Several new identities involving Euler and Bernoulli polynomials

Transcription

1 Bull. Math. Soc. Sci. Math. Roumanie Tome 9107 No. 1, 016, Seveal new identitie involving Eule and Benoulli polynomial by Wang Xiaoying and Zhang Wenpeng Abtact The main pupoe of thi pape i uing the elementay and combinational method to obtain eveal new identitie fo Eule and Benoulli polynomial. A ome application of thee identitie, we give eveal inteeting elationhip between thee polynomial and ome famou equence uch a Fibonacci equence and Luca equence o numbe uch a Meenne numbe and Femat numbe. Key Wod: Eule polynomial; Benoulli polynomial; identity; Fibonacci equence; Luca equence; Pell equence; Meenne numbe; Femat numbe. 010 Mathematic Subject Claification: Pimay 11B37, Seconday 11B83. 1 Intoduction Fo any intege n 0, the famou Eule polynomial E n x and Benoulli polynomial B n x ae defined by e xt e t + 1 E n x t n 1.1 and t e xt e t 1 B n x t n. 1. The fit few value of E n x and B n x ae E 0 x 1, E 1 x x 1, E x x x, E 3 x x 3 3 x + 1 4,. B 0x 1, B 1 x x 1, B x x x+ 1 6, B 3 x x 3 3 x + 1 x,. Thoe E n n E n 1 and Bn B n 0 ae called

2 10 W. Xiaoying and Z. Wenpeng the Eule numbe and the Benoulli numbe epectively. About the popetie of thee polynomial and numbe, many autho had tudied them, and obtained a eie of inteeting eult, een efeence [3]-[9]. Fo example, W. Kejian, S. Zhiwei and P. Hao [7] ued the elementay method and the popetie of powe eie to pove the identity and 1 m 1 m m i0 m i0 m B n+i x 1 n i m E n+i x 1 n i n j0 n j0 n B m+j x j n E m+j x. j Z. Wenpeng [9] ued the elementay numbe theoy method and the popetie of conguence to pove that fo any odd pime p, one ha the conguence { 0 mod p, if p 1 mod 4; E p 1 mod p, if p 3 mod 4. In thi pape, we ue ome new method with ome idea in [9], but diffeent method a in efeence [3]-[9], namely, we ue the elementay method, the popetie of powe eie and the inveion fomula of ome aithmetical function to pove eveal new identitie involving the Eule and Benoulli polynomial. A ome application of thee identitie, we give eveal inteeting elationhip between thee polynomial and ome famou equence uch a Fibonacci equence F n and Luca equence L n, and ome famou numbe uch a Meenne numbe M p p 1 and Femat numbe F n n + 1 ee [1], [10] and [11]. That i, we hall pove the following eveal theoem. Theoem 1. Fo any poitive intege n 1, we have the identitie A. B. n + 1 E x x n+1 x 1 n+1 ; 1 n E 1 x x n x 1 n. 1 Theoem. Fo any poitive intege n 1, we have the identitie C. D. n B x n + 1 x n + x 1 n ; n B 1 x n x n 1 + x 1 n 1. 1

3 Identitie involving Eule and Benoulli polynomial 103 Fom Theoem 1 and Theoem we can alo deduce ome inteeting elationhip between the Eule polynomial and Luca equence L n Fibonacci equence F n, Benoulli polynomial and Luca equence L n Fibonacci equence F n. In fact, taking x 1+ in Theoem 1 and Theoem, and note 1+ that L n n + 1 n, [ Fn 1 1+ n 1 n ], we may immediately deduce the following two coollaie: Coollay 1. Fo any poitive intege n 1, we have the identitie n A. E L n+1 ; B. 1 n 1 n 1 + E 1 F n. 1 Coollay. Fo any poitive intege n 1, we have the identitie n C. B n + 1 L n ; D. 1 n 1 n 1 + B 1 n F n 1. 1 If taking x 1+ in B of Theoem 1 and D of Theoem epectively, then we have ome elationhip between[ Eule polynomial, Benoulli polynomial and 1 Pell equence ee [1] P n 1 n n ] + 1. Namely, Coollay 3. Fo any poitive intege n 1, we have the identitie E. F. n 1 + E n P n ; n 1 + B 1 n 1 4 n 1 P n 1. If we take x in A of Theoem 1 and C of Theoem epectively, then we can alo deduce ome elationhip between Eule polynomial, Benoulli polynomial and Meenne numbe M p p 1, Femat numbe F n n +1. That i, we have the following: Coollay 4. Let p be an odd pime. Then we have the identity p 1 p E p 1 M p.

