Certain families of series associated with the Hurwitz–Lerch zeta function. (English) Zbl 1082.11052

Summary: The history of problems of evaluation of series associated with the Riemann zeta-function can be traced back to Christian Goldbach (1690–1764) and Leonhard Euler (1707-1783). Many different techniques to evaluate various series involving the zeta and related functions have since then been developed. The authors show how elegantly certain families of series involving the Hurwitz-Lerch zeta-function can be evaluated by starting with a single known identity for the Hurwitz-Lerch zeta-function. Some of the special cases of these series identities are also shown to lead to certain known families of summation formulas involving the Hurwitz zeta-function.


11M35 Hurwitz and Lerch zeta functions
Full Text: DOI


[1] Apostol, T.M., Some series involving the Riemann zeta function, Proc. amer. math. soc., 5, 239-243, (1954) · Zbl 0055.06903
[2] Barnes, E.W., On the theory of the multiple gamma function, Philos. trans. roy. soc. London ser. A, 199, 374-439, (1904)
[3] Choi, J., Determinant of Laplacian on S3, Math. japon., 40, 155-166, (1994) · Zbl 0806.58053
[4] Choi, J.; Cho, Y.J.; Srivastava, H.M., Series involving the zeta function and multiple gamma functions, Appl. math. comput., 159, 509-537, (2004) · Zbl 1061.33001
[5] Choi, J.; Srivastava, H.M., Sums associated with the zeta function, J. math. anal. appl., 206, 103-120, (1997) · Zbl 0869.11067
[6] Choi, J.; Srivastava, H.M., Certain classes of series involving the zeta function, J. math. anal. appl., 231, 91-117, (1999) · Zbl 0932.11054
[7] Choi, J.; Srivastava, H.M., An application of the theory of the double gamma function, Kyushu J. math., 53, 209-222, (1999) · Zbl 1013.11053
[8] Choi, J.; Srivastava, H.M., Certain classes of series associated with the zeta function and multiple gamma functions, J. comput. appl. math., 118, 87-109, (2000) · Zbl 0969.11030
[9] Choi, J.; Srivastava, H.M., A certain family of series associated with the zeta and related functions, Hiroshima math. J., 32, 417-429, (2002) · Zbl 1160.11339
[10] Choi, J.; Srivastava, H.M.; Quine, J.R., Some series involving the zeta function, Bull. austral. math. soc., 51, 383-393, (1995) · Zbl 0830.11030
[11] Erdélyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F.G., Higher transcendental functions, Vol. I, (1953), McGraw-Hill Book Company New York, Toronto, London · Zbl 0052.29502
[12] Gradshteyn, I.S.; Ryzhik, I.M., Table of integrals, series, and products, (2000), Academic Press New York, (corrected and enlarged ed. prepared by A. Jeffrey, D. Zwillinger) · Zbl 0981.65001
[13] Hansen, E.R., A table of series and products, (1975), Prentice-Hall Englewood Cliffs, New Jersey · Zbl 0302.65039
[14] Kanemitsu, S.; Kumagai, H.; Yoshimoto, M., Sums involving the Hurwitz zeta function, Ramanujan J., 5, 5-19, (2001) · Zbl 0989.11043
[15] Quine, J.R.; Choi, J., Zeta regularized products and functional determinants on spheres, Rocky mountain J. math., 26, 719-729, (1996) · Zbl 0864.47024
[16] Srivastava, H.M., Sums of certain series of the Riemann zeta function, J. math. anal. appl., 134, 129-140, (1988) · Zbl 0632.10040
[17] Srivastava, H.M., A unified presentation of certain classes of series of the Riemann zeta function, Riv. math. univ. parma (ser. 4), 14, 1-23, (1988) · Zbl 0659.10047
[18] Srivastava, H.M.; Choi, J., Series associated with the zeta and related functions, (2001), Kluwer Academic Publishers Dordrecht, Boston, London · Zbl 1014.33001
[19] Vardi, I., Determinants of Laplacians and multiple gamma functions, SIAM J. math. anal., 19, 493-507, (1988) · Zbl 0641.33003
[20] Voros, A., Special functions, spectral functions and the Selberg zeta function, Comm. math. phys., 110, 439-465, (1987) · Zbl 0631.10025
[21] Whittaker, E.T.; Watson, G.N., A course of modern analysis: an introduction to the general thoery of infinite processes and of analytic functions; with an account of the principal transcendental functions, (1963), Cambridge University Press Cambridge, London, and New York · Zbl 0951.30002
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.