×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.