Adv. Studies Theor. Phys., Vol. 7, 2013, no. 15, 731 - 744 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/astp.2013.3667 A Note on Poly-Bernoulli Polynomials Arising from Umbral Calculus Dae San Kim Department of Mathematics, Sogang University Seoul 121-742, Republic of Korea [email protected] Taekyun Kim Department of Mathematics, Kwangwoon University Seoul 139-701, Republic of Korea [email protected] Sang-Hun Lee Division of General Education Kwangwoon University Seoul 139-701, Republic of Korea [email protected] Copyright c 2013 Dae San 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. Abstract In this paper, we give some recurrence formula and new and interesting 732 Dae San Kim, Taekyun Kim and Sang-Hun Lee identities for the poly-Bernoulli numbers and polynomials which are derived from umbral calculus. 1 Introduction The classical polylogarithmic function Lis(x) are ∞ k x ∈ Lis(x)= s ,s Z, (see [3, 5]). (1) k=1 k In [5], poly-Bernoulli polynomials are defined by the generating function to be −t ∞ n Lik (1 − e ) (k) t ext = eB (x)t = B(k)(x) , (see [3, 5]), (2) − −t n 1 e n=0 n! (k) n (k) with the usual convention about replacing B (x) by Bn (x). As is well known, the Bernoulli polynomials of order r are defined by the generating function to be r ∞ t tn ext = B(r)(x) , (see [7, 9]). (3) t − n e 1 n=0 n! (r) In the special case, r =1,Bn (x)=Bn(x) is called the n-th ordinary Bernoulli (r) polynomial. Here we denote higher-order Bernoulli polynomials as Bn to avoid conflict of notations. (k) (k) If x = 0, then Bn (0) = Bn is called the n-th poly-Bernoulli number. From (2), we note that n n (k) n (k) l n (k) n−l Bn (x)= Bn−lx = Bl x . (4) l=0 l l=0 l Let F be the set of all formal power series in the variable t over C as follows: ∞ tk F ∈ = f(t)= ak ak C , (5) k=0 k! and let P = C[x] and P∗ denote the vector space of all linear functionals on P. L|p(x) denotes the acition of linear functional L on the polynomial p(x), and it is well known that the vector space oprations on P∗ are defined Poly-Bernoulli polynomials arising from umbral calculus 733 by L + M|p(x) = L|p(x) + M|p(x), cL|p(x) = c L|p(x), where c is a complex constant (see [6, 9]). n For f(t) ∈F, let f(t) x = an. Then, by (5), we easily get k n t x = n!δn,k, (n, k ≥ 0), (see [1, 4, 6, 9, 10]), (6) where δn,k is the Kronecker’s symbol. ∞ k tk Let us assume that fL(t)= k=0 L x k! . Then, by (6), we see that n n fL(t) x = L x . That is, fL(t)=L. Additionally, the map L −→ fL(t)is a vector space isomorphism from P∗ onto F. Henceforth, F denotes both the algebra of the formal power series in t and the vector space of all linear func- tionals on P, and so an element f(t)ofF will be thought as a formal power series and a linear functional. F is called the umbral algebra. The umbral calculus is the study of umbral algebra. The order O (f(t)) of the power series f(t) = 0 is the smallest integer for which ak does not vanish. If O(f(t)) = 0, then f(t) is called an invertible series. If O(f(t)) = 1, then f(t) is called a delta series. For f(t),g(t) ∈F, we have f(t)g(t)|p(x) = f(t)|g(t)p(x) = g(t)|f(t)p(x) . (7) Let f(t) ∈F and p(x) ∈ P. Then we have ∞ ∞ tk xk f(t)= f(t)xk ,p(x)= tkp(x) , (see [6, 9]). (8) k=0 k! k=0 k! From (8), we can easily derive k ∞ l d p(x) t p(x) p(k)(x)= = xl−k. (9) k − dx l=k (l k)! Thus, by (8) and (9), we get p(k)(0) = tkp(x) = 1p(k)(x) . (10) Hence, from (10), we have dkp(x) tkp(x)=p(k)(x)= , (see [1, 6, 9]). (11) dxk It is easy to show that eytp(x)=p(x + y), eytp(x) = p(y). (12) 734 Dae San Kim, Taekyun Kim and Sang-Hun Lee Let O(f(t)) = 1 and O(g(t )) = 0. Then there exists a unique sequence sn(x) k of polynomials such that g(t)f(t) sn(x) = n!