 ( ) ( 2 ) 1 ( ) 2 ( ) ( exp( - + + - - + - ∫ - ρ ρ β = u d z iy x tzq z q z e r p i r r r 2 2 0 0 . ) ) 1 ( ) ( 2 ] ) 1 ( ) )( 1 ( [ exp( - + + + - + - ∫ - ρ β and ) 2 ( ) ( . 2 1 2 2 2 1 u u ip u u ip r p i y x - - - = - r r 2 2 1 2 2 2 1 0 2 2 2 1 0 ) 1 ( )) 2 ( ) (( 2 ) ]( ) 1 ( ) )( 1 ( [ z u u i u u tzq u u z q z - + - + + + - + - β 2 2 2 1 u u + = ρ We obtain then ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ - - + + - + + - - + - - - - - + - + = ) ) 1 ( 2 ( ] ) 1 ( ) ) ( 1 ( [ ) 1 ( 2 ) 1 ( 2 ) ) 1 ( 2 ( ] ) 1 (
) )( 1 ( [ 2 0 0 2 0 2 0 2 0 0 x y y x ip z tzq z q z ip z tzq i ip z tzq i ip z tzq z q z X β β The )} ( 4 ) 1 ( )] 1 ( ) 1 {[( ) 1 ( 1 ) det( 0 2 2 2 0 2 y x ip p itzq z p z q z z X + + - + - + + - = r β
We then find the generating function ) , , , (

β p t z G r )} ( 4 ) 1 ( )] 1 ( ) 1 {[( 1 2 ) , , , ( 0 2 2 2 0 y x ip p itzq z p z q z p t z G + + - + - + + = r r β β
In applying the relation (4.4) we find the generating function

) , , ( p t z G r 2 / 3 0 2 2 2 0 0 2 )} ( 4 ) 1 ( ] ) 1 {[( ) 1 ( 2 ) , , ( y x ip p itzq z p q z q z p t z G + + - + + - = r r (4.6)

The wave functions in momentum-space

The development of the function (4.6) gives us the wave functions of the hydrogen atom in momentum space. We derive the basis of momentum-space using the formula

) ( )] , , ( ! 1 [ 1 0 p N p t z G t z n nm nm m m n n r r Ψ = ∂ ∂ ∂ ∂ - (5.1)
In this level we must take ) 2 / 1 /( 1 0 + = n q as to execute the calculations proceed by the following steps:

1 -Derivation with respect to t 2 / 3 2 2 2 0 0 2 0 } ) 1 ( ] ) 1 {[( )] ( 4 [ ) 1 ( ) 2 / 3 ( ) 1 ( 2 )] , , ( + - + + + - - = ∂ ∂ m m y x m m m m m z p q z ip p iq z z p t z G t r r [ We obtain ( ) )) ( ) ( 2 ) (( ) 1 ( ) 1 ( 2 0 2 2 2 0 2 2 0 2 2 2 2 2 0 q p z q p z q p p z z q + + - - + = - + + r r r r , with ] 2 1 )[ ( 2 2 0 2 z zq q p + - + = r ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + - = 2 0 2 2 0 2 q p q p q r r We deduce that m y x m m m m m m m m ip p z zq z z q p q i p t z G t )] [( } 2 1 {[ ) 1 ( ) ( ) 2 / 3 ( ) ( ) 4 ( 2 )] , , ( [ 2 / 3 2 2 2 / 3 2 0 2 1 0 0 + + - - + - = ∂ ∂ + + + r r (5.2)

2-Derivation with respect to z

Using the familiar formula for the generating function of Gegenbauer polynomials

) ( ) 2 1 ( 0 2 q C z z qz m m m α α ∑ ∝ = -= + - We write ( ) q C z z z z zq z z m k k k m m m 2 / 3 0 2 2 / 3 2 2 ) 1 ( ] 2 1 [ ) 1 ( +
∝ = + ∑ - = + - - (5.3) ( ) ( ) [ ] q C q C z m m n m m n n n 2 / 3 2 2 / 3 + - - + - ∝ - = ∑ (5.4) With n m k = + , n m k = + + 2 and ) 2 / 1 /( 1 0 + = n q therefore, × + - = ∂ ∂ ∂ ∂ + + 2 / 3 0 2 1 0 0 ) ( ) 2 / 3 ( ) ( ) 4 ( 2 )] , , ( ! 1 [ m m m m m m n n q p q i p t z G t z n r r ( ) ( ) m y x m m n m m n ip p q C q C )] [( ] [ 2 / 3 2 2 / 3 + - + - - + - (5.5)

