Signed $q$-analogs of Tornheim's double series

Abstract

Abstract. We introduce signed q-analogs of Tornheim's double series and evaluate them in terms of double q-Euler sums. As a consequence, we provide explicit evaluations of signed and unsigned Tornheim double series and correct some mistakes in the literature.

Evaluation of Tornheim’s type of double series

“…For example, the identity ζ 1,1 (1, 1, 3) = 4ζ(5) − 2ζ(2)ζ (3) is well-known. Similar studies have been done in many articles [7,10,12,13,15,17] (see also [9]). We will generalize the above expression to the value ζ a,b (k 1 , k 2 , k 3 ) with k 1 + k 2 + k 3 odd.…”

Double Lerch Value Relations and Functional Relations for Witten Zeta Functions

“…REMARK. After completing the present paper, the author noticed that the above Proposition has been also proved by Xia Zhou, Tianxin Cai and D. Bradley [29] independently. REMARK.…”

On Mordell-Tornheim sums and multiple zeta values

“…We note that the case r = 2 of Theorem 1.2 was proved by Tornheim [19]. Explicit formulas for Tornheim's reduction were given in [15]; see also [23].…”