Remarks on some relationships between the Bernoulli and Euler polynomials

K e y w o r d s B e r n o u l l i polynomials, Euler polynomials, Generating functions, Bernoulli numbers, Euler numbers, Addition theorem, Multiplication theorem~ Generalized Bernoulli polynomials and numbers, Generalized Euler polynomials and numbers. 1. I N T R O D U C T I O N T h e classical Bernoulli polynomials Bn(x) and the classical Euler polynomials En(x) are usual ly def ined by m e a n s of t he fol lowing gene ra t ing funct ions: oo tn t ~ -~ B~ (x) (Itl < 2¢r) (1) e t 1 ~ . n=O and 2eXt oo tn J V I ~ En (x) ~. (Itl < ~), (2) n = 0 The present investigation was completed during the second-named author's visit to the Mathematical Institute of Kossuth Lajos University at Debrecen in July 2003. This work was supported, in part, by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0007353 and, in part, by the Netherlands Organization for Scientific Research (NWO) and the Hungarian Academy of Sciences, by the HNFSR Grants N34001, F34981, and T42985, and by the FKFP Grant 3272-13/066/2001. 0893-9659/04/$ see front matter @ 2004 Elsevier Ltd. All rights reserved. doi: 10.1016/S0893-9659(04)00041-2 Typeset by AA~S-TEX 376 H.M. SRIVASTAVA AND ]k. PINT~R respectively. The corresponding Bernoulli numbers Bn and Euler numbers En are given by B~ := B~ (O) = (-1)'~ B,~ (1) = (21-~I)-I B,~ (1) ( n E N o : = N U { 0 } ; 5 : = { 1 , 2 , 3 , . . . } ) and (3) and t ( n ) B B~ (x + i) = k k (x) (~ e No), (5) k----0 E n ( x + l ) = f i ( n k ) E k ( x ) (n E N0), (6) k = 0 and B,, (x) = k BkE~_k (x) (n e No). (7) k=O (k#l) Both (5) and (6) are well-known (rather classical) results and are obvious special cases of the following familiar addition theorems: