ON FOREST NENETS VIEW ABOUT THE WORLD BEYOND Eva Toulouze

By reading the ethnographers' reports about the Nenets' beliefs, and having done myself fieldwork among the Nenets, I have often felt some kind of suspicion in front of the reconstructions of perfectly coherent systems they have tried to achieve, which often seem to be artificially constructed.

By analysing two recently recorded Forest Nenets tales about the world beyond, I shall not try to build up a closed image of this important element of the Nenets' worldview, but to emphasize some basic characteristic elements.

I shall here concentrate on three Forest Nenets narratives relating a human being's contact with the world beyond. Of the first we have three versions 1 by the same storyteller, a 40-45 years old blind man from the Western area inhabited by the Forest Nenets 2

. The second has been recorded from a 55 years old Nenets, a reindeer herder and a poet with higher education, who is also a fighter for his people's rights 3 . According to his report, this narrative was part of the active repertoire of one of his neighbours, who died in 1995 as an old man 4

. It is only a fragment of a longer story, but the rest has been forgotten. The peculiar feature of this narrative is that the core of it, describing the world beyond, is performed in rhythmical prose, in opposition to the introduction, which is formulated in ordinary prose

5

. The third is another story by the younger storyteller, which has been recorded by himself and literated by his wife; I base my comments on her translation. The texts are presented as appendixes 6 .

I start by summing up their structure: in the first one, the hero, a girl named Atpälha, is the victim of her two stepmothers' jealousy. The women try to kill her by throwing a burning hot thimble in her throat. The girl falls inanimate. Her father, back from hunting, puts her body on a sledge with one single reindeer and sends them where they have to go. Meanwhile the girl sees and hears everything, but she is unable to react. She sees people in the forest, whose strange behaviour makes clear, as 1 One of them has been recorded by Kaur Mägi and myself in autumn 2000 ; another has been recorded in a studio, in Tartu's Literature Museum, in November of the same year. Of the last we have only a written text and its translation into Russian : it is the transcription of the tale as the storyteller told it and recorded it for the Research Institute od Khanty-Mansiysk in the second half of the nineties (we haven't any precise information about the year) made by his wife. I have used here the two latter.

2 His official name, according to his Russian passport, is Aleksandr Čipahevič Logany and he is born in 1960. His Nenets name, by which he is generally known and called, is Tatva. Logany is the Russian version of his clan's name, in Nenets Ngahany. Tatva lives in a camp on the shore of the Porsavar Lake, at 15 km from the village of Num-to. 3 His official name, according to his passport, is Yuri Kylevič Ayvaseda, born on March, 12. 1948. As a matter of fact, his connection to the Ayvaseda clan is but an error of the Soviet administration: his father and his mother were both registered as Ayvaseda, although they belonged respectively to the Vella and Tyott clans. As a poet and public personality Yuri is known as Yuri Vella. At the beginning of the 1990-ies, Yuri moved to the taiga, where he lives presently as a reindeer herder. He belongs to the Eastern group of Nenets and is based at the village of Varyogan.

well as the narrator's comment, that she is in the world beyond 7

. She starts to live there, gets married and during a quarrel her husband hits her in the back. She falls inanimate. When she wakes up, the thimble is on the grass in front of her, she is in a cemetery in the forest with her sledge and her reindeer, and she goes back home. In the second story a man (who is not a Nenets 8

) tries to discover who steals his fish, sets a trap, and follows one morning the traces of blood on the grass. He arrives at a hole and stares into it. He discovers then that it is the upper orifice of a Nenets tent, and what he sees convinces him that he is glancing at the world beyond, and he goes quickly back home.

The structure of the two narratives is slightly different: the first one is a closed round form, in which the trip into the other world constitutes the central part; it is framed by the circumstances of the trip and the return, which form themselves a story. In other words, in this narrative there are two plans, and two stories corresponding to the two worlds. The second narrative is formed by a longer introduction presenting the circumstances of the contact, and a conclusive sentence. In the first story the trip, although central, is but one of the episodes, while in this story, the contact with the world beyond is the core of the whole narrative.

The third narrative is a very long tale, whose story embraces two generations: the plot starts with a young girl's trespassing the prohibition of entering the territory of an evil giant. The giant wants her to marry him, and threatens to kill all her kins if she refuses. Instructed by a young man she met in the forest, she refuses and marries another man. But the giant keeps to his word and kills the husband and his relatives. The young woman goes back to her brothers and gives birth to a son, who soon remains orphan. The child -called Foxhat's Sonis adopted by a Russian merchant, who has a son of his own. Once the Russian's son, Foxhat's Son's companion, plots to kill him, but he is killed instead and the young Nenets must fly away. During his journey, he meets the giant's son and defeats him. After that, he descends to the lower world, where he travels further and when hes journey is over, he returns to the living's world and convinces the Russian merchant that he is innocent of his son's death. In this narrative, the journey to the other world is a marginal element of a much longer and much complicated plot.

1)

The passage point In Atpälha's story the passage point is more developed than in the other tale, where the hero is just an eyewitness and does not enter himself the world beyond. He stays at the border, which is represented by a hole in the ground. Although he does not pass through it, the border is very clearly represented: it is a physically perceptible border, situated in a vertical perspective. Thus is the border in the story of Foxhat's Son: there is a door in the earth, leading downwards and protected by two giants: the hero has to defeat them in order to pursue his journey. In Atpälha's story, the passage is twice achieved by the hero and is not represented by a physically located border, but the by a change in the status of the hero herself.

This change of state and even of status is achieved by external interference and represented by an object, the thimble, whose presence into Atpälha's body is concomitant with her status as nonalive. The two step-mothers, by throwing the thimble into the girl's throat, are cause of the first passage, as well as the husband is the initiator of the second one, by hitting her wife on the back, and provoking the expulsion of the thimble from her body 9

. Both times, the girl falls inanimate: her temporary neutralisation is the narrative sign of the passage.

Since the passage, her position in both worlds is ambiguous: the storyteller tells us that she is dead, but not completely: she sees and hears all as in a dream 10 , but is unable to react, as long as she is still in the camp. As she sets herself in the world beyond, she is still different from the others, as we shall soon see. So after her passage in the other dimension, Atpälha is herself an in-between: she is not alive (she can't move for some time, she sees life in the world beyond), but she is not dead (she sees all what happens around her, she is different from the other inhabitants of the other world). She is in a medium position.

As a hero, Atpälha is a most passive: she merely obeys her stepmothers, who call her into the tent 11 and order her to shut her eyes and open her mouth. Her only initiative in this world is to order the reindeer to go home.

2)

The world beyond as it appears in these tales Before commenting the vision of the world beyond as induced by these tales, I must present the facts as told by the storytellers.

So Atpälha is sent to the place where she is supposed to go. During her journey in the forest, she sees all what happens around her. The sees first people walking towards a camp and carrying wood. But at the camp they are unable to discharge the wood, which remains glued to their arms. The storyteller here observes that these persons were those who, while living, used to steal wood from their neighbours. Then she sees people walking towards a camp carrying on their shoulders fish-traps full of fish. But neither they were able to put the traps on the ground. The storyteller explains that these were the people who, while living, used to steal fish from their neighbour's traps. Then she sees a couple lying under a reindeer skin. Each one pulls the skin to his or her side, so that the skin is permanently moving from one side to another. These were the couples that were unable to live in good harmony

12

. Then the storyteller says that she saw many other people 13 , but he has forgotten 9 No comment in Version 1. Version 2: "the thimble was blocked in the throat. That's what prevented her living". 10 Version 1: "Atpälha fell on the ground. She was dead. She fell. (…) She hears all, but she is not strong enough to speak (…). During her journey, she opened her eyes"; version 2: "And she died, but it was as if she were not completely dead. (…) Although her eyes were closed, at the same time she sees all as in a dream". 11 In version 2, she just steps in the tent with the wood she has just been chopping. 12 In version 2, the order is slightly different: she sees first of all the couple, then the people carrying wood and fish.

their stories. Finally she arrives to a big "town", a camp with many tents. The reindeer stops in front of one tent. A woman comes out and rejoices seeing the girl: she helps her stepping from the sledge and welcomes her home, announcing that she has been given a bride for her son. Atpälha is given a place in the tent. Her mother-in-law is very attentive to the girl's comfort: she knows Atpälha must be hungry, and gives her soup. But the soup is full of worms and fly grubs, and the girl refuses to eat, saying she is not hungry. After some days, her stepmother starts to be concerned for her not eating at all and says to her son: "She is not used to our food. Go and slaughter your best reindeer, let her have good fresh meat". Her son obeys, slaughters their best reindeer and gives Atpälha a whole leg. But when she starts to skin the leg, she sees under the skin the same worms and fly grubs, and throws the leg away from her. Then her husband gets angry against his capricious wife and hits her on the back, Atpälha falls and so on.

In the second tale, the hero looks through the hole and understands that it is but the upper orifice of a tent. He sees a woman bending over a cradle. In the cradle lies a very old man, wounded;

on the other side of the tent, two naked children, a boy and a girl, play. The woman says to the man in the cradle: "Look what you have done! I have always told you that the humans' food is not good for us.

Heaven and earth have set that our food is worms. If you go on like that, you will never grow as old as your grandfather and your grandmother, who are playing so quietly on the ground". Hearing these words, the man understood that he was glancing at the world beyond, where time runs the other way round.

In Foxhat's Son tale, the young Nenets arrives at the door after having defeated numerous enemies. He has to defeat the two giants at the door and while he pursues his journey in the dead's world, he fights many other supernatural beings -"He fought many evil spirits and gods of diseases".

Then he arrives near a tree supporting two eagles' eyrie: the eagles assist the evil spirits and bring them victims among the living, from the earth. Foxhat's Son accepts not to kill them, if they bring him further. They fly over six sulphur rivers, then they see the Russian merchant's son, who walked against the sun, but he refuses to answer Foxhat's Son's call. After the seventh river he arrives near a huge house made of iron: it is the dwelling of the emperor of all diseases and evil spirits. The emperor receives him as his son and asks for support, but Foxhat's Son's fights with him until the emperor promises to give him back his mother and the other dead. Then the young man returns to the earth, where indeed he finds his mother and his other relatives alive.

