Elliptic extensions of the Apostol-Bernoulli and Apostol-Euler polynomials

In this paper, we investigate the elliptic analogues of the Apostol-Bernoulli and Apostol-Euler polynomials and obtain the closed expressions of sums of products for these elliptic type polynomials. Some interesting special cases are also shown.

[1]  Mehmet Cenkci,et al.  An explicit formula for generalized potential polynomials and its applications , 2009, Discret. Math..

[2]  Tom M. Apostol,et al.  On the Lerch zeta function. , 1951 .

[3]  Hari M. Srivastava,et al.  Some generalizations of the Apostol–Bernoulli and Apostol–Euler polynomials , 2005 .

[4]  Kwang-Wu Chen.  Sums of products of generalized Bernoulli polynomials , 2003 .

[5]  Ken Kamano.  Sums of products of hypergeometric Bernoulli numbers , 2010 .

[6]  Eugene P. Wigner,et al.  Formulas and Theorems for the Special Functions of Mathematical Physics , 1966 .

[7]  A. Levin.  Elliptic polylogarithms: An analytic theory , 1997, Compositio Mathematica.

[8]  Hari M. Srivastava,et al.  Some relationships between the generalized Apostol–Bernoulli polynomials and Hurwitz–Lerch Zeta functions , 2006 .

[9]  Hari M. Srivastava,et al.  Some relationships between the Apostol-Bernoulli and Apostol-Euler polynomials , 2006, Comput. Math. Appl..

[10]  D. Zagier.  Periods of modular forms and Jacobi theta functions , 1991 .

[11]  Bernoulli Numbers and Polynomials via Residues , 1999 .

[12]  Karl Dilcher,et al.  Sums of Products of Bernoulli Numbers , 1996 .

[13]  Qiu-Ming Luo,et al.  Fourier expansions and integral representations for the Apostol-Bernoulli and Apostol-Euler polynomials , 2009, Math. Comput..

[14]  André Weil,et al.  Elliptic Functions according to Eisenstein and Kronecker , 1976 .

[15]  Leonard Carlitz,et al.  Multiplication formulas for products of Bernoulli and Euler polynomials , 1959 .

[16]  Lee-Chae Jang,et al.  On the q-Extension of Apostol-Euler Numbers and Polynomials , 2008 .

[17]  Hari M. Srivastava,et al.  Some q-extensions of the Apostol-Bernoulli and the Apostol-Euler polynomials of order n, and the multiple Hurwitz zeta function , 2008, Appl. Math. Comput..

[18]  Qiu-Ming Luo,et al.  The multiplication formulas for the Apostol–Bernoulli and Apostol–Euler polynomials of higher order , 2009 .

[19]  Khristo N. Boyadzhiev.  Apostol-Bernoulli functions, derivative polynomials and Eulerian polynomials , 2007 .

[20]  E. Stein,et al.  Complex Analysis , 2003 .

[21]  Kronecker's double series and exact asymptotic expansions for free models of statistical mechanics on torus , 2001, cond-mat/0102470.

[22]  E. Hansen.  A Table of Series and Products , 1977 .

[23]  Kazuya Kato,et al.  Number Theory 1 , 1999 .

[24]  Tianming Wang,et al.  Some results on the Apostol-Bernoulli and Apostol-Euler polynomials , 2008, Comput. Math. Appl..

[25]  Hari M. Srivastava,et al.  Series Associated with the Zeta and Related Functions , 2001 .

[26]  Hari M. Srivastava,et al.  Some formulas for the Bernoulli and Euler polynomials at rational arguments , 2000, Mathematical Proceedings of the Cambridge Philosophical Society.

[27]  Qiu-Ming Luo,et al.  APOSTOL-EULER POLYNOMIALS OF HIGHER ORDER AND GAUSSIAN HYPERGEOMETRIC FUNCTIONS , 2006 .