On evaluations of infinite double sums and Tornheim’s double series. (English) Zbl 1196.11122

Kuba, Markus

Sémin. Lothar. Comb. 58, B58d, 11 p. (2007).

Summary: We consider generalizations of a sum, which was recently analyzed by R. Pemantle and C. Schneider [Am. Math. Mon. 114, No. 4, 344–350 (2007; Zbl 1226.11134)] using the computer software Sigma, and later also by A. Panholzer and R. Prodinger [Sémin. Lothar. Comb. 55, B55a, 3 p., electronic only (2005; Zbl 1088.11066)]. Our generalizations include Tornheim’s double series as a special case. We also consider alternating analogs of Tornheim’s series. For Tornheim’s double series and its alternating counterparts we provide short proofs for evaluation formulas, which recently appeared in the literature. We introduce finite Tornheim double sums and alternating analogs, and provide relations to finite multiple zeta functions, similarly to the infinite case. Besides, we discuss the evaluation of another double series, which also generalizes Tornheim’s double series.

MSC:

11M41	Other Dirichlet series and zeta functions
40B05	Multiple sequences and series (should also be assigned at least one other classification number in this section)
11Y60	Evaluation of number-theoretic constants

Keywords:

Tornheim’s double series; Euler sums; alternating double series