Let us see what features of the world beyond these tales reveal.

a) First of all, in these stories, the world beyond is not presented as a whole system: in both cases, the heroes see only a fragment of the life that is led there. In one story, it is the interior of a single household. In the other, the vision is more complex, for during her journey in the forest, Atpälha sees different persons in different situations, representing each a type of misbehaviour and its punishment in the world beyond. The story-teller informs us that the list was much longer, but he doesn't remember the other cases. Anyhow, the result is a traditional structure in three episodes. So we can identify in this tale a starting point for a global vision of the world beyond, which is not systematically developed. The place she arrives to is a city 14 : but this broader frame is just mentioned, and the main episode of Atpälha's story, where she is an actor and no long a spectator, is in a single tent, where the characters apparently live as in real life. In Foxhat's Son's tale there is a much more developed topography: after seven sulpher rivers, there is an iron house where the god of diseases dwells.

b) This is the second structural feature about the world beyond: it is mostly represented like ordinary life. In Atpälha's story, the forest is an ordinary forest and the people's activities are like all other forest people's ones: they carry fish traps or wood, like anybody else in real life. The tent is an ordinary tent, where people live and eat ordinary food: soup, reindeer marrow, raw meat. So the organisation of life is a reproduction of the real world's way of living. People have reindeers and act like ordinary people. There is apparently nothing strange, nothing supernatural in the world beyond.

The third representation differs substantially from these two: there are supernatural beings presented as such, eagles who carry the hero and sulphur rivers and a house of ironall of these elements are far from realistic. We may even wonder, whether this "other" world is the same as the one appearing in the first two narratives: the aspect emphasised by the narrator is not life in the "other" world, after death, but the place where evil spirits dwell. Still, the two are very thoroughly connected, as revealed by the presence of the young Russian killed by Foxhat's son in self-defence.

But even in the cases where this world is similar to the ordinary world, this is only the surface.

For some elements are most essentially different, and the kind of difference is in all cases of the same nature: the reverse. The character that is mentioned as a little child has the appearance of an elder man and the grandparents are small naked children. Only the woman, a middle aged person, is as she is supposed to be, for in both calculations of time she is in the midst. So the appearance of the characters is inverted comparing to what we use to have in the human's world.

In Atpälha's case, the difference is still more subtle, and appears only in connection with food: first of all, people eat things that for humans are disgusting; only after a while we have the full revelation of the situation: even the reindeer, which look alive, are as a matter of fact but rot meat.

People eat what for humans is the symbol of death, for, in spite of appearances, they are dead themselves. The fact Atpälha does not accept worms as normal food is the proof that she does not belong to the world of the dead.

Even in Foxhats' Son story there is a very small indication of this principle: the Russian's son is walking in the opposite direction comparing to the Sun's, and this element is just connected with the minor aspect of this "other" world, the one connected to the dead.

3) Some deductions from these two tales

If we try to get some general features of the understanding of the world beyond on the basis of these two tales, we may distinguish the following features:



The world beyond is no extraordinary place. We have no fantastic constructions, but the simple surface reproduction of ordinary life, as far as there is no confusion with the location where gods live.



The passage moments in the tales show contradictory locations: in Atpälha's story there is a spatial continuum between the home forest and the forest inhabited by dead people. There is no sign of verticality. In the two other tales however verticality is patent and the world beyond is represented as being under the real world of the living humans. In Foxhat's Son, the doors lead down, but this lower world is itself flat.



The main feature distinguishing the two worlds is the fact that some elements in the world beyond are the reverse of those of the living's world: as to which elements are inverted, the two stories differ in some way. In one tale, it is space, a movement in the "wrong" direction; in another tale, it is time. In Atpälha's story, there are no space-time discrepancies between the two worlds: Atpälha's husband and stepmother are in this regard as they would be in the human's world. But there is a distinctive element common to two of the narratives:

food. On one case, the woman nursing the old man in the cradle says that human food is no good for them and that they eat worms. In the other story, the girl is given unacceptable food, and even the slaughtering of a reindeer, supposed to provide fresh meat, does not change the nature of the meat.



In one of the tales the dimension of the world beyond as a place of punishment for misbehaviours committed in lifetime is present, while in the other this aspect is completely unknown. In Foxhat's son, the world beyond for the dead corresponds spatially to the place where evil spirits live.

How can we explain the differences between the three tales? Do they express complementary views, all acceptable to the Nenets? Is it possible to establish a chronology between them, and explain the divergent elements by a chronological evolution of the Nenets' worldview? Some elements seem to prove that both Tatva's stories -Atpälha, as it has been told us in 2000, and Foxhat's Sonare of more recent origin and express a more syncretistic way of understanding the world beyond.

The first clue is connected with genre. The tale where time measuring is inverted belongs to a genre called wanL, and characterised, according to Nenets genre taxonomy, by its connection with truth. It has been transmitted by an old man at the end of is life, so we may suppose that it has circulated for decades 15

. It is true that we no nothing about Atpälha's story and its origin. It has been 15 According to Tatva, this story is not a wanL, but a šotpyaws, a fiction tale. The difference between the name originally given to it and Tatva's understanding is directly connected with the evolution of beliefs: what for Auli, born at the beginning of the 20 th century, would be object of belief, is clearly no more so for Tatva, born in the second half of the same century.

told by a younger storyteller, who says that it is šotpjaws, alias a fiction tale 16 . Foxhat's Son is also presented as a tale and does not pretend to have any connection with real life. As a matter of fact, it may even by classified among the "belief tales" (kahäj šotpjaws): we are informed at the end that Foxhat's Son became himself a god.

A second clue suggesting the possibility of Atpälha being a more recent item is its symmetrical construction, with at least one ternary episode, the three visions in the forest. True enough, the storyteller observes that he has forgotten the others, suggesting thus that the tale has been longer.

But such as it is told nowadays by Tatva, the structure is closed: Incidentpassage to the world beyonda ternary vision sequence and a sequence about life in that worldreturn to the world of humanspunishment of the culprits. As far as we know from historical sources, this kind of structure is not traditional in Nenets narrative. We have no extensive comparative material from Forest Nenets folklore: the larger collection of Forest Nenets texts is the one due to Toivo Lehtisalo (Lehtisalo 1947), who collected language samples and folklore among the Forest Nenets in 1914. We find there some short tales and one somewhat longer. But this is not enough to have a comprehensive view of oral traditions at the beginning of the 20 th century.

On the other hand, we are allowed to compare Forest Nenets folklore with Tundra Nenets oral tradition: the two peoples are very closely connected to one another, although nowadays phonetic evolutions prevent mutual understanding and the way of life present divergent features. Tundra

Nenets folklore has been thoroughly analysed, especially the so-called "epic" genres, some samples have been also published by N. Tereshchenko [START_REF] Tereshchenko ; Терещенко | [END_REF]). These materials are not of recent origin, for they have all been collected in the first half of the 20 th century. There is in Tereshchenko's texts no trace of circular structure. We may therefore suppose that this arrangement of the material is of relatively recent origin. The narrative about Foxhat's son is not constructed according the same pattern: it is a linear development of the story, in which the journey to the other world is only the culmination.

The third clue is a very intriguing aspect of Atpälha's story, the crime and punishment aspect.

It is completely absent from Yuri Vella's story. In Atpälha's it is concentrated in one series of episodes and does not affect the rest of the story. Most of the specialised literature omit this aspect in the conception of the world beyond in the Nenets' traditional world view, while this element is on the other hand essential in the Western understanding in Christianity as well as in the elder mythologies. We may suppose that this is a complementary element of later origin. Some original aspects are also present in Foxhat's Son: some details as the sulphur rivers cannot but remind of the traditional connection in Christian folklore between the devil and sulphur. On the other hand, the world of the dead coincides with the world of the evil spirits. The moral element is present in this superposition.

If we assume that this chronological distribution is justified, we may identify an evolution in the Nenets way of understanding the world beyond, where the basic elements remain unchangedan 16 I must mention that during the recording of the 2. version, a discussion aroused between our two informants, Yuri and Tatva. They stressed, Yuri actively and Tatva passively, that depending on the interpretation of what happened to Atpälha, this story could be also understood as a wanL: this could be the case if we consider that all what Atpälha saw and experienced in the other world is but a dream. Anyhow there is in this story an interesting mixture of both emic genres' elements.

apparent reproduction of the human's life context with reverse elementsand some features may fade in more recent treatment, as the thorough verticality and the time aspect. More developed contacts with the Russian world may have introduced a moral element, which was absent in the traditional view. It is interesting to notice that Tatva, who is blind, has not been to school and is illiterate. But he is a very open-minded Nenets, interested by all phenomena in the world and not at all reticent about progress; he has introduced in his own life many changes comparing to tradition. He may well have given to this tale some of its peculiar features, especially the ternary aspect, which is clearly present in many of his other tales.

I could materialise the chronological evolution I suggest in the followin way: on an axis, I would place Auli's legend as the older text, followed by Foxhat's Son and later by Atpälha. The reason I put Foxhat's Son in-between is because some elements as verticality, and the world understood as formed of different layers seem to be connected with the way ethnographers have presented the classical Nenets understanding of the world beyond. Still, some details, as the sulphur rivers and the merging of two understandings of the other world (both as the dead's and the evil spirits' dwelling)

indicate that new elements are becoming essential. As far as we know, the vertical construction of the world (the earthbeing in the middle, between heavens and underworld), gives the evil spirits and the diseseas' master the underworld as main place; the dead are also supposed to be inserted at the same place, but with a different status and seemingly different local dispositionthese two elements are melted in Tatva's tale.

If these items are the only Forest Nenets ones that we have recorded presenting a view of the world beyond, there are Tundra Nenets folklore texts on this subject (Homič 1995 : 264). They are all much older than our recordings, and present a slightly different image from the one we have just summed up. In these texts, the inhabitants of the world beyond live like the humans, only during the nightat daylight, they are invisible, and their camp is a traditional cemetery (Idem). As we have seen, there is in Atpälha's story an allusion to the cemetery, where she wakes up after coming back to life. But there is no distinction between day and night -Atpälha is sent away by her father during the dayand in Tundra Nenets' texts the deceased do not see the living humans coming among them. As in our texts, the world beyond is either located on the same level as the humans' world, either placed under the ground.

But other Tundra Nenets materials see to confirm the chronological evolution I proposed: Leonid Lar's material comes clearly 17 from his fieldwork at the end of the 20 th century. The conception he presents, based on "myths and tales" about living people's journeys to the other world, differs very deeply from those based on earlier texts: if verticality is absolute and not discussed, it as also accompanied by a very thorough dual principle of good versus evil. According to Lar, the dead's soul has to accomplish a long and difficult journey in the other world, passing through different villages where evil spirits attack him. He is protected by a shaman and by two spirits, who accompany him until he arrives to the place where he will be judged: there, his companions report his good and his bad deeds. If the good prevales, the fate of his soul is decided by Num, who may send it in a kind of paradise, in the village of his ancestors or back on the earth as an animal (Lar 1998 : 23-24). In another chapter, commenting the funeral rituals performed by Nenets shamans, he just observs that "the life of good people in the other world is absolutely similar to their life on the earth. Their souls find their friends, their kins and their acquaintances" (Lar 1998 : 35). If we put in perspective this assertion with Tatva's tale, it is to be noticed that there is no opposition between the world of the people punished and another virtual space for "good" people. Nothing suggests that the people who receive Atpälha were "good" people in life: the moral element is not structural, it apears only in one episode.

