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Selberg's integral and linear forms in zeta values

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Niveau: Supérieur, Doctorat, Bac+8
Selberg's integral and linear forms in zeta values Tanguy Rivoal Abstract Using Selberg's integral, we present some new Euler-type integral rep- resentations of certain nearly-poised hypergeometric series. These integrals are also shown to produce linear forms in odd and/or even zeta values that generalize previous work of the author. 1 Selberg's integral and nearly-poised series Much work has been devoted to evaluating multiple hypergeometric integrals after Beukers' proof of Apery's theorem “?(3) is irrational”, in which he used the following integrals equations [Be]: ∫ 1 0 ∫ 1 0 xn(1? x)nyn(1? y)n (1? (1? x)y)n+1 dxdy = an?(2) + bn and ∫ 1 0 ∫ 1 0 ∫ 1 0 xn(1? x)nyn(1? y)nzn(1? z)n (1? (1? (1? x)y)z)n+1 dxdydz = An?(3) + Bn for some (explicitly computable) rational numbers an bn, An and Bn. See in particular the work of Hata [Ha], Rhin and Viola [RV1, RV2], Vasilyev [Va], Sorokin [So], Zudilin [Zu].

  • rational numbers

  • hypergeometric series

  • integrals equations

  • trivial transformation

  • selberg's integral

  • hypergeometric integrals after

  • now define

  • integrals

  • numbers such

  • partial fractions


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Tanguy Rivoal
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5Rb7RfT'gidtRTf6b6dPdR6fbyp-eigRPgRfiRg Much work has been devoted to evaluating multiple hypergeometric integrals after Beukers’ proof of Apery’s theorem “ehtdesuehhcihwnitional",3)isirra( following integrals equations [Be]: Z Z ) )n nn n I(1I)J(1J) dIdJ=/n(2) +>n n+) (1(1I)J) ( ( and Z Z Z ) ) )n nn nn n I(1I)J(1J)z(1z) dIdJdz=.n(3) +Bn n+) (1(1(1I)J)z) ( ( ( for some (explicitly computable) rational numbers/n>n,.nandBnin. See particular the work of Hata [Ha], Rhin and Viola [RV1, RV2], Vasilyev [Va], Sorokin[So],Zudilin[Zu].Similarly,inordertoprovethatin nitelymanyodd zeta values are irrational, the following multiple integral was used in [Ri] and [BR] : Za nrn I j=(j(1Ij) r,n J:= dI(· · ·dIaδ(1) a (+r+))n++ (1I· · ·I) a+1 [(,)] (a wheren0,/δ r1 are integers such that (/+ 1)n >(2r+ 1)n+ 2.This is interesting principally because there exist explicitly computable rational numbers r,n B(forA= 0δ γ γ γ δ /) such that j,a a r,n r,nr,n J=B+B (A) a(,a j,a j=+ 1