Discrete Dynaics in Nature and Society Volue 202, Article ID 927953, pages doi:055/202/927953 Research Article Soe Forulae of Products of the Apostol-Bernoulli and Apostol-Euler Polynoials Yuan He and Chunping Wang Faculty of Science, Kuning University of Science and Technology, Kuning 650500, China Correspondence should be addressed to Yuan He, hyyhe@yahoococn Received 7 May 202; Revised 3 July 202; Accepted 5 July 202 Acadeic Editor: Cengiz Çinar Copyright q 202 Y He and C Wang This is an open access article distributed under the Creative Coons Attribution License, which perits unrestricted use, distribution, and reproduction in any ediu, provided the original work is properly cited Soe forulae of products of the Apostol-Bernoulli and Apostol-Euler polynoials are established by applying the generating function ethods and soe suation transfor techniques, and various known results are derived as special cases Introduction The classical Bernoulli polynoials B n x and Euler polynoials E n x are usually defined by eans of the following generating functions: te xt e t B n x tn t < 2π, 2e xt e t E n x tn t <π In particular, B n B n 0 and E n 2 n E n /2 are called the classical Bernoulli nubers and Euler nubers, respectively These nubers and polynoials play iportant roles in any branches of atheatics such as cobinatorics, nuber theory, special functions, and analysis Nuerous interesting identities and congruences for the can be found in any papers; see, for exaple, 4 Soe analogues of the classical Bernoulli and Euler polynoials are the Apostol- Bernoulli polynoials B n x; and Apostol-Euler polynoials E n x; μ They were
2 Discrete Dynaics in Nature and Society respectively introduced by Apostol 5 see also Srivastava 6 for a systeatic study and Luo 7, 8 as follows: te xt e t 2e xt e t B n x; tn E n x; tn t < 2π if ; t < log otherwise, 2 t <π if ; t < log otherwise 3 Moreover, B n B n 0; and E n 2 n E n /2; are called the Apostol-Bernoulli nubers and Apostol-Euler nubers, respectively Obviously B n x; and E n x; reduce to B n x and E n x when Soe arithetic properties for the Apostol-Bernoulli and Apostol-Euler polynoials and nubers have been well investigated by any authors For exaple, in 998, Srivastava and Todorov 9 gave the close forula for the Apostol-Bernoulli polynoials in ters of the Gaussian hypergeoetric function and the Stirling nubers of the second kind Following the work of Srivastava and Todorov, Luo 7 presented the close forula for the Apostol-Euler polynoials in a siilar technique After that, Luo 0 obtained soe ultiplication forulas for the Apostol-Bernoulli and Apostol-Euler polynoials Further, Luo showed the Fourier expansions for the Apostol-Bernoulli and Apostol-Euler polynoials by applying the Lipschitz suation forula and derived soe explicit forulae at rational arguents for these polynoials in ters of the Hurwitz zeta function In the present paper, we will further investigate the arithetic properties of the Apostol-Bernoulli and Apostol-Euler polynoials and establish soe forulae of products of the Apostol-Bernoulli and Apostol-Euler polynoials by using the generating function ethods and soe suation transfor techniques It turns out that various known results are deduced as special cases 2 The Restateent of the Results For convenience, in this section we always denote by δ, the Kronecker sybol given by δ, 0 or according to / or, and we also denote by axa, b the axiu nuber of the real nubers a, b and by x the axiu integer less than or equal to the real nuber x We now give the forula of products of the Apostol-Bernoulli polynoials in the following way Theore 2 Let and n be any positive integers Then, B x; B n y; μ n k B k k y x; Bnk y; μ n k n B k x; μ kb n k y x; μ k! n! B n y x; 2
Discrete Dynaics in Nature and Society 3 Proof Multiplying both sides of the identity e u e u μe v e u μe v μe uv 22 by uve xuyv,weobtain ue xu e u It follows fro 23 that ve yv uex yu μe v v e u uv ex yu u v e u ey xv μe v e yuv μe uv uvey xv μe v e xuv μe uv uexu e u ve yv uex yu e yuv μe v v e u μe uv δ,μ u v u vey xv e xuv μe v μe uv δ,μ u v 23 24 By the Taylor theore we have e xuv μe uv δ,μ u v n e xu u n μe u δ,μ v n u 25 Since B 0 x; when andb 0 x; 0 when / see eg, 8,by2 and 25 we get e xuv μe uv u v 0 B n x; μ n u! vn 26 Putting 2 and 26 in 24, with the help of the Cauchy product, we derive uv ex yu ey xv u v e u μe v ] B nk y; μ u B k k x y; n k! vn 0 ] n B k x; μ u kb n k y x; μ 0 0 u B x; B n y; μ! vn k! vn 27
4 Discrete Dynaics in Nature and Society If we denote the left-hand side of 27 by M and ve yv ue xu M 2 δ, μe v δ,μ μe u ve yv ue xu δ, μe v δ,μ e u δ,, 28 then applying 2 to 28, in light of 27, we have M M 2 0 k B k x y; B nk y; μ n k n n B k x; μ B n k n k y x; μ k B x; Bn ] y; μ u n! vn 29 On the other hand, a siple calculation iplies M M 2 0 when μ / and uv M M 2 ex yu u v e u δ, u e y xv /e v δ,/ v 20 when μ Applying u n n n k u v k v n k to 2, in view of changing the order of the suation, we obtain ex yu e u δ, u B n x y; n u v k k v n k nk nk n! B n x y; n! B n x y; n n u v k v n k k vn 2
Discrete Dynaics in Nature and Society 5 It follows fro 2, 20, 2, and the syetric relation for the Apostol-Bernoulli polynoials B n x; n B n x;/ for any nonnegative integer n see eg, 8 that M M 2 uv nk nk k B n y x;/ n! k B n y x;/ n! 0 k nk 0 n u v k v n k k n k k B n y x;/ n! k k u v n 0 n k k B n y x;/ u v n n! 0 n!b n2 y x;/ u n 2!! u v n vn 22 Thus, by equating 29 and 22 and then coparing the coefficients of u v n,we coplete the proof of Theore 2 after applying the syetric relation for the Apostol- Bernoulli polynoials It follows that we show soe special cases of Theore 2 By setting x y in Theore 2, we have the following Corollary 22 Let and n be any positive integers Then, B x; B n x; μ n k Bnk x; μ B k k n k n B k x; μ kb n k μ k! n! B n 23 It is well known that the classical Bernoulli nubers with odd subscripts obey B /2and B 2n 0 for any positive integer n see, eg, 2 Setting μ in Corollary 22, we iediately obtain the failiar forula of products of the classical Bernoulli polynoials due to Carlitz 3 and Nielsen 4 as follows
6 Discrete Dynaics in Nature and Society Corollary 23 Let and n be any positive integers Then, B xb n x ax/2,n/2 { n 2k! n! B n n B n 2k x B 2k} 2k n 2k 24 Since the Apostol-Bernoulli polynoials B n x; satisfy the difference equation / xb n x; nb n x; for any positive integer n see, eg, 8, by substituting x y for x in Theore 2 and then taking differences with respect to y, we get the following result after replacing x by x y Corollary 24 Let and n be any positive integers Then, k B k k y x; B n k y; μ B x; B n y; μ n B n k n k y x; μ B k x; μ n B n y; μ B x; 25 Setting x t and y t in Corollary 24, byb n x; n B n x;/ for any nonnegative integer n, we get