3-The wave function in momentum space

With the help of the recurrences formula [START_REF] Erdelyi | Higher transcendal functions[END_REF] )] (

) ( )[ 2 / 1 ( ) ( ) 2 / 1 ( 2 / 3 2 2 / 3 2 / 1 q C q C m q C n m m n m m n m m n + - - + - + - - + = + (5.6)
We find:

× + - + + = ∂ ∂ ∂ ∂ + + 2 / 3 2 0 2 1 0 0 ) ( ) 2 / 3 ( ) ( ) 4 ( ) 1 2 ( ) 1 2 ( 2 )] , , ( ! 1 [ m m m m m m n n q p q i m n p t z G t z n r r m y x m m n ip p q C
)] [( ) (

p q C q p q i m n N p )] [( ) ( ) ( ) 2 / 3 ( ) ( ) 4 ( 2 ) 1 2 ( ) 1 2 ( ) ( 2 / 1 2 / 3 0 2 1 0 + + - + + = Ψ + - + + r r (5.7) It is important to note that ) ( 2 / 1 q C m m n + -
is a particular function of Gegenbauer polynomials. This function has a connexion with the associated Legendre polynomials.

The wave functions in momentum space in terms of spherical harmonics

The connection of Gegenbauer polynomials with the associated Legendre polynomials is given by the formula [START_REF] Gradshteyn | table of integrals series and products[END_REF] )

( )! 2 ( 2 ! ) 1 ( ) 1 ( ) ( ! )! 1 2 ( 1 ) ( 2 / 2 2 / 1 t P m m t dt t P d m t C m n m m m m n m m m n - + - - - = - = (6.1)
We derive finally the wave functions in momentum space:

2 / 3 2 0 2 1 0 ) ( ) 2 / 3 ( ) ( ) 4 ( 2 ) 2 / 1 ( ) 2 / 1 ( ) ( + + + - + + = Ψ m m m m nm nm q p q i m n N p r r m y x m n m m m ip p q P m m q )] [( ) ( )! 2 ( 2 ! ) 1 ( ) 1 ( 2 / 2 + - - - (6.2) But m m m m m m q p q p q p q p q ) /( ] [ ] [ 2 ) ( ) ( 1 ) 1 ( 2 0 2 2 / 2 0 2 / 2 2 / 2 2 0 2 2 0 2 2 / 2 + = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + - - = - r r r r (6.3) And ! 4 )! 1 2 ( ) 2 / 3 ( m m m m + = (6.4)
Replacing (6.3) and (6.4) in (6.2) we find the wave functions in momentum space in terms of spherical harmonics:

p im m n m nm e q P q p q m n m n i p ϕ π ) ( 2 )! ( )! ( 2 ) ( ) ( 2 / 3 2 0 2 0 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + - = Ψ r r (6.5)
It is clear that we obtain by an elementary method not only the wave function in momentum representation [START_REF] Parfitt | [END_REF] but also the phase factor and the calculation in the case of 2-dimensions. It is particular and differs from the treatment of 3-dimensions.

The new generating function for the associated Legendre functions

We start from the expressions (5.3) and (5.4). We write that: Finally, we note that the Fourier transformation for the hydrogen atom transforms the radial part into angular which explains the Fock transformation. We also observe that our method for the 2-dimensional case is important not only for the physical applications [START_REF] Parfitt | [END_REF][START_REF] Parfitt | [END_REF][START_REF] Aquilanti | The d-dimensional hydrogen atom: Hyperspherical harmonics as momentum space orbital and alternative sturmian basis sets[END_REF] but also for educational applications because our result can also be an introduction to Octonion algebra and of course to Fock's transformations.

  and n = 0, 1, 2, . . . The solutions of the equation (2.7) are the associated Laguerre polynomials

5 )

 5 To calculate this expression (4.1) we must use the (u) representation and the formula (3.5):

  (5.4) and(5.1) gives us the wave function in momentum space:

  By connecting Gegenbauer polynomials with the associated Legendre polynomials we find a new generating function of these last polynomials