a note on eulerian polynomials

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2012, Article ID 269640, 10 pages doi:10.1155/2012/269640 Research Article A Note on Eulerian Polynomials D S Kim,1 T Kim,2 W J Kim,3 and D V Dolgy4 Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea Hanrimwon, Kwangwoon University, Seoul 139-701, Republic of Korea Correspondence should be addressed to T Kim, tkkim@kw.ac.kr Received 29 May 2012; Accepted 25 June 2012 Academic Editor: Josef Dibl´ık Copyright q 2012 D S Kim et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited We study Genocchi, Euler, and tangent numbers From those numbers we derive some identities on Eulerian polynomials in connection with Genocchi and tangent numbers Introduction As is well known, the Eulerian polynomials, An t , are defined by generating function as follows: 1−t exp x t − eA t x −t ∞ An t n with the usual convention about replacing An t by An t that A t t−1 n − tAn t xn , n! 1.1 see 1–18 From 1.1 , we note − t δ0,n , 1.2 where δn,k is the Kronecker symbol see Thus, by 1.2 , we get A0 t 1, An t n−1 n Al t t − t−1l l n−l , n≥1 1.3 Abstract and Applied Analysis By 1.1 , 1.2 , and 1.3 , we see that m i n i n ti n tm An−l t l m l t − n−l n l −1 l −1 n t tm − t−1 n An t , 1.4 where m ≥ and n ≥ see The Genocchi polynomials are defined by 2t et ext ∞ eG x t Gn x n tn , n! 1.5 see 6–18 In the special case, x 0, Gn Gn are called the nth Genocchi numbers see 14, 17, 18 It is well known that the Euler polynomials are also defined by et see 1–5, 19–24 Here x have ext En x n 0, then En E0 ∞ eE x t 1, tn , n! 1.6 En is called the nth Euler number From 1.6 , we E n En 2δ0,n , 1.7 see 3–5, 19–23 As is well known, the Bernoulli numbers are defined by 1, B0 B n − Bn δ0,n , 1.8 see 5, 18, 19 , with the usual convention about replacing Bn by Bn From 1.8 , we note that the Bernoulli polynomials are also defined as n Bn x l see 5, 18, 19 The tangent numbers T2n−1 expansion of tan x: ∞ tan x n n Bl xn−l l B x n, 1.9 n ≥ are defined as the coefficients of the Taylor T2n−1 x2n−1 2n − ! x 1! x3 3! x5 16 5! ··· , see 1–3, In this paper, we give some identities on the Eulerian polynomials at t with Genocchi, Euler, and tangent numbers 1.10 −1 associated Abstract and Applied Analysis Witt’s Formula for Eulerian Polynomials In this section, we assume that Zp , Qp , and Cp will, respectively, denote the ring of p-adic integers, the field of p-adic numbers, and the completion of algebraic closure of Qp The padic norm is normalized so that |p|p 1/p Let q be an indeterminate with |1 − q|p < Then the q-number is defined by x − qx , 1−q q x − −q q −q x 2.1 , see 6–18 Let C Zp be the space of continuous functions on Zp For f ∈ C Zp , the fermionic p-adic q-integral on Zp is defined by I−q f Zp f x dμ−q x pN −1 lim