Extended Zeilberger's Algorithm for Identities on Bernoulli and Euler Polynomials

We present a computer algebra approach to proving identities on Bernoulli polynomials and Euler polynomials by using the extended Zeilberger's algorithm given by Chen, Hou and Mu. The key idea is to use the contour integral definitions of the Bernoulli and Euler numbers to establish recurrence relations on the integrands. Such recurrence relations have certain parameter free properties which lead to the required identities without computing the integrals.

[1]  Ira M. Gessel.  Applications of the classical umbral calculus , 2001 .

[2]  C. Jacobi,et al.  De usu legitimo formulae summatoriae Maclaurinianae. , 1834 .

[3]  Kwang-Wu Chen.  A summation on Bernoulli numbers , 2005 .

[4]  Chris F. Woodcock,et al.  Convolutions on the ring of p-adic integers , 1979 .

[5]  Doron Zeilberger,et al.  The Method of Creative Telescoping , 1991, J. Symb. Comput..

[6]  Zhi-Wei Sun,et al.  Combinatorial identities in dual sequences , 2003, Eur. J. Comb..

[7]  Hiroo Miki,et al.  A relation between Bernoulli numbers , 1978 .

[8]  Saulo Alves de Araujo,et al.  Identification of novel keloid biomarkers through Profiling of Tissue Biopsies versus Cell Cultures in Keloid Margin specimens Compared to adjacent Normal Skin , 2010, Eplasty.

[9]  Philippe Flajolet,et al.  On the Non-Holonomic Character of Logarithms, Powers, and the nth Prime Function , 2005, Electron. J. Comb..

[10]  Carsten Schneider,et al.  Computer proofs of a new family of harmonic number identities , 2003, Adv. Appl. Math..

[11]  F. Stan.  Computer-assisted proofs of special function identities related to Poisson integrals , 2009 .

[12]  Manuel Kauers,et al.  Summation algorithms for Stirling number identities , 2007, J. Symb. Comput..

[13]  Zhi-Wei Sun,et al.  Identities concerning Bernoulli and Euler polynomials , 2004 .

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

[15]  Kurt Wegschaider,et al.  Computer Generated Proofs of Binomial Multi-Sum Identities , 1997 .

[16]  Takashi Agoh,et al.  Convolution identities and lacunary recurrences for Bernoulli numbers , 2007 .

[17]  George E. Andrews,et al.  Plane partitions VI: Stembridge's TSPP theorem , 2005, Adv. Appl. Math..

[18]  Masanobu Kaneko,et al.  A Recurrence Formula for the Bernoulli Numbers , 1995 .

[19]  Michael Karr,et al.  Theory of Summation in Finite Terms , 1985, J. Symb. Comput..