Further on, life in the forest (as the one led by the "sinners")except for punishmentis as similar if not more similar Atpälha's way of life than life in a large group of people.

Further on, according to Lar, If evil prevales, the soul is sent to the lowest circle of the underworld and has to serve its god Nga until he decides of its fate and its punishment. So in this understanding, the moral element is structural and reward for deeds in life is fundamental. Lar presents a legendary journey, which reminds very much of some elements in Dante's elaborate construction: the visitor has guides, he sees the people punished and his own acquaintances and he arrives to the lowest circle, and then comes back to the middle-world. As Lar himselt observes, the influence of Christianity is here evident. We may regret that Lar does not put his information into a wider context, presenting sources, or informing about its recurrence in similar material: he just affirms that such is the Nenets' understanding.

If in conclusion we compare the results of this analysis with the materials collected by ethnographers, we may observe that some elements are clearly confirmed by fieldwork data.

All ethnographical data concerning the world beyond show that the representation of a world that is the opposite than the human's world is generalised. According to the narrator of Atpälha's story, everything is the reverse: first of all time, but also the moon and the sun -the moon is the deceased's sun 18

. The habit of burying dead Nenets is a coffin on which a broken sledge is disposed is generally explained by the same fact: broken things, useless here, are useful there. This usage is still fully practised at the North of Western Siberia, in the Yamal Peninsula, by the Yamal Nenets, and we have photos from the late nineties proving the maintenance of this custom. According to a traveller in the Agan region in the 1930-ies, "according to the beliefs of the Ostyaks19 and the Samoyeds20 all the objects offered to the gods broken are given in the "gods'" world their original appearance" (Homič 1972 : 209). The Forest Nenets bury their deceased in the ground, but the belief in reverse features of the world beyond seems indeed to be very solid and lasting. As the two tales show, the reverse elements may be of different kind, but they exist. It is interesting to notice that the world described in the two stories is not totally ruled by opposition. This may even be only marginally mentioned, as in Foxhat's Son. We have the impression that in the narratives, the presence of at least one of these inversed features is a sufficient sign to assert the nature of the whole world. It does not alter in essentials the similarity with the human's world. This last element is also confirmed by recent interviews with Forest Nenets, who assert that the life beyond is no better neither worse that life in this world: the +/-dimension is, according to them, entirely absent 21 . This confirms Homič's statement about the Nenets' view of the world beyond: "In the Nenets' representation, the world beyond was very similar to the world around them and to everyday life. After his death, the Nenets hunted, used reindeer sledges, carried and prepared fur clothes, ate, drank, etc." (Homič 1995 : 226). Foxhat's Son, where the world beyond is very different from everyday life, seems to be an exception. But in his journey, he is but very little interested by the dead: his and the narrator's main focus is on the other aspect of the world beyond, the dwelling of evil spirits, which is totally absent from the other tales.

A very interesting question is the topological disposition of the worlds, for ethnographers have presented different schemes. According to Homič and to Lar the Nenets' understanding of the world is strictly vertical, with the seven heavens and the seven levels below the ground, which were inhabited by hostile spirits, by diseases [START_REF] Homič ; Хомич | Некоторые особенности хозяйства и культуры лесных ненцев" -Охотники, собиратели[END_REF]: 181, Lar 1998 : 24) and although by the most powerful of these spirits, Nga, the divinity opposite to the heaven's god Num, as confirmed by Černecov's fieldwork in the 1920-ies (Istočniki 1987 : 88). Golovnyov, on the bases of his fieldwork, emphasises this vertical organisation, and remarks that both divinities are in contact with the intermediate world and observe it through different holes. These holes are materialised in the structure of the tent, forming an axis in its centre, from the hole on which the stove is disposed up to the upper orifice (Golovnyov 1995 : 398-399). This reminds us of the disposition appearing Auli's story, suggesting that the world beyond is the world below, the humans' world the intermediate one and the gods' world the upper one. The hero glances at the world beyond, the world of the dead, through the upper orifice of the tent, thus reproducing, at human scale, the way Num is supposed to look at the human's world.

The sacred number seven is present also in Foxhat's son, but we have the impression that its seven rivers are disposed on one single layer and not plunging deeper and deeper into the abyss.

Yet there are two different localisations of the deceased's world: in the other version, it is on the same level that this one. Ethnographical data mention also this occurrence. Golovnyov, presenting the Tundra Nenets' funeral rituals, observes that as the dead are buried on the surface, the ground cannot be a symbolical border between the two worlds, but is represented on the horizontal level by distance in space (Golovnyov 1995 : 205). As a matter of fact, in Atpälha's story, the forest is an ambiguous space: on one hand it is a space of the human world, surrounding the camp. On the other hand, it is already part of the other world, where the "sinners" bear their punishment, before the town inhabited by the people who have no punishment to bear. The contact point, as we happen to know at the end of Atpälha's journey, is the cemetery, the necropolis where she founds herself when she returns to life.

According to ethnographic data, this empire of the dead is located northwards. Here we have no indications of direction: we only know that Atpälha's father, when her daughter is seen coming back home, is incredulous and exclaims: "Those who are gone in this direction never come back!" There is also another detail that shows concordance between this story and Golovnyov's data, according to which the journeys of the living humans and the dead in the tundra cross twice a year, at the moment they change the direction of their migrations. This happens in spring and in autumn, which are the transition seasons from winter to summer and vice-versa (Golovnyov 1995 : 206). Our 1 st version indicates clearly that Atpälha's journey starts at the beginning of spring and she comes back when spring is already confirmed. True, Golovnyov does not present any sources for his assertions, but they are clearly based on Tundra Nenets' evidence, hence only for them the change of direction in migrations is of real significance: the Forest Nenets' migrations are much more limited and their direction depends concretely on the disposition of reindeer pastures. Tatva lives in a border zone with Tundra Nenets migrations, and contacts have left their trace on Num-to oral culture [START_REF] Toulouze | [END_REF]. This chronological coincidence in Atpälha's story may be one of them. On the other hand, the 2 nd version does not confirm this point. There are a few indications about seasons: at the beginning of the story, Atpälha's father works outdoors, it is summertime. He comes back from fishing, but no detail is given on the season. But when Atpälha wakes up, the thimble shines on the snow and the narrator adds, when the girl arrives at the camp, that her father was working outdoors and it was late autumn, perhaps winter. So the data are completely different. The only point we may observe in this version is that Atpälha has been absent during the change of seasons. Anyhow, the variation on this point shows that this element is not a basic one for the narrator, and the choice of the season may be, from his point of view, indifferent.

Anyhow, we notice that ethnographic literature as well as our narratives confirm the idea that there is a double location of the deceased's world. It is difficult to assess in which way these two dimensions are linked. We must consider that both understandings are accepted by the Nenets, not only as complementary but even as concomitant, without trying to build on any artificial system. I am not sure that this duality is felt as being contradictory. To conclude on that point, I may add two remarks: first of all, the vertical disposition of the Nenets' universe has been confirmed by many travellers and scholars since the 19 th century, in connection with very precise information about location of different spirits and gods and the respective shamanistic hierarchy. But during the whole of the 20 th century the religious representations and even the knowledge connected to it have deeply suffered from Soviet repression of shamans as well as from the people's atheistic surroundings and has not been regularly transmitted. This is well illustrated by Juri Vella's asserting that he learnt from the ethnographers that Num was not only the heaven and the weather, but also the Nenets' main god (Golovnyov 1995 : 380). We must not be surprised to find, at the beginning of the 21 st century, vague and contradictory reports. On the other hand, Atpälha's story may reflect Tundra Nenets influences.

For the Tundra Nenets, who occupy only permafrost territories, the representation of a vertical world could be imaginary, but much more difficult to find concrete illustration as required by the narratives.

When needing to create visual representation, the deepness is just transposed into horizontal space.

In other words: the beliefs in the location of Num and Nga and their assistants are part of a most sophisticated system, which is certainly quite remote for nowadays Nenets. On the other hands, contact with death is a permanent concern and must be dealt with in everyday life: the border has to be materialised, and it is easier to define it according to their concrete perception of the surrounding world.

Finally, the dimension of crime and punishment is practically absent from the sources, excepted from missionaries' and travellers' accounts in the European zone of Russia, where syncretistic legends were told and reported. In Europe, we may well explain them by early contacts with Christianity, but this explanation is not very convincing when Forest Nenets are concerned, for their contacts with missionaries are very unlikely. The Forest Nenets were scarcely known at the end of the 19 th century, and during the 20 th century Christian religion disappeared from the public life of the peoples of the North. It is more verisimilar to explain the presence of such elements by indirect sources, by later contacts with Russians who, not being themselves missionaries or even active Orthodox, had preserved a traditional worldview of the Christian world beyond, based on punishment for the sins. He sins presented here were certainly the most common among the Forest Nenets: while reindeer theft was one of the main "sins" of the Tundra Nenets, wood and fish for the Forest Nenets represent the basic elements in everyday life. So the elements appearing in Atpälha may show an evolution of the Forest Nenets' understanding of the world beyond during the 20 th century. It may be due to increased contacts with the Russian population, but we canno exclude a cumulated pressure both directly from Russian culture and indirectly through Tundra Nenets contacts, if we presume, following Lar, that the latter understanding of the world beyond in the 20 th century has been very much influenced by Christian moral duality.

As a conclusion, the present analysis allows to present formulate the following hypotheses:

 Auli's storyor the fragment we knowis probably on its present shape the older of the two, and is much more homogeneous that Tatva's narrative. It reflects a view on the world beyond that is mostly coherent with what we know of the Nenets' traditional beliefs.

 Atpälha's story is probably more recent and presents traces of different influences: one of them is the Western worldview, with the idea of punishment in the other world, and a very neat circular structure. On the other hand, some elements may reflect Tundra Nenets influences.

 We observe that some elements in the Nenets' view of the other world are stable, as the principle of ordinary live with reverse elements. The idea that the human's food is not acceptable for the dead and vice-versa seems to be a general feature. The idea of inverted time measuring, generally emphasised in interviews, does not appear in both narratives,

showing that one inverted element is enough to show the audience that the border has been passed.

