In this paper we present some new binomial identities for multiple harmonic sums whose indices are the sequences ({1} a , c, {1} b ), ({2} a , c, {2} b ) and prove a number of congruences for these sums modulo a prime p. The congruences obtained allow us to find nice p-analogues of Leshchiner's series for zeta values and to refine a result due to M. Hoffman and J. Zhao about the set of generators of the multiple harmonic sums of weight 7 and 9 modulo p. As a further application we provide a new proof of Zagier's formula for ζ * ({2} a , 3, {2} b ) based on a finite identity for partial sums of the zeta-star series.License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 3132 KH. HESSAMI PILEHROOD, T. HESSAMI PILEHROOD, AND R. TAURASOGlaisher [8] in 1900, and Lehmer [21] in 1938, proved that even the multiple harmonic sums H p−1 (m) modulo a higher power of a prime p ≥ m + 3 are related to the Bernoulli numbers:The systematic study of MZVs began in the early 1990s with the works of Hoffman [16] and Zagier [33]. The set of the MZVs has a rich algebraic structure given by the shuffle and the stuffle (harmonic shuffle or quasi-shuffle) relations. These follow from the representation of multiple zeta values in terms of iterated integrals and harmonic sums, respectively. There are many conjectures concerning multiple zeta values, and despite some recent progress, lots of open questions still remain to be answered. Licensed to New York Univ, Courant Inst. Prepared on Tue Feb 3 02:49:34 EST 2015 for download from IP 128.122.253.228. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use NEW PROPERTIES OF MULTIPLE HARMONIC SUMS 3133Let Z w denote the Q-vector space spanned by the set of multiple zeta values ζ(s 1 , . . . , s r ) with s r ≥ 2 and the total weight w = s 1 + · · · + s r , and let Z denote the Q-vector space spanned by all multiple zeta values over Q. A conjecture of Zagier [33] states that the dimension of the Q-vector space Z w is given by the Perrin numbers d w defined for w ≥ 3 by the recurrencewith the initial conditions d 0 = 1, d 1 = 0, d 2 = 1. The upper bound dim Z w ≤ d w was proved independently by Goncharov [9] and Terasoma [28].It is easy to see that the Perrin number d w is equal to the number of multiple zeta values ζ(s 1 , . . . , s r ) with s 1 + · · · + s r = w and each s j ∈ {2, 3}. While investigating the deep algebraic structure of Z, Hoffman [17] conjectured that the MZVs ζ(s 1 , . . . , s r ) of weight w with s j ∈ {2, 3} span the Q-space Z w . Very recently, this conjecture was proved using motivic ideas by Brown [4]. So the main problem which remains open is proving that the numbers ζ(s 1 , . . . , s r ) with s j ∈ {2, 3} are linearly independent over Q.According to Zagier's conjecture, a basis for Z w for 2 ≤
We prove approximation formulas for the logarithms of some infinite products, in particular, for Euler's constant γ , log 4 π and log σ , where σ is Somos's quadratic recurrence constant, in terms of classical Legendre polynomials and partial sums of their series expansions. We also give conditional irrationality and linear independence criteria for these numbers. The main tools are Euler-type integrals, hypergeometric series, and Laplace method.
International audience We prove generating function identities producing fast convergent series for the sequences beta(2n + 1); beta(2n + 2) and beta(2n + 3), where beta is Dirichlet's beta function. In particular, we obtain a new accelerated series for Catalan's constant convergent at a geometric rate with ratio 2(-10); which can be considered as an analog of Amdeberhan-Zeilberger's series for zeta(3)
In this paper, we prove a new identity for values of the Hurwitz zeta function which contains as particular cases Koecher's identity for odd zeta values, the Bailey-Borwein-Bradley identity for even zeta values and many other interesting formulas related to values of the Hurwitz zeta function. We also get an extension of the bivariate identity of Cohen to values of the Hurwitz zeta function. The main tool we use here is a construction of new Markov-WZ pairs. As application of our results, we prove several conjectures on supercongruences proposed by J. Guillera, W. Zudilin, and Z. W. Sun.
Combinatorics International audience By application of the Markov-WZ method, we prove a more general form of a bivariate generating function identity containing, as particular cases, Koecher's and Almkvist-Granville's Apéry-like formulae for odd zeta values. As a consequence, we get a new identity producing Apéry-like series for all ζ(2n+4m+3),n,m ≥ 0, convergent at the geometric rate with ratio 2−10.
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