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Hypergeometry inspired by irrationality questions. (English) Zbl 1450.11072
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 $$\pi^2$$.

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.)
HYP
Full Text:
References:
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