APPENDIXES Atpälha (Version 1)

I don't know precisely in what circumstances this tale has been recorded. Anyhow, Tatva recorded it himself on cassette, in order to give it to the Research Institute in Khanty-Mansiysk. His wife transliterated the Nenets text and provided it with Russian translation. A copy of the Nenets text has been given to the Estonian National Museum, and I had the opportunity of copying the Nenets original text as well as the translation from the paper book where it has been written. This is a translation from the Russian translation, as we have still not worken on the original text. The translation is not very precise from the linguistic point of viewit is difficult to take it as a basis for a morphological analysis. I consider that for our purpose, it is reliable enough. Probably the more developed text is du to Tatva's wish to give their Institute the best possible material.

Once lived an man. He had a daughter. His daughter was called "similar to a swan". The mother of his daughter had died long before, and now, as he got older, he married again and took two wives, this man. They lived handsomely. During the day, as he made things, he sang a song. It was summer probably, or winter. He sings:

"Like a swan one day you shall be at the head of the caravan. When you will be a bride". His daughter was clearheaded, that's why she was called "similar to a swan". Her stepmothers did not like the words of the song. They protest:

-Why does he sing about her? They did not feed well Atpälha. When her father was not at home, they did not give her any food at all. In daytime, Atpälha always chops wood outdoors.

Once in spring, just before thaw, Atpälha shopped wood outdoors as she always did. Her stepmothers were in the tent and her father was hunting savage reindeers.

The stepmothers were angry -Why does he sing about her? They cried:

-Atpälha, come into the tent, we give you bone marrow! She stepped into the tent. The poor girl wanted to eat. As soon as she camp in, they said:

-Atpälha, open your mouth, we throw you marrow. They had been broking on purpose reindeer bones, for her to hear. The poor girl opened her mouth and they threw her a copper thimble that had been heated on the fire.

The thimble ran into the throat with a hiss. Atpälha fell on her back. And she died. She fell. The stepmothers come around the girl and weep:

-Hi-hi-hi-hi! The man came back. Atpälha hears all, but she had not the strength to speak.

-What happened to you? -Oh, our Atpälha is dead. She had an attack. The man replied:

-But she was in good health when I left. What a quick disease took her away! The man wept soundly, and the stepmothers also weep soundly, or pretended to weep. Atpälha's father said:

-I live with acute distress, take this male reindeer! They had one only male reindeer. The man harnessed it, wrapped up his daughter in clothes and put her on the sledge. He said to the reindeer:

-Bring her to the world beyond. Don't abandon her at half-way. Eh-eh! The reindeer moved, she was fastened to the sledge. The reindeer went. It was alive when his master sent him. On the way, Atpälha opened her eyes. She looked around. Suddenly, she noticed people coming from somewhere with wood. They wanted to throw the wood near their tents, but the wood is glued to their arms. It does not want to get loose. She thinks:

-How strange. Why is it so? That's what they did in lifetime. They stole wood to other people near to their homes. And after their death they are tormented this way. And then again some people go with fish traps on their shoulders. The try to take off the trap from their shoulders, the trap does not detach, as if it were glued. These are the people who stole from fish traps in their lifetime. The same punishment is applied. Then she saw a man and a woman in the forest lying without a tent, the sleep on the bare ground, and they cover themselves with a fur mantle. They pull this mantle, once one, once the other. These are men and wives, who always quarrelled in lifetime. Until she got to the other world, she saw many things on her way. But all finished, and she found herself right in a large camp. There were many tents around, everywhere. Either from the first tent either from a tent in the middle, a woman came out, ran quickly when the girl's reindeer stopped.

-Oh! A bride has come for me! She unfastened Atpälha, the girl jumped down the sledge, the relived. She has no disease at all. The woman took her by the hand and brought her into her tent. She rejoiced:

-A bride has come to me! There are many people in this camp. Atpälha does not know what there is in the other tents. The woman brought the girl in her tent, in the tent there was the woman's son. He lives on the other side of the tent. She made Atpälha sit by her son.

-Look, a wife has come for you. Live together.

When they started to prepare the meal, the girl thought: -I hope I shall have a good meal, I am very hungry. They put meat in a wooden dish. She started to eat, but in the food there are worms. She puts the meat away.

-Mother this meat is full of worms! She does not eat at all. Enormous worms move in the dish, fly grubs. This way they lived some days.

-Soon my daughter-in-law shall starve to death, said once the woman to her son. You must slaughter your transport reindeer.

It was certainly a good transport animal. Probably they had no other male. They slaughtered this animal. Again the girl thought! -Now, I shall be able to eat! The woman came indoors with a leg and gave it to her daughter-in-law:

-Take this leg, you shall eat marrow. You are getting thinner and thinner. The rest of the meat was put in the tent, close to the fire. The girl rejoiced, thinking:

-How good to eat fresh reindeer meat! She cut the skin on the leg and suddenly she saw the same enormous worms moving under the skin. Again she threw away the leg:

-Again this leg is full of worms. How is it possible, the reindeer was alive a moment ago! Her husband then hit her on her bask, between the shoulders, crying with anger:

-What a wife I have! And for her I slaughtered my best reindeer! When he hit her, something came out of her mouth, fell soundly on the ground. She did not remember anything. It is as if she fell asleep.

Suddenly she woke up, there is no tent, she is lying on her bask. She thinks:

-What fell? She started looking around, the day was bright. Near her the copper thimble shined in the sun, it laid on the ground. When her husband hit her on her back, the thimble jumped out of her mouth. The looked very attentively around, there are many many graves. She was in a cemetery. There was no human being alive. All had been buried long ago. Her reindeer had woken up like her. It was still harnessed to the sledge. She said to her reindeer:

-Lead us there, where we came from. Bring us home. The reindeer stepped slowly, quietly. He went all the time, he did not stop. She travels back in daytime, with sunlight. She did not meet any people with wood, either with fish traps or pulling clothes. She travels back as a living being.

Water has already flowed everywhere. The spring has come. She sees her tent. As always, her father is sitting outdoors, he makes something out of wood. Far away, the stepmothers cry:

-Eh-eh! Atpälha has come! On the sledge. On which she had gone long ago. But her father just wept and said:

-Can those who went away over their come back? Why do you mock me? Who shall bring her back, my poor Atpälha. For she has gone to the world of the dead.

He was sitting turning his back to her. But he glanced once and he saw her:

-Kylyo!!! It's true, he is here! His daughter came closer, she fastened the reins and she went to her father. He asked her: -My little girl, how did you come back? -How did I comelook!and she showed him the thimble. My stepmothers killed me, saying they would throw bone marrow into my mouth. But they threw the thimble, heated on fire. And now in the world of the dead, I was given a husband, he hit me on my back and the thimble jumped out. So I relived.

Her father got angry, he stepped into the tent. He caught his wives at once: he caught one by her leg, put his foot on the other leg, and broke her. He killed them, they were probably witches, they were not ordinary persons.

They started to live together. They looked for other people, for neighbours. Then she grew up and became a young woman. She married somebody.

When they move, on the first sledge on the caravan, people recognise Atpälha from far away, she is so white, similar to a swan. As her father sang when she was a little girl. This is the end.

Atpälha (Version 2)

This version was recorded at the Estonian Literature Museum at the end of October 2000 by Tatva in dialogue with Juri Vella. This tale was the first item Tatva recorded in studio. He was certainly not so motivated to furnish the most complete version, although he considered that it was very important to record his stories. Certainly the fact that none of our team, at that moment, was able to understand Nenets may have influed in the sense of neglect. Some days later, this record was listened and translated into Russian with the help of both Juro Vella and Tatva himself. All the sentences were translated with the utmost precision. The present version is based on this Russian translation, and has been verified with the help of the Nenets texte by Kaur Mägi.

My story, the one that puts the story in movement, is called Atpälha. Atpälha means "similar to a swan", with a child's pronunciation. Once lived an old man, he has a daughter, and his daughter was called Atpälha. She was called Atpälha because she had a clear, complexion and clear hair, she was "similar to a swan".

The old man had been married with the girl's mother, bat his wife had died and after her he had taken two wives, who lived with them. So they lived, and in summer, when the weather was warm, he worked outdoors, he made sledges, traps for the fish, and while sitting and working, he sings a song to his daughter. He says:

"Oh, Atpälha, my daughter Atpälha, one day you shall sing your song like this. When you'll be a bride, you'll drive at the head of the caravan. You, Atpälha, you will be at the head of the caravan".

At the same time the little girl played, she listened to the song and played near her father. But at home, the two younger wives of her father do not like the old man's song, they do not like the words.

They say:

-Why does he always sing about her? And thy started to be jealous, and say to one another:

-Why couldn't we lose (=kill) her? And once, when her father had gone fishing, the girl started to chop wood (she was somewhat older, she could already chop wood). When her father was not at home, his wives did not give her any food, they laughed with one another and while the girl was outdoors, they heated a thimble on the fire. The heated it until he became red, and when the girl stepped inside, they said:

-Come on, open your mouth, we have just broken bones, we are going to throw into your mouth bone marrow! This shows that their husband was a hunter of savage reindeers. Atpälha was hungry of course and she opened her mouth, she really thought that they were going to throw her marrow. But instead of marrow, they threw the hot thimble. And the thimble flied into her throat with a hiss. Atpälha fell immediately, she broke down.

And she died, but it was as she had not completely died, her ears still heard some sounds, and when her father came back, the two stepmothers wept. They said:

-Now she is dead, a sudden disease broke her, she could not even wait the water in the pan to boil, she died so quickly! Her father said:

-How did it happen? When I went from home, she was fine, she was healthy, why did it happen so?

And her father wept, and the girl felt how he put her on the sledge, and they had a male reindeer, and he sent his reindeer somewhere. He said to the reindeer:

-Lead her to the right place, don't leave her on her way, lead her over there, he said to his reindeer.

And she was lying on the sledge, and during the journey her eyes were closed, and yet she saw everything like in a dream, she sees those who are close to her.

And once she saw two persons quarrelling, pulling their clothes, one pulls to one side, the other to the other, they are not able to share. They do it eternally. These are the persons who in lifetime quarrelled with one another and could not share their clothes and being dead they eternally pull clothes and cannot share them.

Then she saw people who took wood near other people's tent, the put it on their shoulders and near to their tents they try to throw it on the ground, but the wood remains on their shoulders and they can't put it down. These are the persons who during lifetime stole wood to their neighbours: in the other life, they are tormented in this waythey carry wood, but the wood remains glued to their shoulders.

That's why we tell the young people that they must live righteously, not steal wood. And she saw people, who took out from the river a fish trap full of fish, put it on their shoulders, bring it home, they come near to their place, try to put the trap on the ground, but it remains glued to their shoulders: these are those people, who stole from the traps of other people without permission, took out the fish from other people's traps.

And she saw all kind of people, how they stole things from one another, people who did not live righteously, she saw them all, until it finished. And she arrived in a long long city, where she saw plenty of houses.

