Hypergeometry inspired by irrationality questions. (English) Zbl 1450.11072

Krattenthaler, Christian; Zudilin, Wadim

Kyushu J. Math. 73, No. 1, 189-203 (2019).

Let \(\zeta (k)=\sum_{n=1}^\infty \frac 1{n^k}\) be a value of Riemann’s zeta function. Then the authors prove that for any \(\lambda\in\mathbb R\), each of the sets
\[ \Bigl\{\zeta(2m+1)-\lambda \frac{2^{2m}(2^{2m+2}-1)\mid B_{2m+2}\mid}{(2^{2m+1}-1)(m+1)(2m)!} \pi^{2m+1};\quad m=1, \ldots, 19 \Bigr\}\]
and
\[ \Bigl\{\zeta(2m+1)-\lambda \frac{2^{2m}(2^{2m}-1)\mid B_{2m}\mid}{(2^{2m+1}-1)m(2m)!} \pi^{2m+1};\quad m=1,\ldots,21 \Bigr\}\]
contains at least one irrational number. Here \(B_{2m}\) denotes the \(2m\)-th Bernoulli number. The paper also includes some interesting identities concerning \(\log 2\), Catalan’s constant and \(\pi^2\).

Reviewer: Jaroslav Hančl (Ostrava)

MSC:

11J72	Irrationality; linear independence over a field
11M06	\(\zeta (s)\) and \(L(s, \chi)\)
11Y60	Evaluation of number-theoretic constants
33C20	Generalized hypergeometric series, \({}_pF_q\)
33D15	Basic hypergeometric functions in one variable, \({}_r\phi_s\)
33F10	Symbolic computation of special functions (Gosper and Zeilberger algorithms, etc.)

Keywords:

irrationality; Riemann zeta function; \(\pi\); Catalan constant; log 2; hypergeometric series

Software:

HYP

References:

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