Some Series for the Riemann-Hurwitz function

PANZONE, PABLO; PABLO ANDRES PANZONE

Revista de la Unión Matemática Argentina

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Lugar: Bahía Blanca (BsAs); Año: 1998 vol. 41 p. 89 - 107

The author derives several formulas for the Riemann zeta function (s) when s is a positive integerwith s 2. One of these is the formula of M. Hjortnaes,(3) =52X1n=1(−1)n−1n3􀀀2nn ,made famous by R. Ap´ery [cf. A. van der Poorten, Math. Intelligencer 1 (1978/79), no. 4, 195–203;MR0547748 (80i:10054)] in his famous proof of the irrationality of (3). The author establishesseveral general formulas but concentrates on representations for (3) and (5). Appearing invarious formulas are the central binomial coefficient􀀀2nn, other rational functions of n, partialsums of the harmonic series or similar sums, and values of the hypergeometric function 5F4. Aprimary ingredient in some of the proofs is the formulaXNk=1(a1a2 · · · ak−1)j(x(x+a1) · · · (x+ak))jXj−1m=0xj−mamkjm=1xj−(a1a2 · · · ak)j(x(x+a1) · · · (x+ak))j ,which generalizes a formula of Ap´ery.