And from one of the houses located in the midst of the other houses, a woman came out crying:

-Oh, a bride has come to me, a bride. And she took Atpälha from the sledge, she lifted her, brought her to the house and put her on a reindeer skin. She had a son and she put Atpälha near him on the skin and said:

-This is your place, close to my son. Her husband said that his bride should be given food, she had a long journey. Food was brought, but it was full of worms and fly grubs, different kinds of flies. She spent there a few days, but she was unable to start eating, for her it was unfamiliar food.

As her mother-in-law noticed that she did not eat, she said, after some days:

-Our bride will soon grow very thin indeed. And to the girl's husband, to her son, she said: -You have a good reindeer, a castrate. I suppose this is the animal you have to slaughter. So they slaughtered that reindeer, they used to harness to the sledge, and the girl thought:

-Well, I shall eat fresh meat too! Coming indoors, her mother-in-law brought her a leg from the reindeer and said to her daughter-in-law:

-Here you have a fresh leg! She had certainly a knife.

-Skin and cut it yourself, eat fresh meat, otherwise you will get very thin. She has probably, brought her knife, it was on the sledge. She took her knife and started to skin the leg. She cut the skin, opened it and under the skin she saw enormous shining worms. Then she threw the leg to the place where wood is stored, -Take it away, she said, this is no fresh meat, there are plenty of worms, of fly grubs. Is it really the rear leg of the reindeer?

-Yes, of course, it is really the rear leg of the reindeer, it is nothing else. Then the husband got angry against his young wife. He cried:

-What a capricious wife I have been given! And he took her, and he hit her strongly between the shoulders, he was angry to have such a capricious wife and he wanted to teach her.

And as her husband hit her between her shoulders, something flew out of her mouth, fell on the ground and she fell too inanimate.

She seemed to sleep. She woke up, there was no house, she was simply in the forest or in the tundra, in some natural landscape, she was lyingit was already winter.

She thinks:

-What came out of my mouth? She rose, looked around, there was light everywhere, the sun shined. And there she saw shining on the snow the bronze thimble. It was the same thimble her step-mothers had thrown into her throat, it was that thimble. It was this thimble that was blocked in her throat and prevented her living, now it had jumped out of her, that's why she was conscious again. And she relived, that thimble was the only reason why she was in a state as before death.

She took the thimble and kept it in her mitt, and her reindeer rose up, it was still alive, it shook off the snow on it, he was alive and even harnessed, he was just harnessed to the sledge, and around the reindeer there were many graves.

-So I have come to a cemetery! And she addressed her reindeer and said: -I don't remember wherefrom we came here with you, if you remember, bring us back to the place we come from.

The reindeer turned back on the same route from where they had arrived. And now it was full day, it was a real white day, and around the sledge she did not see anybody, no persons pulling clothes, no persons having stolen wood, no persons unable to get rid of the fish-traps on their shoulders, she did not see anybody at all.

And then she saw her home, and her father outdoors working at the same place where he used to work, he made some tool, it was either autumn or winter and he worked outdoors.

His two wives saw the girl and said:

-Oh, Atpälha has come again! But her father turned his back to the side from where Atpälha was coming, he wept, he did not see her for he was turned the other way round, and with a weeping voice he said:

-Why do you mock me? You know that those who went that way do never come back. Yet he finally turned and watched, and saw his reindeer, his male reindeer coming home and her daughter, her child coming with the reindeer.

And he asked her:

-How did you get to come here back? And she started to tell her story:

-As I chopped wood and brought it home, my step-mothers told me they would give me marrow, I had to open my mouth and they threw something and I fainted. I went through some places and arrived to a place where my dead husband (she had understood that she had been to the deceased) hit me, this thimble came out of my throat, the thimble prevented me living.

And she showed her father the thimble: -This thimble almost killed me! His father got angry, he went to the house. His wives were probably witches, he got angry and caught them by one leg, he put his foot on the other leg, and broke them. And he killed them, he broke them into pieces.

Then they lived some time together, and indeed she got married, and indeed she went at the head of the caravan, and in her song she sings that she is always at the head of the caravan. And her father accompanied her and started to live with their family.

And she lived among the living people, she lived a clear life, and this is perhaps even not a tale, it talks about the life of people, perhaps it is only a story, and this is the end.

Comments after the end of the narration:

Yuri -(In Russian) Now, as Yuri has translated you this story, you may write it from the tape on the paper, or translate it into your language and you will tell it to children. Anyhow, is it a tale or a legend?

Tatva -(In Nenets) A legend? I don't know! A tale! Yuri -(In Nenets) Or a legend? Or a tale? Tatva -(In Nenets) Or a legend… The story was longer, but I have forgotten, there were more episodes, I only told three of them, but there were more of them.

Auli Yusi. Auli's camp was at some 30 km from Yuri's own camp in the taiga.

A more extended analysis of this narrrative feature may be found in Mägi,[START_REF] Toulouze | [END_REF]

The records are available in the CD « Discovering Siberia : the Songs of the Forest Nenets » (with booklet by Kaur Mägi, Triinu Ojamaa, Eva Toulouze), Published by the Estonian Literature Museum.

Version 1: "She wondered: -How strange! Why is it so? That's what they did when they were alive. They stole wood near the tents of other people. And after their death they are tormented in that way."; version 2: "These people are co occupied eternally. These are the people who during their life… and when they are dead in the other life they go on doing the same".

Often the heroes of tales are not Nenets. They are presented by the word kapi, which nowadays means usually "Khanty", but is known to have been used for other indigenous peoples. Russians are not included under the word kapi.

Version 1 : "She saw all kind of things on her way, until she arrived to the other world"; version 2: "And she saw any kind of people, how they stole things from one another, all those who did not live in a normal way, until all this finished".

Version 1: "They arrive at a big camp. There were many many tents, tents everywhere"; version 2: "And she arrived at a long, long town, where she saw many houses".

Unfortunately Lar systematically omits to refer to his sources. But he happens to mention " I have heard a similar story reported by …"(Lar 1998 : 24).

According to Kaur Mägi's (Tartu University) fieldwork inNum-to, in autumn 2001.

Old name for the Khantys.

Old name for the Nenets. In the scientific terminology, the Samoyeds are a group of peoples speaking genetically connected languages; the Nenets, along with the Enets, and the Nganasans, form the Northern branch of this group.

According to Kaur Mägi's (Tartu University) fieldwork inNum-to, in autumn 2001.


Guo-Niu Han

Hankel Determinant Calculus for the Thue-Morse and related sequences

Keywords: Hankel determinant, continued fraction, automatic sequence, Thue-Morse sequence, reduction modulo p, Stieltjes algorithm, integer sequence, grafting technique, chopping method 2010 Mathematics Subject Classification. 05A10, 05A15, 11B50, 11B65, 11B85, 11C20, 11J82, 11Y65, 15A15, 30B70

Hankel determinant calculus for the Thue-Morse and related sequences

Introduction

Let x be a parameter. We identify a sequence a = (a 0 , a 1 , a 2 , . . .) and its generating function f = f (x) = a 0 + a 1 x + a 2 x 2 + • • •. Usually, a 0 = 1. For each n ≥ 1 and k ≥ 0 the Hankel determinant of the series f (or of the sequence a) is defined by .

Let H n (f ) := H (0) n (f ), for short; the sequence of the Hankel determinants of f is defined to be:

In some cases Hankel determinants can be evaluated by using basic determinant manipulation, LU -decomposition, or Jacobi continued fraction (see, e.g., [START_REF] Krattenthaler | Advanced determinant calculus[END_REF][START_REF] Krattenthaler | Advanced determinant calculus: A complement[END_REF][START_REF] Flajolet | Combinatorial aspects of continued fractions[END_REF][START_REF] Wall | Analytic theory of continued fractions[END_REF][START_REF] Muir | The theory of determinants in the historical order of development[END_REF]). However, the Hankel determinants of several power series f related to automatic sequences do not seem to have closed-form expressions, as will be seen in this paper. The following result by Allouche, Peyrière, Wen and Wen [APWW] in 1998, has strongly motivated the present paper.

Theorem 1.1 [APWW]. Let P 2 = P 2 (x) = ∞ k=0 (1 -x 2 k ) be the ±1 Thue-Morse sequence. Then H n (P 2 ) = 0 for every positive integer n.

The first values of the coefficients and Hankel determinants of P 2 (x) are: 1,1,1,1,1,1,1, -1, . . .) H(P 2 ) = (1, 1, -2, 4, 8, -16, -32, -64, 128, -256, -1536, -3072, . . .)

P 2 = (1,
A combinatorial proof of Theorem 1.1 was recently derived by Bugeaud and the author [START_REF] Bugeaud | A combinatorial proof of the non-vanishing of Hankel determinants of the Thue-Morse sequence[END_REF].

Let u = (u 1 , u 2 , . . .) and v = (v 0 , v 1 , v 2 , . . .) be two sequences. Recall that the Jacobi continued fraction attached to (u, v), or J-fraction, for short, is a continued fraction of the form

(1.2) v 0 1 + u 1 x - v 1 x 2 1 + u 2 x - v 2 x 2 1 + u 3 x - v 3 x 2 . . .
, also denoted by

J u v = J[u/v] = J u 1 , u 2 , • • • v 0 , v 1 , v 2 , • • • .
The basic properties on J-fractions, we now recall, can be found in [START_REF] Flajolet | Combinatorial aspects of continued fractions[END_REF][START_REF] Wall | Analytic theory of continued fractions[END_REF][START_REF] Viennot | Une théorie combinatoire des polynômes orthogonaux généraux[END_REF]. The J-fraction of a given power series f exists (i.e., f = J[u/v]) if and only if all the Hankel determinants H n (f ) of f are nonzero. The first values of the coefficients u n and v n in the J-fraction expansion can be calculated by the Stieltjes Algorithm. Also, Hankel determinants can be calculated from the J-fraction by means of the following fundamental relation:

(1.3) H n J u 1 , u 2 , • • • v 0 , v 1 , v 2 , • • • = v n 0 v n-1 1 v n-2 2 • • • v 2 n-2 v n-1 .
Conversely, the coefficients u n and v n in the J-fraction can be calculated using the Hankel determinants by means of the following relations, when all denominators are nonzero.

u n = - 1 H (1) n-1 H n-1 H (1) n H n + H n H (1) n-2 H n-1 , (n ≥ 2) (1.4) v n = H n H n-2 (H n-1 ) 2 . (n ≥ 2) (1.5) Relation (1.
3) is an efficient method for evaluating Hankel determinants.

Let us try to evaluate the Hankel determinants for the Thue-Morse sequence by using the J-fraction. By the Stieltjes algorithm, we get

P 2 (x) = J u v = J 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, • • • 1, -2, 1, -1, -1, -1, 1, -1, 1, -3, 1 3 , -1 3 , -3, • • • .
The top coefficients u n seem to be very simple. However, we are not able to guess any closed-form expression for the bottom coefficients v n , which are even rational numbers. Therefore, we cannot prove anything about the Hankel determinants.

