arXiv subjects
Publications and source records attributed to T. Hessami Pilehrood.
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 $ζ(2n+4m+3),$ $n,m\ge 0,$ convergent at the geometric rate with ratio $2^{-10}.$
Using the WZ method we present simpler proofs of Koecher's, Leshchiner's and Bailey-Borwein-Bradley's identities for generating functions of the sequences $\{ζ(2n+2)\}_{n\ge 0}, \{ζ(2n+3)\}_{n\ge 0}.$ By the same method we give several new representations for these generating functions yielding faster convergent series for values of the Riemann zeta function.
We study a problem of finding good approximations to Euler's constant $γ=\lim_{n\to\infty}S_n,$ where $S_n=\sum_{k=1}^n\frac{1}{n}-\log(n+1),$ by linear forms in logarithms and harmonic numbers. In 1995, C. Elsner showed that slow convergence of the sequence $S_n$ can be significantly improved if $S_n$ is replaced by linear combinations of $S_n$ with integer coefficients. In this paper, considering more general linear transformations of the sequence $S_n$ we establish new accelerating convergence formulae for $γ.$ Our estimates sharpen and generalize recent Elsner's, Rivoal's and author's results.