4 104 W. Xiaoying and Z. Wenpeng Coollay. Fo any poitive intege n, we have the identity n + 1 Seveal lemma n 1 n + 1 B n + 1 F n. In thi ection, we hall give eveal ample Lemma, which ae neceay in the poof of ou theoem. Heeinafte, we hall ue ome elementay numbe theoy content and popetie of powe eie, all of thee can be found in efeence [1] and [], o they will not be epeated hee. Lemma 1. Let function fn and gn ae defined fo all intege n 0. Then n fn g if and only if gn 0 n 1 n f. Poof. Fo any intege n 0, if function fn and gn atify the identity fn n g, then we have n 1 n n 1 n n f n 1 n 0 n 1 n n 1 n!n! g!n! g n 0!!! g n! n!! g 1 n g 1 1 n gn..1

5 Identitie involving Eule and Benoulli polynomial 10 If gn atify the identity then we have gn 1 n 0 n g n!n! n f, 1 0!!! f f n 1!n! n!!n! f 0 n n n f 1 0 n f 1 1 n fn.. Combining.1 and. we may immediately deduce Lemma 1. Lemma. Fo any intege n 0, we have the identity n x n E n x E x. Poof. Fom the geneating function 1.1 of E n x we have e xt x n t n E n x 1 t n + e t E n x t n E n x + t n 1 tn E n x n t n 1 + n E x t n..3 Compaing the coefficient of t n in.3 we may immediately deduce the identity n x n E n x + E x o x n E n x n E x.

6 106 W. Xiaoying and Z. Wenpeng Thi pove Lemma. Lemma 3. Fo any intege n 0, we have the identity n n x n 1 + B n x B x. Poof. Fom the geneating function 1. of B n x we have t e xt x n B n x e tn+1 t n t 1 B n x t n B n x + t n 1 tn B n x t n 1 + n n B x t n n+1 1 n + 1 n + 1! B x t n+1 B n+1 x n + 1! tn+1..4 Compaing the coefficient of t n in.4 we may immediately deduce the identity Thi pove Lemma 3. 3 Poof of the theoem n x n 1 + B n x n B x. In thi ection, we hall complete the poof of ou theoem. Fit we pove Theoem 1. Fo any intege n 0, taking fn x n E n x and gn E n x. Fom Lemma 1 and Lemma we have n E n x 1 n x E x x 1 n n 1 n E x. 3.1 Applying 3.1 and Lemma we have x n x 1 n 1 1 n n E x. 3.

7 Identitie involving Eule and Benoulli polynomial 107 It i clea that 3. implying and n + 1 E x x n+1 x 1 n n E 1 x x n x 1 n Now Theoem 1 follow fom 3.3 and 3.4. Similaly, taking fn nx n 1 + B n x and gn B n x in Lemma 1, note that the identity n x 1 n 1 fom Lemma 3 we have n B n x 1 n n 1 n 1 n x 1 n n n x 1, x 1 + B x x n 1 n n Combining 3. and Lemma 3 we may immediately deduce that n x n 1 + x 1 n 1 It i clea that 3.6 implying and n B x n B x B x n n B x. 3.6 n + 1 x n + x 1 n 3.7 n B 1 x n x n 1 + x 1 n Now Theoem follow fom 3.7 and 3.8. The coollaie 1- follow fom Theoem 1 and Theoem with ome pecial value of x. Thi complete the poof of ou all eult.

8 108 W. Xiaoying and Z. Wenpeng Some note: Uing ou Lemma 1 we can alo give anothe poof fo the identitie in [7]. But eult in ou pape eem no contact with efeence [3]-[9]. Acknowledgement. The autho would like to thank the efeee fo hi vey helpful and detailed comment, which have ignificantly impoved the peentation of thi pape. Thi wok i uppoted by the N. S. F and P. N. S. F. 013JM1017 of P. R. China. Refeence [1] Tom M. Apotol, Intoduction to Analytic Numbe Theoy, Spinge-Velag, New Yok, [] Tom M. Apotol, Mathematical Analyi, nd ed., Addion-Weley Publihing Co., Reading, Ma.-London-Don Mill, Ont., [3] M. Kaneko, A ecuence fomula fo the Benoulli numbe, Poc. Japan Acad. Se. A, Math. Sci., , [4] H. Momiyama, A new ecuence fomula fo Benoulli numbe, The Fibonacci Quately, , [] K. Dilche, Sum of poduct of Benoulli numbe, Jounal of Numbe Theoy, , [6] Glenn J. Fox, Conguence elating ational value of Benoulli and Eule polynomial, The Fibonacci Quately, , 0-7. [7] W. Kejian, S. Zhiwei and P. Hao, Some identitie fo Benoulli and Eule polynomial, The Fibonacci Quately, 4 004, [8] L. Guodong, Some identitie involving Benoulli numbe, The Fibonacci Quately, 43 00, [9] Z. Wenpeng, Some identitie involving the Eule and the cental factoal numbe, The Fibonacci Quately, , [10] J. Young and D. A. Buell, The twentieth Femat numbe i compoite, Math. Comp., , [11] C. Pomeance, On pimitive divio of Meenne numbe, Acta Aith., , [1] W. Tingting and Z. Wenpeng, Some identitie involving Fibonacci, Luca polynomial and thei application, Bull. Math. Soc. Sci. Math. Roumanie, , Received: Revied:.0.01 Accepted: School of Mathematic, Nothwet Univeity Xi an, Shaanxi, P. R. China xdwxy@163.com wpzhang@nwu.edu.cn