Coons [START_REF] Coons | On the rational approximation of the sum of the reciprocals of the Fermat numbers[END_REF], using the method described in [APWW], proved the following theorem.

Theorem 1.2 [Coons]. Let

S 2 = S 2 (x) = 1 x ∞ n=0 x 2 n 1 -x 2 n .
Then H n (S 2 ) ≡ 1 (mod 2). Again, we are not able to guess any closed-form expression for the Hankel determinants of S 2 , as the first values of the coefficients of the series, the Hankel determinants and the J-fraction of S 2 read:

S 2 = (1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 4, . . .) H(S 2 ) = (1, 1, -3, -1, 21, 1, -3, -9, 945, 9, -3, -1, 21, 9, -243, . . .) S 2 = J -2, 7 3 , 23 3 , -167 21 , -169 21 , 7, 7, -629 105 , -631 105 , 7, 7, -57 7 , -55 7 , • • • 1, -3, -1 9 , -63, -1 441 , -63, -1, -35, -1 11025 , -35, -1, -63, • • •
The main idea to solve the problem is to proceed as follows:

Let p be a prime number and f a sequence. We want to prove that H n (f ) = 0 (mod p); if, apparently, there is no closed-form for the coefficients in the J-fraction of f , we try to find a sequence g ≡ f (mod p), such that the Hankel determinants of g have a closed form. As it is easy to prove that H n (f ) ≡ H n (g) (mod p), it is very likely that some properties on the Hankel determinants of f can be established.

Question. How to find a nice sequence g such that g ≡ f for which each coefficient in the J-fraction of g has a closed-form expression?

By observing the occurrences of the factor 2 in the coefficients of the Jfraction of S 2 given in Theorem 1.2, we guess the following "nice" sequence

g = J 0, 1, 1, 1, 1, 1, . . . 1, 1, 1, 1, 1, 1, 1, . . . ,
whose Hankel determinant is H n (g) = 1 = 0. For proving Theorem 1.2, it remains to prove that S 2 ≡ g (mod 2). For Theorem 1.1, it is more complicated; we need the so-called grafting technique. The proofs of Theorems 1.1-2 are given in Section 2 with further examples. In Section 3 we derive two J-fractions by using the chopping method (Proposition 3.2 ′′ and Theorem 3.3) and prove that the Hankel determinant sequences of several power series are periodic (Propositions 3.6-8).

On the one hand, we provide short proofs of results established in the papers [START_REF] Allouche | Hankel determinants of the Thue-Morse sequence[END_REF][START_REF] Coons | On the rational approximation of the sum of the reciprocals of the Fermat numbers[END_REF], on the other hand, we obtain several new results. In particular, we should like to single out the following theorem.

Theorem 1.3. Let P 3 = P 3 (x) = k≥0 (1 -x 3 k ). Then H n (P 3 ) ≡ (-1) n (mod 3) for every positive integer n.

Notice that the sequence P 3 is obtained from the Thue-Morse sequence P 2 by modifying the exponent of x from 2 to 3. It is worth mentioning that, when m ≥ 4, the Hankel determinants for the sequence k≥0 (1-x m k ) are not all nonzero. A self-contained and short proof of Theorem 1.3 is found in Section 4. The following result, that could be called "one sequence, two modulos", is also proved in Section 4.

Theorem 1.4. We have

1 (1 -x)(1 + 3x) ≡ ∞ k=0 (1 -x 2 k ) (mod 4), 1 (1 -x)(1 + 3x) ≡ ∞ k=0
(1 -x 3 k ) (mod 3).

Hankel determinants modulo p and the grafting technique

Let p be a prime number. For a given power series f we present some methods for guessing and calculating the J-fraction of f , and also proving properties mod p for its Hankel determinants. An ultimately periodic sequence is written in contracted form by using the star sign. For instance, the sequence a = (1, (3, 0) * ) represents (1, 3, 0, 3, 0, 3, 0, . . .), that is, a 0 = 1 and a 2k+1 = 3, a 2k+2 = 0 for each positive integer k. Two sequences a and b are said to be congruent modulo p if a k ≡ b k (mod p) for all k.

For each integer z we have (x + z) p ≡ x p + z p (mod p) and derive the following lemma.

Lemma 2.1. Let f (x) be a power series with integral coefficients. Then

(2.1) f (x) p ≡ f (x p ) (mod p). Let a 1 , b 1 , a 2 , b 2 be four integers such that (p, b 1 ) = 1 and (p, b 2 ) = 1. The two fractions a 1 /b 1 and a 2 /b 2 are said to be congruent modulo p if a 1 b 2 ≡ a 2 b 1 (mod p). We write a 1 /b 1 ≡ a 2 /b 2 (mod p). This fractional congruence is closed under addition and multiplication. Let a 1 /b 1 ≡ c 1 (mod p) and a 2 /b 2 ≡ c 2 (mod p), then a 1 /b 1 + a 2 /b 2 ≡ c 1 + c 2 (mod p) and a 1 /b 1 × a 2 /b 2 ≡ c 1 c 2 (mod p).
The fractional congruence for power series is also closed under addition and multiplication. Also, the ring of formal power series with rational coefficients modulo p is an integral domain.

Lemma 2.2. Let f and f be two power series with rational coefficients and

J[u, v] = f, J[û, v] = f be their J-fraction expansions. Then (1) If f ≡ f (mod p), then H(f ) ≡ H( f ) (mod p).
(2) If u ≡ û (mod p) and v ≡ v (mod p), then f ≡ f (mod p).

(3

) If v ≡ v (mod p), then H(f ) ≡ H( f ) (mod p).
Proof. (1) The Hankel determinants are expressed in terms of the coefficients of the power series by using only addition and multiplication.

(2) The coefficients of the power series are expressed in terms of the coefficients in the J-fraction by using only addition and multiplication. (3) By the fundamental relation (1.3).

Remark. The converse of (1) is not true. A counter-example is the following pair with p = 2:

f = 1 -1 -4x 2 1-x 2x 2 and f = 1 -1 -4x 1-x 2x .
Let f be a power series and g be a J-fraction. If the two sequences u and v in g are ultimately periodic with the same period, we can check that f and g are equal or not. For example, we claim that the J-fraction of the power series

f = (1 -x)(1 + 2x) -(1 -x)(1 -2x)(1 + 3x)(1 + 2x -4x 2 ) 4x 2 (1 -x)
is equal to

g = J (-1 2 , -1 2 , 2) * 1, ( 1 4 , 2, 2) * .
To see this, we check that f verifies the following quadratic functional equation

f = 1 1 -1 2 x - 1 4 x 2 1 -1 2 x - 2x 2 1 + 2x -2x 2 f .
Moreover, the first values of f and g are the same, namely, (1, 1 2 , 1 2 , . . .). Hence, the two power series f and g are equal. Later in the paper this kind of proof will not be reproduced, as it can be done automatically: the sentence "we can prove" replaces the full proof.

Proof of Theorem 1.2 [Coons]. We have

(xS 2 (x)) 2 ≡ x 2 S 2 (x 2 ) = ∞ n=1 x 2 n 1 -x 2 n = xS 2 (x) - x 1 -x (mod 2) so that xS 2 (x) 2 ≡ S 2 (x) - 1 1 -x ≡ S 2 (x) - 1 1 + x (mod 2)
and

S 2 (x) - 1 + 1-3x 1+x 2x S 2 (x) - 1 -1-3x 1+x 2x ≡ 0 (mod 2).
We get

S 2 (x) ≡ 1 -1-3x 1+x 2x (mod 2).
Let g be the right-hand side of the above equation. We can prove

g = J 0, (-1) * (1) * .
Hence, H n (g) ≡ 1 (mod 2), so does H n (S 2 ) by Lemma 2.2(1).

Proof of Theorem 1.3. We successively have

P 3 (x) = (1 -x)P 3 (x 3 ) ≡ (1 -x)P 3 (x) 3 (mod 3), P 3 (x)(1 -(1 -x)P 3 (x) 2 ) ≡ 0 (mod 3), 1 -(1 -x)P 3 (x) 2 ≡ 0 (mod 3), P 3 (x) 2 ≡ 1 1 -x (mod 3), P 3 (x) - 1 1 -x P 3 (x) + 1 1 -x ≡ 0 (mod 3), P 3 (x) ≡ 1 1 -x (mod 3).
Notice that P 3 (x) has integral coefficients, but 1 1-x has rational coefficients. We can prove that

1 1 -x = J (-1/2) * 1, 1/8, (1/16) * .
The above J-fraction itself is congruent to g = J

(1) * 1, -1, (1) * modulo 3, by Lemma 2.2(2), knowing that 1/2 ≡ -1 (mod 3). We have

H n (g) = (1, -1) * . Hence, H n (P 3 ) ≡ H n (g) ≡ (1, -1) * (mod 3).
There is also a proof without using fractional congruence, see Section 4. Notice that H(P 3 (x)) = H(P 3 (-x)) by (1.3). That means H(g) = 0 for g = k≥0 (1 + x 3 k ).

For proving Theorem 1.1, we need a technique, called "grafting". Let F (x) and G(x) be two J-fractions

F (x) = J u 1 , u 2 , u 3 , • • • v 0 , v 1 , v 2 , v 3 , • • • and G(x) = J a 1 , a 2 , a 3 , • • • b 0 , b 1 , b 2 , b 3 , • • • such that b 0 = 1. For each k ∈ N the grafting of G(x) into F (x) of order k, denoted by F (x)| r G(x)
, is defined to be the following J-fraction

F (x)| k G(x) = J u 1 , u 2 , • • • , u k , a 1 , a 2 , a 3 , • • • v 0 , v 1 , v 2 , • • • , v k , b 1 , b 2 , b 3 , • • • . Let F |G := F | 1 G and F ||G := F | 2 G, for short.
If u i , v i (mod p) exists and v i ≡ 0 (mod p) for all i ≥ k +1, we define

Ḡ := J u k+1 (mod p), u k+2 (mod p), u k+3 (mod p), • • • 1, v k+1 (mod p), v k+2 (mod p), v k+3 (mod p), • • • and F = F | r Ḡ.
Then the Hankel determinants of F and F have the following relation

(2.2) H n (F ) H n ( F ) ≡ 1 (mod p)
in view of the fundamental relation (1.3).