the following Corollary 25 Let and n be any positive integers Then, k B k k 2t; B n k t; μ B t; B n t; μ n n k k B n k 2t; B k t; μ μ n B t; B t; μ 26 In particular, the case μ in Corollary 25 gives the following generalization for Woodcock s identity on the classical Bernoulli nubers, see 5, 6, Corollary 26 Let and n be any positive integers Then, k B k k 2tB n k t B tb n t n n k k B n k 2tB k t n B ntb t 27 We next present soe ixed forulae of products of the Apostol-Bernoulli and Apostol- Euler polynoials and nubers
Discrete Dynaics in Nature and Society 7 Theore 27 Let and n be non-negative integers Then, E x; E n y; μ 2 k E k k y x; Bnk y; μ n k n B k x; μ 2 ke n k y x; μ k! 2 n! E n y x; 28 Proof Multiplying both sides of the identity e u e u μe v e u μe v μe uv 29 by 2e xuyv,weobtain 2 2e xu e u 2e yv μe v 2ex yu e u e yuv μe uv 2ey xv μe v e xuv μe uv 220 It follows fro 220 that 2ex yu u v e u 2ey xv μe v 2 2e xu e u 2e yv 2ex yu e yuv μe v e u μe uv δ,μ u v 2ey xv e xuv μe v μe uv δ,μ u v 22 Applying 3 and 26 to 22, in view of the Cauchy product, we get 2ex yu u v e u 2ey xv μe v 2 E B nk y; μ x; E n y; μ E k k x y; n k 0 ] n B k x; μ u ke n k y x; μ k! vn 222
8 Discrete Dynaics in Nature and Society On the other hand, since the left-hand side of 222 vanishes when μ /, it suffices to consider the case μ Applying u n n n k u v k v n k to 3, in view of changing the order of the suation, we have 2ex yu e u E n x y; n u v k k v n k nk nk E n x y; v n E n x y; n u v k v n k k 223 It follows fro 3, 223, and the syetric relation for the Apostol-Euler polynoials E n x; n E n x;/ for any non-negative integer n see, eg, 7 that 2ex yu u v e u 2ey xv μe v nk nk k E n y x;/ k E n y x;/ 0 k nk 0 n 0 n u v k v n k k n k k E n y x;/ k k u v n 0 n k k E n y x;/ u v n!e n y x;/ n! u! vn u v n 224 Thus, by equating 222 and 224 and then coparing the coefficients of u v n, we coplete the proof of Theore 27 after applying the syetric relation for the Apostol- Euler polynoials Next, we give soe special cases of Theore 27 By setting x y in Theore 27, we have the following
Discrete Dynaics in Nature and Society 9 Corollary 28 Let and n be non-negative integers Then, E x; E n x; μ 2 k E k k 0; Bnk x; μ n k n B k x; μ 2 ke n k 0; μ k! 2 n! E n 0; 225 Since the classical Euler polynoials E n x at zero arguents satisfy E 0 0, E 2n 0 0, and E 2n 0 2 2n B 2n /n for any positive integer n see, eg, 2, by setting μ in Corollary 28, we obtain the following Corollary 29 Let and n be non-negative integers Then, E xe n x 2 ax/2,n/2 k B nn2 2kx n 2 2k { 2k 2! n! K,n, } n 2 2k B 2k 2k k 226 where K,n 2 2 n2 B n2 / n 2 when n 0 od 2 and K,n 0 otherwise Theore 20 Let be non-negative integer and n positive integer Then, E x; B n y; μ n 2 k E k k y x; E nk y; μ n B k n k y x; μ Ek x; μ 227 Proof Multiplying both sides of the identity e u e u μe v e u μe v μe uv 228 by 2ve xuyv,weobtain 2e xu e u ve yv μe v v 2 2ex yu e u 2e yuv vey xv μe uv μe v 2e xuv μe uv 229
0 Discrete Dynaics in Nature and Society By 3 and the Taylor theore, we have 2e xuv μe uv 0 E n x; μ u! vn 230 Applying 2, 3,and230 to 229, weget 0 2 u E x; B n y; μ! vn E ] u k k x y; Enk y; μ 0 n ] u kb n k y x; μ Ek x; μ 0! vn! vn, 23 which eans 0 E x; B n y; μ 0 n u! vn E 2 k k x y; Enk y; μ n n n B ] u k n k y x; μ Ek x; μ! vn 232 Thus, by coparing the coefficients of u v n in 232, we conclude the proof of Theore 20 after applying the syetric relation for the Apostol-Euler polynoials Obviously, by setting x y in Theore 20,wehavethefollowing Corollary 2 Let be non-negative integer and n positive integer Then, E x; B n x; μ n 2 k E k k 0; E nk x; μ n B k n k μ Ek x; μ 233 Since B /2, E 0 0, E 2n 0 0, and E 2n 0 2 2n B 2n /n for any positive integer n, by setting μ in Corollary 2, we obtain the following
