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VERY WELL POISED HYPERGEOMETRIC SERIES AND THE DENOMINATORS CONJECTURE

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13 Pages
English

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Niveau: Supérieur, Doctorat, Bac+8
VERY-WELL-POISED HYPERGEOMETRIC SERIES AND THE DENOMINATORS CONJECTURE TANGUY RIVOAL Laboratoire de Mathematiques Nicolas Oresme, CNRS UMR 6139, Universite de Caen, BP 5186, 14032 Caen cedex, France email: This survey deals with the recent appearance of very-well-poised hypergeometric series as a tool for studying the diophantine nature of the values of the Riemann zeta function at positive integers. In this context, we give examples of an important and general ex- perimental phenomenon known as the Denominators Conjecture, and we explain the ideas behind its proof in the case presented here, recently obtained by C. Krattenthaler and the author. It is an expanded version of the talk given at the symposium “Analytic Number Theory and Surrounding Areas”, held at the RIMS of the University of Kyoto in october 2004. I would like to thank the organizer M. Amou and also N. Hirata-Kohno for their invitation to come to Japan, under the auspicies of the JSPS program. 1. Notations Let us first remind the reader of the definition of hypergeometric series (or functions). These are power series defined by q+1Fq [?0, ?1, . . . , ?q ?1, . . . , ?q ; z ] = ∞∑ k=0 (?0)k (?1)k · · · (?q)k k! (?1)k · · · (?q)k z k, where ?j ? C, ?j ? C\Z≤0 and

  • very-well-poised hypergeometric

  • apery's great

  • rational function

  • cn sn

  • irrationality proof

  • sn ?

  • ex- perimental phenomenon


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VERY-WELL-POISED HYPERGEOMETRIC SERIES AND THE DENOMINATORS CONJECTURE
TANGUY RIVOAL
LaboratoiredeMathe´matiquesNicolasOresme,CNRSUMR6139, Universit´edeCaen,BP5186,14032Caencedex,France email: rivoal@math.unicaen.fr
This survey deals with the recent appearance of very-well-poised hypergeometric series as a tool for studying the diophantine nature of the values of the Riemann zeta function at positive integers. In this context, we give examples of an important and general ex-perimental phenomenon known as the Denominators Conjecture , and we explain the ideas behind its proof in the case presented here, recently obtained by C. Krattenthaler and the author. It is an expanded version of the talk given at the symposium “Analytic Number Theory and Surrounding Areas”, held at the RIMS of the University of Kyoto in october 2004. I would like to thank the organizer M. Amou and also N. Hirata-Kohno for their invitation to come to Japan, under the auspicies of the JSPS program. 1. Notations Let us first remind the reader of the definition of hypergeometric series (or functions). These are power series defined by q +1 F q · α 0 β, 1 α, 1 .,....,.β, q α q ; z ¸ = k = X 0 ( αk 0 !)( k β ( 1 α ) 1 ) k ∙ ∙ ( β ( q α ) kq ) k z k , k ∙ ∙ ∙ where α j C , β j C \ Z 0 and ( x ) m = x ( x + 1) ∙ ∙ ∙ ( x + m 1) is the Pochhammer symbol. It can be proved that such series converge for all z C such that | z | < 1, and for z = ± 1, provided that Re( β 1 + ∙ ∙ ∙ + β q ) > Re( α 0 + α 1 + ∙ ∙ ∙ + α q ). The literature (see [3, 8, 14]) contains various special kind of hypergeometric series whose parameters satisfy particular relations. For example, a hypergeometric series is said to be: balanced if α 0 + ∙ ∙ ∙ + α q + 1 = β 1 + ∙ ∙ ∙ + β q ; nearly-poised (of the first kind) if α 1 + β 1 = ∙ ∙ ∙ = α q + β q ; well-poised if α 0 + 1 = α 1 + β 1 = ∙ ∙ ∙ = α q + β q ; very-well-poised if it is well-poised and α 1 = 21 α 0 + 1. We will show that the very-well-poised case is of special importance. 1