For instance, the first values of the J-fraction of the Thue-Morse sequence P 2 are

P 2 = J (1, -1) * 1, -2, 1, -1, -1, -1, 1, -1, 1, -3, 1 3 , -1 3 , -3, 1, -1, 1, 1, -3, • • • .
We see that the previous sequences u, v contain only one even number, -2, and it occurs at position v 1 . Delete (v 1 , u 1 ), which means that we define the following J-fraction g

g = J (-1, 1) * 1, 1, -1, -1, -1, 1, -1, 1, -3, 1 3 , -1 3 , -3, 1, -1, 1, 1, -3, • • •
, so that all the Hankel determinants of g are odd fractional numbers by (1.3).

Proof of Theorem 1.1. Define the sequence g by

P 2 = 1 1 + x + 2x 2 g , or g = 1 2x 2 ( 1 P 2 -1 -x).
By Theorem 1.4 the following identities hold:

1/P 2 ≡ (1 -x)(1 + 3x) (mod 4), g ≡ 1 2x 2 (1 + x -(1 -x)(1 + 3x)) (mod 2).
We can prove that the right-hand side ḡ of the above equation has a simple J-fraction ḡ = J (1) * (1) * .

Let P2 be the grafting of ḡ into P 2 P2 = P 2 |ḡ = 1 1 + x + 2x 2 ḡ , so that H n ( P2 ) = (-2) n-1 from (1.3). Hence, H n (P 2 )/2 n-1 ≡ 1 (mod 2) by (2.2).

Let P 2 = ∞ n=0 η n x n be the Thue-Morse sequence. We now evaluate the Hankel determinants of the following two sequences

δ n = (η n -η n+1 )/2, (2.3) γ n = (η n -η n+2 )/2. (2.4)
The following result was proved in [APWW, Proposition 2.2(2)].

Proposition 2.3. The Hankel determinants of the sequence (δ n ) n=0,1,2,... are odd integral numbers.

Proof. The generating function for the sequence (γ n ) is equal to

f = 1 -(1 -x)P 2 2x ,
which is congruent to

g := 1 -(1 -x) 1 (1-x)(1+3x) 2x (mod 2)
by Theorem 1.4. We can prove that g has the following J-fraction expansion

g = 1 - 1-x 1+3x 2x = J 2, (1) *
(1) * . Hence, H n (g) = 1 and H n (f ) ≡ 1 (mod 2).

Proposition 2.4. The Hankel determinants of the sequence (γ n ) n=0,1,2,... are odd integral numbers.

Proof. The generating function for the sequence (γ n ) is equal to

f = 1 -x -(1 -x 2 )P 2 2x 2 ,
which is congruent to

g := - 1 -x -(1 -x 2 ) 1 (1-x)(1+3x) 2x 2 (mod 2)
by Theorem 1.4. We can prove that g has the following J-fraction expansion

g = - 1 -x -(1 + x) 1-x 1+3x 2x 2 = J (3, -1) * 1, (-1) * . Hence, H n (g) ≡ 1 (mod 2) and H n (f ) ≡ 1 (mod 2). Proposition 2.5. Let f = 3 ∞ n=1 (1 -x 3 n ) - 2 1 -x .
Then, H k (f ) = 0 for all k.

Remark. When replacing the factor 1 -x 3 n by 1 + x 3 n in the above formula, experimental calculation of the first values suggests that all the Hankel determinants are still nonzero. However, we are not able to prove that the latter Hankel determinants do not vanish.

Proof. We have

f = J 2, -7/2,
f = 1 1 + 2x + 6x 2 g , or (2.6) (1 + 2x + 6x 2 g) 3 ∞ n=0 (1 -x 3 n ) -2 = 1 -x.
By (1.3) we have

(2.7) H n (f ) = (-6) n-1 H n-1 (g).
for all n. By Theorem 1.4 identity (2.6) becomes

(1 + 2x + 6x 2 g) 3 1 (1 -x)(1 + 3x) -2 ≡ 1 -x (mod 9) or g ≡ (1 + 2x) 1 (1-x)(1+3x) -1 -x x 2 (mod 3).
Let h be the right-hand side of the above equation. Then, h = J (-1) * 1, (4, -1/2, -1/2) * , so that H n (h) ≡ 1 (mod 3). Hence, H n (g) ≡ 1 (mod 3). By (2.7) we have H n (f ) = 0. Proposition 2.6. Let f be the sequence obtained from P 3 by deleting the first term, i.e., f = (1 -P 3 )/x. Then, H(f ) ≡ (1) * (mod 3).

Proof. By Theorem 1.4,

f ≡ 1 - 1 (1-x)(1+3x)
x (mod 3).

Let g be right-hand side of the above equation. Then, g has the J-fraction

g = J 3, (-2, 5/2, 5/2) * 1, (-2, -2, 1/4) * ≡ J 0, (-1, 1, 1) * (1) * (mod 3).
Hence, H(g) ≡ (1) * (mod 3), so does H(f ).

Theorem 2.7. Let

(2.8) f = f (x) = k≥0 (1 -x 3 k -x 2•3 k ).
Then H n (f ) = 0.

Proof. We successively have

f (x 3 ) = k≥1 (1 -x 3 k -x 2•3 k ); f = f (x) = k≥0 (1 -x 3 k -x 2•3 k ) = (1 -x -x 2 )f (x 3 ); f ≡ (1 -x -x 2 )f 3 (mod 3); 1 ≡ (1 -x -x 2 )f 2 (mod 3); f ≡ 1/(1 -x -x 2 ) (mod 3).
The right-hand side of the above equality has the following J-fraction expansion:

1 1 -x -x 2 = J (-1/2) * 1, 5/8, (5/16) * ≡ J (1) * 1, 1, (-1) * (mod 3), so that (2.9) H(f ) ≡ (1, (1, 1, 2, 2) * ) (mod 3).
Remark. The sequence f defined in (2.8) is a {1, -1}-sequence.

Continued fraction and the chopping method

When the two coefficients in the J-fraction are ultimately periodic with the same period, the corresponding power series is easy to obtained. However this is not always the case, as shown in the following Proposition.

Proposition 3.1. Let f (x) = 1 -1 -4x 4 1-x 2 2x 4 . Then f = J (0) * 1, 1, 1, -1, -1, 2, 1/2, -1/2, -2, 3, 1/3, -1/3, -3, . . . . In other words, if f = J[u/v], then u k = 0 and v 4k+1 = k, v 4k+2 = 1/k, v 4k+3 = -1/k, v 4k+4 = -k for every positive integer k.
The proof of Proposition 3.1 is based on the following generalization with one more parameter z. Proposition 3.1 ′ becomes Proposition 3.1 when z = 1.

Proposition 3.1 ′ . Let f = f (x; z) = 1 -(2z -1)x 2 -(1 -x 2 )(1 -x 2 -4x 4 ) 2x 2 ((1 -z) + (1 -z + z 2 )x 2 -x 4 ) . Then f = J (0) * 1, z, 1/z, -1/z, -z, z + 1, 1/(z + 1), -1/(z + 1), -(z + 1), . . . . In other words, if f = J[u/v], then u k = 0 and v 4k+1 = z + k, v 4k+2 = 1/(z+k), v 4k+3 = -1/(z+k), v 4k+4 = -(z+k)
for every positive integer k.

Proof. We need to check that f (x; z) verifies the following functional equation:

f (x; z) = 1 1 - zx 2 1 - 1 z x 2 1 + 1 z x 2 1 + zx 2 f (x; z + 1)
.

Let us explain how to get Proposition 3.1 ′ from Proposition 3.1. Let

f 1 = 1 -1 -4x 4 1-x 2 2x 4
and

f 1 = J (0) * 1, 1, 1, -1, -1, 2, 1/2, -1/2, -2, 3, 1/3, -1/3, -3, . . . .
Define f 2 by deleting the first four pairs u i , v i (i = 1, 2, 3, 4) from the J-fraction of f 1 . In other words,

f 2 = J (0) * 1, 2, 1/2, -1/2, -2, 3, 1/3, -1/3, -3, . . . .
By the very definition of the continued fraction we get the first values of f 2 f 2 = (1, 0, 2, 0, 5, 0, 12, 0, 30, 0, 75, 0, 190, 0, 483, 0, 1235, 0, 3167, . . .).

With the help of a computer algebra system (see [START_REF] Rubey | Extended Rate, more GFUN[END_REF] for example), we observe that f 2 satisfies the equation

(x 6 -3x 4 + x 2 )f 2 2 + (-3x 2 + 1)f 2 -1 = 0.
Define f 3 by deleting the first four pairs u i , v i (i = 1, 2, 3, 4) from the J-fraction of f 2 and repeat these steps, we successively get

(x 6 -7x 4 + 2x 2 )f 2 3 + (-5x 2 + 1)f 3 -1 = 0, (x 6 -13x 4 + 3x 2 )f 2 4 + (-7x 2 + 1)f 4 -1 = 0, • • •
and guess the general equation valid for every z

(x 6 -(z 2 -z + 1)x 4 + (z -1)x 2 )f 2 z + (-(2z -1)x 2 + 1)f z -1 = 0.
Solving the above equation yields the series f (x; z), defined in Proposition 3.1 ′ . The above procedure of finding generalization of J-fraction will be called the chopping method.

Proposition 3.2. Let

f (x) = 1 -1 -4x 4 1+x 2x 4 . Then H(f (x)) = (1, 1, 0, 0, -1, -1, 0, 0) *
As H 3 (f (x)) = 0, the traditional method fails. We then have to find a polarization, as stated in the following example, which becomes Proposition 3.2 when y = -1 and z = 0. Notice that Proposition 3.1 is also a special case of Proposition 3.2 ′ by taking y = 0 and z = 1.

Proposition 3.2 ′ . Let f (x; y, z) = 1 -1 - 4x 4 1-yx-zx 2 2x 4 . Then f (x; y, z) = J (-y, 0, 0, 0) * 1, z, 1/z, -1/z, -z, 2z, 1/(2z), -1/(2z), -(2z), . . . .
By Proposition 3.2 ′ and the fundamental relation (1.3), the Hankel determinants of f (x; y, z) are

H(f (x; y, z)) = (1, 1, z, z, -1, -1, -2z, -2z, 1, 1, 3z, 3z, . . .).
When z = 0 and y = -1 we get H(f (x; -1, 0)) = (1, 1, 0, 0, -1 -1, 0, 0) * . Proposition 3.2 is proved.

However we are not able to prove Proposition 3.2 ′ directly. By using the chopping method we find and prove the following generalization of Proposition 3.2 ′ . Letting t = 0 in Proposition 3.2 ′′ we get Proposition 3.2 ′ . Proposition 3.2 ′′ . Let

f (x) = - 2 ztx 2 + zx 2 + yx -1 + (4 x 4 + yx -1 + zx 2 ) (yx -1 + zx 2 ) 2x 2 (-zx 4 -x 3 y + x 2 + x 2 z 2 t + x 2 z 2 t 2 + yztx -zt)
Then f = J (-y, 0, 0, 0) * 1, (t + 1)z, 1 (t+1)z , -1 (t+1)z , -(t + 1)z, (t + 2)z, 1 (t+2)z , . . .