x f x −q , pN N →∞ 2.2 −q x see 7, 10–13 From 2.2 , we can derive the following: q−1 I−q−1 f1 where f1 x f x Let us take f x e−x q t q−1 f 0, 2.3 Then, by 2.3 , we get e− q q I−q−1 f q t Zp e−x q t dμ−q−1 x q−1 2.4 Thus, from 2.4 , we have Zp e−x q t dμ−q−1 x e− ∞ q q t q An −q n tn n! 2.5 tn n! 2.6 By Taylor expansion on the left-hand side of 2.5 , we get ∞ n −1 n Zp xn dμ−q−1 x q n ∞ n! n nt An −q Comparing coefficients on the both sides of 2.6 , we have Zp xn dμ−q−1 x −1 n q Therefore, by 2.7 , we obtain the following theorem n An −q 2.7 Abstract and Applied Analysis Theorem 2.1 For n ∈ Z , one has Zp −1 xn dμ−q−1 x n q −q , n An 2.8 where An −q is an Eulerian polynomials It seems interesting to study Theorem 2.1 at q I−1 f1 where f1 x f x Zp I−1 f By 2.3 , we get 2f , 2.9 From 2.9 , we can derive the following equation: f x −1 n dμ−1 x n−1 n−1 Zp f x dμ−1 x −1 n−l f l , 2.10 l where n ∈ Z see 5–13 From 2.9 , we can derive the following: Zp sin a x cos a Zp 1 dμ−1 x Zp cos a x cos a Zp sin axdμ−1 x dμ−1 x Zp Zp sin axdμ−1 x sin a Zp cos axdμ−1 x , 2.11 cos axdμ−1 x cos axdμ−1 x − sin a Zp sin axdμ−1 x By 2.11 , we get Zp sin axdμ−1 x − sin a cos a a − tan 2.12 From 1.10 and 2.12 , we have ∞ n T2n−1 a 2n − ! 2n−1 − ∞ Zp sin axdμ−1 x n −1 n a2n−1 2n − ! Zp x2n−1 dμ−1 x 2.13 Abstract and Applied Analysis By comparing coefficients on the both sides of 2.13 , we get Zp −1 x2n−1 dμ−1 x n T2n−1 , 22n−1 for n ∈ N, 2.14 where T2n−1 is the 2n − th tangent number Therefore, by 2.14 , we obtain the following theorem Theorem 2.2 For n ∈ N, one has Zp n T2n−1 , 22n−1 2.15 −1 n An −1 2n 2.16 −1 x2n−1 dμ−1 x where T2n−1 is the 2n − th tangent numbers From Theorem 2.1, one has Zp xn dμ−1 x Therefore, by Theorem 2.2 and 2.16 , we obtain the following corollary Corollary 2.3 For n ∈ N, one has A2n−1 −1 −1 n−1 2.17 T2n−1 From 1.6 and 2.9 , we have Zp ext dμ−1 x ∞ et tn , n! 2.18 n T2n−1 22n−1 2.19 En n E2n−1 −1 see Thus, by 2.16 and 2.18 , we get Zp x2n−1 dμ−1 x Therefore, by Corollary 2.3 and 2.19 , we obtain the following corollary Corollary 2.4 For n ∈ N, one has E2n−1 −1 n T2n−1 22n−1 − A2n−1 −1 22n−1 2.20 Abstract and Applied Analysis By 1.5 and 2.9 , we get t 2t 2t et − 2t −1 e −1 ext dμ−1 x Zp e2t ∞ ∞ n Bn n n tn − t n! n n! Bn n ∞ Bn n − Bn 2n 2.21 tn n! By 2.21 , we get Zp Bn xn dμ−1 x 1/2 − Bn n 1 2n 2.22 Thus, from 2.19 , Theorem 2.2 and Corollary 2.3, we have B2n 1/2 − B2n 22n 2n n T2n−1 22n−1 −1 − A2n−1 −1 22n−1 2.23 Therefore, by 2.23 , we obtain the following theorem Theorem 2.5 For n ∈ N, one has B2n 1/2 − B2n 22n n n T2n−1 22n−2 −1 − A2n−1 −1 22n−2 2.24 From 1.5 , we note that t Zp ∞ 2t ext dμ−1 x et tn n! 2.25 2δ1,n , 2.26 Gn n see 13, 14 Thus, by 2.25 , we get G0 0, G n Gn see 13, 14 , with the usual convention about replacing Gn by Gn From 1.5 and 2.9 , one has t Zp ext dμ−1 x et ∞ n 2t t − 2t −1 e −1 tn Bn − Bn n! n 2.27 Abstract and Applied Analysis Thus, by 2.27 , we get Zp xn dμ−1 x Bn − 2n Bn n 1 2.28 From 2.28 , we have G2n 2n Zp B2n − 22n B2n , n x2n−1 dμ−1 x for ∈ N 2.29 Therefore, by 2.19 , Corollary 2.3 and 2.29 , we obtain the following theorem Theorem 2.6 For n ∈ N, we have G2n B2n − 22n B2n 2.30 In particular, −1 A2n−1 −1 22n−1 −1 n T2n−1 22n−1 G2n 2n 2.31 Further Remark In complex plane, we note that tan x i eix − e−ix eix e−ix i 1− ∞ n 1 i 1− ∞ n En n n n i x n! 2e−ix eix e−ix i − ∞ n En n n n i x n! 3.1 −1 n E2n−1 22n−1 x2n−1 2n − ! By 1.10 and 3.1 , we also get T2n−1 −1 n E2n−1 22n−1 , for n ∈ N 3.2 Abstract and Applied Analysis From 1.5 , we have ∞ ∞ t2n G2n 2n ! n n it 2n −1 n G2n 2n ! 2it − it eit it e−it/2 − eit/2 it − eit eit eit/2 eit/2 − e−it/2 /2i t e−it/2 eit/2 e−it/2 /2 3.3 t t tan Thus, by 1.10 and 3.3 , we get ∞ n t2n G2n 2n ! ∞ t t tan t n t/2 2n−1 T2n−1 2n − ! ∞ n t2n T2n−1 2n − !22n−1 3.4 From 3.4 , we have 22n−2 G2n nT2n−1 22n−1 − 22n B2n 3.5 i n tn n! 3.6 By 1.1 , we see that e−2it ∞ An −1 n Thus, we note that ∞ in−1 An −1 n tn n! −1 i e−2it ∞ tan t T2n−1 n 1 − e−2it e−2it i eit − e−it /2 eit e−it /2 i 3.7 t2n−1 2n − ! From 3.7 , we have A2n −1 0, A2n−1 −1 −1 n−1 n≥1 3.8 −1 m − An −1 2n 3.9 T2n−1 , It is easy to show that m k kn −1 k −1 m n k n Ak −1 n−k m − k 2k Abstract and Applied Analysis For simple calculation, we can derive the following equation: eix − e−ix eix e−ix i tan x 1− e2ix − e4ix − 3.10 By 3.10 , we get x tan x 4ix 2ix − e2ix − e4ix − −ix ∞ n −1 n B2n 4n − 4n 2n x 2n ! 3.11 Thus, from 3.11 ,we have ∞ tan x n −1 n B2n 4n − 4n 2n−1 x 2n ! 3.12 By 1.10 and 3.12 , we get T2n−1 −1 n B2n 4n − 4n , 2n for n ∈ N 3.13 From Corollary 2.3 and 3.13 , we can derive the following identity: A2n−1 −1 − B2n 22n−1 − 4n n 3.14 Acknowledgments This research was supported by Basic Science Research Program through the National Research Foundation of Korea NRF funded by the Ministry of Education, Science and Technology 2012R1A1A2003786 Also, the authors would like to thank the referees for their valuable comments and suggestions References L Euler, Institutiones calculi differentialis cum eius usu in 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and tangent numbers 1.10 −1 associated Abstract and Applied Analysis Witt’s Formula for Eulerian Polynomials In this section, we assume... 19 S Araci, D Erdal, and J J Seo, ? ?A study on the fermionic p-adic q-integral representation on Zp associated with weighted q-Bernstein and q-Genocchi polynomials, ” Abstract and Applied Analysis,... Abstract and Applied Analysis, vol 2012, Article ID 784307, 15 pages, 2012 T Kim, “Euler numbers and polynomials associated with zeta functions,” Abstract and Applied Analysis, vol 2008, Article