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Modular relation interpretation of the series involving the Riemann zeta values. (English) Zbl 1225.11103

Authors’ summary: “We locate M. Katsurada’s results [Publ. Inst. Math., Nouv. Sér. 62(76), 13–25 (1997; Zbl 0885.11052)], in our framework of modular relations, on two series involving the values of the Riemann zeta-function, which are decisive generalizations of earlier results of S. Chowla and D. Hawkins [J. Indian Math. Soc., New Ser. 26, 115–124 (1962; Zbl 0112.30203)] and of R. G. Buschman and H. M. Srivastava [Monatsh. Math. 115, No. 4, 291–298 (1993; Zbl 0779.11033)] et al. We elucidate these results as an improper or a proper modular relation according as the involved parameter \(\nu\) exerts effects on the series or not, eventually indicating that they are disguised form of modular relations as given by Theorem 4 in §3.”
Katsurada’s Theorem 1 is a special case of the following more general theorem, which in turn is a simple corollary of Theorem 4.
Theorem 3: \[ \begin{split} \frac{1}{\Gamma(1-b)}\sum_ {n\geq 0} \frac{(b)_ n}{n!}\zeta(n-\nu)x^ n= x^ {\nu+1}\frac{\Gamma(-\nu-1)}{\Gamma(-\nu-b)}\\ +x^ {\nu+1}\sum_ {k\geq 1}\bigg\{G_ {1,2}^ {2,0}\left(2i\pi kx\left|\frac{-b-\nu}{0,-\nu-1}\right.\right) +G_ {1,2}^ {2,0}\left(-2i\pi kx\left|\frac{-b-\nu}{0,-\nu-1}\right.\right)\bigg\} \end{split} \] where \(G_ {p,q}^ {m,n}\) stands for the Meijer \(G\)-function.

MSC:

11M06 \(\zeta (s)\) and \(L(s, \chi)\)
33C60 Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions)
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References:

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[2] S. Chowla and D. Hawkins, Asymptotic expansions of some series involving the Riemann zeta function, J. Indian Math. Soc. (N.S.) 26 (1962), 115-124. · Zbl 0112.30203
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