. By using the chopping methods, we derive the following continued fraction.

Theorem 3.3. Let g = -2zx 2 -(sx -x 2 y -1) -(sx -x 2 y -1) 2 -(2x 2 ) 2 2x 2 (x 2 + x 2 z 2 + z(sx -x 2 y -1)
) .

Then

g = J (-s, 0) * v 0 , v 1 , v 2 , . . . = 1 1 -sx - α 1 α 0 x 2 1 - α 0 α 1 x 2 1 -sx - α 2 α 1 x 2 1 - α 1 α 2 x 2 . . . where v 2k+1 = α k+1 /α k , v 2k+2 = α k /α k+1 and α n is defined by n α n x n = 1 + zx 1 + yx + x 2 .
Thus the Hankel determinants are H(g) = (α 0 , α 0 , α 1 , α 1 , α 2 , α 2 , α 3 , α 3 , . . .).

The proof of Theorem 3.3 is based on the following generalization.

Theorem 3.4. Let a n , b n , d n , α n be numbers defined by the following generating functions

n≥0 a n x n = (1 -yz + z 2 )(1 -(y 2 -2)x + x 2 ) (1 -x)(1 -(y 2 -2)x + x 2 ) , n≥0 b n x n = -z + y(yz -1)x -z(yz -1)x 2 (1 -x)(1 -(y 2 -2)x + x 2 ) , n≥0 d n x n = -1 -(1 + z 2 -2yz)x -z 2 x 2 (1 -x)(1 -(y 2 -2)x + x 2 ) , n α n x n = 1 + zx 1 + yx + x 2 and f n (x) by (3.1) (a n x 4 -sb n x 3 +b n x 2 )f n (x) 2 +((yd n -2b n )x 2 -sd n x+d n )f n (x) = d n . Then f n (x) = 1 1 -sx - α n+1 α n x 2 1 - α n α n+1 x 2 1 -sx - α n+2 α n+1 x 2 1 - α n+1 α n+2
x 2

. . .

When n = 0, we have a(0) = 1-yz+z 2 ; b(0) = -z; d(0) = -1. Solving (3.1) yields Theorem 3.3. For proving Theorem 3.4, we first convert it to Theorem 3.5, in which the coefficients are given by explicit formulas. This conversion is done by the following change of variables:

(3.2) t + t -1 = y; t = y + y 2 -4 2 ; K = (-t) n . Theorem 3.5. Let a(K), b(K), d(K), α(K) be numbers defined by a(K) = z 2 -(t + t -1 )z + 1 b(K) = - t 2 + 1 γ 2 γ 3 γ 2 1 - tγ 2 2 K 2 γ 2 1 - K 2 γ 2 3 t γ 2 1 d(K) = - 2tγ 2 γ 3 γ 2 1 - γ 2 2 K 2 γ 2 1 - K 2 t 2 γ 2 3 γ 2 1 α(K) = - γ 2 (1 -t 2 )K - Kγ 3 t 1 -t 2
where γ 1 = (t -1)(t + 1); γ 2 = zt -1; γ 3 = t -z, and f (x; K) by

a(K)x 4 -sb(K)x 3 + b(K)x 2 f (x; K) 2 (3.3) + ((t + t -1 )d(K) -2b(K))x 2 -sd(K)x + d(K) f (x; K) = d(K). Then f (x; K) = 1 1 -sx - α(-tK) α(K) x 2 1 - α(K) α(-tK) x 2 1 -sx - α(t 2 K) α(-tK) x 2 1 - α(-tK) α(t 2 K) x 2 . . . . Proof. Solving (3.3) yields f (x; K) = (sx -1)tQ 1 -(t -1) (t + 1) Q 2 x 2 + Q 1 √ Q 0 Q 1 2x 2 t(1 -sx)Q 3 Q 1 + K 2 (t -1) 2 (t + 1) 2 (zt -1) (t -z) x 2
where

Q 0 = sxt -t -x 2 -2 x 2 t -x 2 t 2 sxt -t -x 2 + 2 x 2 t -x 2 t 2 ; Q 1 = -1 + zt + K 2 t 2 -K 2 zt; Q 2 = K 2 t 2 -zt -K 2 zt + 1; Q 3 = -K 2 z + zt 2 -t + K 2 t.
Then, we can verify

f (x; K) = 1 1 -sx - α(-tK) α(K) x 2 1 -α(-tK) α(K) x 2 f (x; -tK) . Proposition 3.6. Let f = f (x) = 1 x 4 ∞ k=1 x 2 k+1 1 -x 2 k .
Then, H(f ) ≡ (1, 1, 1, 1, 1, 1, 0, 0) * (mod 2).

Proof. We successively have

x 8 f (x 2 ) = ∞ k=2 x 2 k+1 1 -x 2 k = ∞ k=1 x 2 k+1 1 -x 2 k - x 4 1 -x 2 = x 4 f (x) - x 4 1 -x 2 , x 4 f (x 2 ) = f (x) - 1 1 -x 2 , x 4 f (x) 2 ≡ f (x) - 1 1 -x 2 (mod 2), f (x) ≡ 1 -1 -4x 4 1-x 2 2x 4 (mod 2).
Let g be the right-hand side of the above equation. By Proposition 3.1 we have

H(g) = (1, 1, 1, 1, -1 -1, -2, -2, 1, 1, 3, 3, -1, -1, -4, -4, 1, 1, 5, 5, . . .).
In other words,

H 4k (g) = H 4k+1 (g) = (-1) k , H 4k+2 (g) = H 4k+3 (g) = (-1) k (k + 1), so that H(f ) ≡ H n (g) ≡ (1, 1, 1, 1, 1, 1, 0, 0) * (mod 2). Proposition 3.7. Let f = f (x) = 1 x 4 ∞ k=0 x 2 k+2 1 + x 2 k .
Then, H(f ) ≡ (1, 1, 0, 0) * (mod 2).

Proof. Using the method described in the proof of Proposition 3.6, we derive

f (x) ≡ 1 -1 -4x 4 1+x 2x 4 (mod 2).
Let g be the right-hand side of the above equation. By Proposition 3.2 H(g) = (1, 1, 0, 0, -1, -1, 0, 0) * . In other words, H 4k (g) = H 4k+1 (g) = (-1) k , H 4k+2 (g) = H 4k+3 (g) = 0.

Hence, H(f ) ≡ H n (g) ≡ (1, 1, 0, 0) * (mod 2).

Proposition 3.8. Let

f = f (x) = 1 x 2 ∞ k=0 x 2 k+1 1 + x 2 k+1 .
Then H(f ) ≡ (1, 1, 0, 0, 1, 1) * (mod 2).

Proof. Using the method described in the proof of Proposition 3.6, we derive

f (x) ≡ 1 -1 -4x 2 1+x 2
2x 2 (mod 2).

Lemma 4.1. We have

√ 1 -4x ≡ 1 + 2 ∞ k=0
x 2 k (mod 4).

Proof. The following expansion is well known (See [St99, WiCa] for example)

1

- √ 1 -4x 2x = ∞ k=0 C n x n ,
where C n = 1 n+1 2n n is the Catalan number. It is easy to see that C n ≡ 1 (mod 2) if and only if n = 2 k -1 for some integer k [AK73], knowing, for instance, that C n is the number of binary trees with n vertices.

Proof of Theorem 1.4. We need to prove

1 (1 -x)(1 + 3x) ≡ ∞ k=0
(1 -x 2 k ) (mod 4), (4.3)

1 (1 -x)(1 + 3x) ≡ ∞ k=0
(1 -x 3 k ) (mod 3). (4.4)

The second equality is just relation (4.1). For proving the first equality let f (x) be the left-hand side of (4.3). By Lemma 4.1, we get

(1 -x)f (x) = 1 -x 1 + 3x = 1 - 4x 1 + 3x ≡ 1 + 2 ∞ k=0 x 1 + 3x 2 k (mod 4), (1 -x)f (x) ≡ 1 + 2 ∞ k=0 x 1 + x 2 k
(mod 4), and

(1 -x 2 )f (x 2 ) ≡ 1 + 2 ∞ k=0 x 2 1 + x 2 2 k (4.5) ≡ 1 + 2 ∞ k=0 x 1 + x 2 k+1
[By Lemma 2.1] ≡ (1 -x)f (x) -2x 1 + x (mod 4).

Let P 2 (x) be the right-hand side of (4.3). Then,

(1 -x)P 2 (x 2 ) = P 2 (x), which implies on one hand (1 -x)(P 2 (x)) 2 ≡ P 2 (x) (mod 2), hence

P 2 (x) ≡ 1 1 + x (mod 2),
and on the other hand (1 -x 2 )P 2 (x 2 ) = (1 + x)(1 -x)P 2 (x 2 ) = (1 + x)P 2 (x).

Hence, (4.6) (1 -x 2 )P 2 (x 2 ) -(1 -x)P 2 (x) ≡ 2x 1 + x (mod 4).

Taking the difference of (4.6) and (4.5) yields

(1 -x 2 )f (x 2 ) -(1 -x 2 )P 2 (x 2 ) ≡ (1 -x)(f (x) -P 2 (x)) (mod 4), and (4.7) f (x) -P 2 (x) ≡ (1 + x)(f (x 2 ) -P 2 (x 2 )) (mod 4).

By applying (4.7) recursively we get f (x) -P 2 (x) ≡ 0 (mod 4), since f (0) = P 2 (0) = 1.

a

  k+n-1 a k+n . . . a k+2n-2

Acknowledgements. The author should like to thank Zhi-Ying Wen who suggested that I study the Hankel determinants of the Thue-Morse sequence back to 1991, and who invited me to Tsinghua University where the paper was finalized. The author also thanks Yann Bugeaud for reanimating the topic and valuable discussion.

Let g be the right-hand side of the above equality and let α n be defined by

By Theorem 3.3 we have

Hence

H(f ) ≡ H(g) ≡ (1, 1, 0, 0, 1, 1) * (mod 2).

4. One sequence, two modulos Theorem 1.3 is proved in Section 2 by using the fractional congruence. In fact, the fractional congruence can be avoided.

Proof of Theorem 1.3. We successively have

≡ 0 (mod 3).

We then have (4.1)

by using the value of P (0). The right-hand side of the above equation has integral coefficients and its J-fraction is equal to:

so that H n (P 3 ) ≡ 2 n-1 ≡ (-1) n-1 (mod 3) by (1.3).

Next we will prove the "one sequence, two modulos" theorem 1.4. We need the following lemma.