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The evaluation of Tornheim double sums. I. (English) Zbl 1168.11033
The Tornheim double series is defined for $$a,b,c\in \mathbb C$$ as $T(a,b,c)=\sum_{n=1}^\infty\sum_{m=1}^\infty\frac{1}{n^am^b(n+m)^c}.$ The authors provide an analytic expression for $$T(a,b,c)$$ in terms of the integrals $I(a,b,c)=\int_0^1\zeta(1-a,q)\zeta(1-b,q)\zeta(1-c,q)\,dq$ and $J(a,b,c)=\int_0^1\zeta(1-a,q)\zeta(1-b,q)\zeta(1-c,1-q)\,dq,$ where $$\zeta(z,q)$$ is the Hurwitz zeta function. In the case that $$a,b,c$$ are natural numbers they show that $$T(a,b,c)$$ can be expressed in terms of definite integrals of triple products of Bernoulli polynomials and the Bernoulli function $$A_k(q):=k\zeta'(1-k,q)$$.

##### MSC:
 11M35 Hurwitz and Lerch zeta functions 11B68 Bernoulli and Euler numbers and polynomials
##### Keywords:
Hurwitz zeta function; Bernoulli polynomial
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