Rational series for multiple zeta and log gamma functions

Abstract Text We give series expansions for the Barnes multiple zeta functions in terms of rational functions whose numerators are complex-order Bernoulli polynomials, and whose denominators are linear. We also derive corresponding rational expansions for Dirichlet L-functions and multiple log gamma functions in terms of higher order Bernoulli polynomials. These expansions naturally express many of the well-known properties of these functions. As corollaries many special values of these transcendental functions are expressed as series of higher order Bernoulli numbers. Video For a video summary of this paper, please click here or visit http://youtu.be/2i5PQiueW_8 .

[1]  M. Rubinstein.  Identities for the Hurwitz zeta function, Gamma function, and L-functions , 2012, 1206.1992.

[2]  T. R. Riley,et al.  The Absence of Efficient Dual Pairs of Spanning Trees in Planar Graphs , 2006, Electron. J. Comb..

[3]  F. T. Howard.  CONGRUENCES AND RECURRENCES FOR BERNOULLI NUMBERS OF HIGHER ORDER , 1994 .

[4]  N. E. Nörlund.  Vorlesungen über Differenzenrechnung , 1924 .

[5]  Gergýo Nemes.  An Asymptotic Expansion for the Bernoulli Numbers of the Second Kind , 2011 .

[6]  Llan Vardi,et al.  Determinants of Laplacians and multiple gamma functions , 1988 .

[7]  Philippe Flajolet,et al.  Singularity Analysis of Generating Functions , 1990, SIAM J. Discret. Math..

[8]  Steven Roman.  The Umbral Calculus , 1984 .

[9]  Károly Jordán.  Calculus of finite differences , 1951 .

[10]  Renzo Sprugnoli,et al.  The Cauchy numbers , 2006, Discret. Math..

[11]  Michael O. Rubinstein.  Identities for the Riemann zeta function , 2008 .

[12]  Helmut Hasse,et al.  Number Theory , 2020, An Introduction to Probabilistic Number Theory.

[13]  Henri Cohen,et al.  Number Theory: Volume II: Analytic and Modern Tools , 2007 .

[14]  Larry Ericksen,et al.  Sum of powers from uniform distribution moments , 2002 .

[15]  Simon Ruijsenaars,et al.  On Barnes' Multiple Zeta and Gamma Functions , 2000 .

[16]  T. Apostol.  Introduction to analytic number theory , 1976 .

[17]  E. Alkan.  Values of Dirichlet L-functions, Gauss sums and trigonometric sums , 2011 .

[18]  L. Washington.  Introduction to Cyclotomic Fields , 1982 .

[19]  Feng-Zhen Zhao.  Some Properties of a New Class of Generalized Cauchy Numbers , 2012 .

[20]  Philippe Flajolet,et al.  A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics , 2006, Electron. J. Comb..