Discrete Dynaics in Nature and Society Corollary 22 Let be non-negative integer and n positive integer Then, E xb n x ax/2,n/2 k E n x { 2 2k } n n B 2k 2k 2k 2kE n 2kx 234 Reark 23 For the equivalent fors of Corollaries 29 and 22, the interested readers ay consult 4 Acknowledgents The authors are very grateful to anonyous referees for helpful coents on the previous version of this work They express their gratitude to Professor Wenpeng Zhang who provided the with soe suggestions This paper is supported by the National Natural Science Foundation of China Grant no 06794 References Y He and Q Liao, Soe congruences involving Euler nubers, The Fibonacci Quarterly, vol 46/47, no 3, pp 225 234, 2008/09 2 Y He and W Zhang, Soe syetric identities involving a sequence of polynoials, Electronic Cobinatorics, vol 7, no, Note 7, p 7, 200 3 D S Ki, T Ki, S-H Lee, D V Dolgy, and S-H Ri, Soe new identities on the Bernoulli and Euler nubers, Discrete Dynaics in Nature and Society, vol 20, Article ID 85632, pages, 20 4 H M Srivastava and Á Pintér, Rearks on soe relationships between the Bernoulli and Euler polynoials, Applied Matheatics Letters, vol 7, no 4, pp 375 380, 2004 5 T M Apostol, On the Lerch zeta function, Pacific Matheatics, vol, pp 6 67, 95 6 H M Srivastava, Soe forulas for the Bernoulli and Euler polynoials at rational arguents, Matheatical Proceedings of the Cabridge Philosophical Society, vol 29, no, pp 77 84, 2000 7 Q-M Luo, Apostol-Euler polynoials of higher order and Gaussian hypergeoetric functions, Taiwanese Matheatics, vol 0, no 4, pp 97 925, 2006 8 Q-M Luo and H M Srivastava, Soe generalizations of the Apostol-Bernoulli and Apostol-Euler polynoials, Matheatical Analysis and Applications, vol 308, no, pp 290 302, 2005 9 H M Srivastava and P G Todorov, An explicit forula for the generalized Bernoulli polynoials, Matheatical Analysis and Applications, vol 30, no 2, pp 509 53, 988 0 Q-M Luo, The ultiplication forulas for the Apostol-Bernoulli and Apostol-Euler polynoials of higher order, Integral Transfors and Special Functions, vol 20, no 5-6, pp 377 39, 2009 Q-M Luo, Fourier expansions and integral representations for the Apostol-Bernoulli and Apostol- Euler polynoials, Matheatics of Coputation, vol 78, no 268, pp 293 2208, 2009 2 M Abraowitz and I A Stegun, Handbook of Matheatical Functions with Forulas Graphs, and Matheatical Tables,, vol 55, National Bureau of Standards, Washington, DC, USA, 964 3 L Carlitz, Note on the integral of the product of several Bernoulli polynoials, the London Matheatical Society, vol 34, pp 36 363, 959 4 N Nielsen, Traité éléentaire des nobres de Bernoulli, Gauthier-Villars, Paris, France, 923 5 Y He and W Zhang, Soe su relations involving Bernoulli and Euler polynoials, Integral Transfors and Special Functions, vol 22, no 3, pp 207 25, 20 6 H Pan and Z-W Sun, New identities involving Bernoulli and Euler polynoials, Cobinatorial Theory A, vol 3, no, pp 56 75, 2006
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