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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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##### References:
 [1] Adamchik, V., Polygamma functions of negative order, J. comput. appl. math., 100, 191-198, (1998) · Zbl 0936.33001 [2] Adamchik, V., Multiple gamma function and its application to computation of series, Ramanujan J., 9, 271-288, (2005) · Zbl 1088.33014 [3] Apostol, T., Introduction to analytic number theory, (1976), UTM, Springer Berlin · Zbl 0335.10001 [4] Berndt, B., On the Hurwitz zeta function, Rocky mountain J., 2, 151-157, (1972) · Zbl 0229.10023 [5] Boros, G.; Espinosa, O.; Moll, V., On some families of integrals solvable in terms of polygamma and negapolygamma functions, Integral transforms special funct., 14, 3, 187-203, (2003) · Zbl 1032.33002 [6] Borwein, J.; Bailey, D.; Girgensohn, R., Experimentation in mathematics: computational paths to discovery, (2004), A.K. Peters · Zbl 1083.00002 [7] Boyadzhiev, K.N., Evaluation of euler – zagier sums, Internat. J. math. sci., 27, 407-412, (2001) · Zbl 1015.11046 [8] Boyadzhiev, K.N., Consecutive evaluation of Euler sums, Internat. J. math. sci., 29, 555-561, (2002) · Zbl 0994.40003 [9] Carlitz, L., Note on the integral of the product of several Bernoulli polynomials, J. London math. soc., 34, 361-363, (1959) · Zbl 0086.05801 [10] Crandall, R.; Buhler, J., On the evaluation of Euler sums, Exp. math., 3, 275-285, (1994) · Zbl 0833.11045 [11] Espinosa, O.; Moll, V., On some integrals involving the Hurwitz zeta function: part 1, Ramanujan J., 6, 159-188, (2002) · Zbl 1019.33001 [12] Espinosa, O.; Moll, V., On some integrals involving the Hurwitz zeta function: part 2, Ramanujan J., 6, 449-468, (2002) · Zbl 1156.11333 [13] Espinosa, O.; Moll, V., A generalized polygamma function, Integral transforms special funct., 15, 101-115, (2004) · Zbl 1052.33002 [14] O. Espinosa, V. Moll, The evaluation of Tornheim double sums, Part 2, in preparation. · Zbl 1188.11042 [15] Gosper, R.Wm., $$\int_{n / 4}^{m / 6} \ln \operatorname{\Gamma}(z) \mathit{dz}$$. in special functions. $$q$$-series and related topics, (), 71-76 · Zbl 0870.33001 [16] Huard, J.G.; Williams, K.S.; Zhang, N.Y., On Tornheim’s double series, Acta arith., 75, 105-117, (1996) · Zbl 0858.40008 [17] D. Kreimer, Knots and Feynman diagrams, Cambridge Lecture Notes in Physics, 2000. · Zbl 0964.81052 [18] Matsumoto, K., On the analytic continuation of various multiple zeta-functions, (), 417-440 · Zbl 1031.11051 [19] Sitaramachandrarao, R.; Subbarao, M.V., Transformation formulae for multiple series, Pacific J. math., 113, 471-479, (1984) · Zbl 0549.10031 [20] Tornheim, L., Harmonic double series, Amer. J. math., 72, 303-314, (1950) · Zbl 0036.17203 [21] Tsumura, H., On Witten’s type zeta values attached to $$\mathit{SO}(5)$$, Arch. math., 82, 145-152, (2004) · Zbl 1063.40004 [22] E. Whittaker, G. Watson, A Course of Modern Analysis, fourth ed., reprinted, Cambridge University Press, Cambridge, MA, 1963. · Zbl 0108.26903 [23] D. Zagier, Values of zeta functions and their applications, First European Congress of Mathematics, vol. 2, Paris, 6-10 July, 1992, Birkhauser Verlag, Basel, pp. 497-512.
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