δn,k, for n, k ≥ 0. The se- quence sn(x) is called a Sheffer sequence for (g(t),f(t)) which is denoted by sn(x) ∼ (g(t),f(t)). The Sheffer sequence sn(x) for (g(t),t) is called the Appell sequence for g(t). For p(x) ∈ P, f(t) ∈F, we have f(t)|xp(x) = ∂tf(t)|p(x) = f (t)|p(x) , (see [9, 10]). (13) Let sn(x) ∼ (g(t),f(t)). Then the following equations are known: ∞ ∞ k h(t) sk(x) k g(t)f(t) p(x) h(t)= g(t)f(t) ,p(x)= sk(x), (14) k=0 k! k=0 k! where h(t) ∈F, p(x) ∈ P, ∞ k 1 yf¯(t) t ∈ ¯ e = sk(y) , for all y C, (15) g f(t) k=0 k! where f¯(t) is the compositional inverse for f(t) with f¯(f(t)) = t, and f(t)sn(x)=nsn−1(x). (16) As is well known, the Stirling numbers of the second kind are also defined by the generating function to be ∞ l ∞ t m t m! l+m e − 1 = m! S2(l, m) = S2(l + m, m)t . (17) l=m l! l=0 (l + m)! Let sn(x) ∼ (g(t),t). Then the Appell identity is given by n n n n−k n k sn(x + y)= sk(y)x = sn−k(y)x , (see [6, 9]), (18) k=0 k k=0 k and g(t) sn+1(x)= x − sn(x), (see [6.9]). (19) g(t) For sn(x) ∼ (g(t),f(t)), rn(x) ∼ (h(t),l(t)), we have n sn(x)= rm(x)cn,m, (20) m=0 Poly-Bernoulli polynomials arising from umbral calculus 735 where ¯ 1 h f(t) m n cn,m = l f¯(t) x , (see [9]). (21) m! g f¯(t) The equations (20) and (21) are important in deriving our main results of this paper. In this paper, we give some recurrence formula and new and interesting iden- tities for the poly-Bernoulli numbers and polynomials which are derive from umbral calculus. 2 Poly-Bernoulli numbers and polynomials 1−e−t Let gk(t)= −t . Then, by (2) and (15), we get Lik(1−e ) (k) Bn (x) ∼ (gk(t),t) . (22) (k) That is, poly-Bernoulli polynomial Bn (x) is an Appell sequence. By (1), we easily get d 1 (k) d (k) (k) Lik(x)= Lik−1(x),tB(x)= B (x)=nB (x). (23) dx x n dx n n−1 From (2) and (15), we have −t (k) 1 n Lik (1 − e ) n Bn (x)= x = −t x . (24) gk(t) 1 − e Let k ∈ Z and n ≥ 0. Then we have −t ∞ −t m−1 Lik (1 − e ) (1 − e ) B(k)(x)= xn = xn (25) n 1 − e−t mk m=1 ∞ ∞ m 1 − −t m n 1 − j m −jt n = k 1 e x = k ( 1) e x m=0 (m +1) m=0 (m +1) j=0 j n m 1 − j m − n = k ( 1) (x j) . m=0 (m +1) j=0 j 736 Dae San Kim, Taekyun Kim and Sang-Hun Lee By (17) and (25), we get n ∞ (k) 1 − a m! a+m n Bn (x)= k ( 1) S2(a + m, m)t x (26) m=0 (m +1) a=0 (a + m)! n n−m 1 − a m! n−a−m = k ( 1) S2(a + m, m)(n)a+mx m=0 (m +1) a=0 (a + m)! n n−l − n−m−l ( 1) n − l = k m!S2(n l, m) x , l=0 m=0 (m +1) l where (a)n = a(a − 1)(a − 2) ···(a − n + 1). Now, we use the well-known transfer formula for Appell sequences (see equation (19)). By (19) and (22), we get (k) gk(t) (k) Bn+1(x)= x − Bn (x), (27) gk(t) where gk(t) −t −t = (log gk(t)) = log 1 − e − log Lik 1 − e (28) gk(t) −t − −t e − Lik−1 (1 e ) = − −t 1 − −t 1 e Lik (1 e ) −t −t 1 Lik (1 − e ) − Lik−1 (1 − e ) = t −t . e − 1 Lik (1 − e ) From (27) and (28), we have (k) (k) gk(t) (k) Bn+1(x)=xBn (x) − Bn (x) (29) gk(t) −t −t t Lik (1 − e ) − Lik−1 (1 − e ) = xB(k)(x) − xn. n et − 1 t (1 − e−t) Here, we note that −t −t ∞ Lik (1 − e ) − Lik−1 (1 − e ) 1 1 m−1 = − 1 − e−t (30) 1 − e−t mk mk−1 m=2 1 1 = − t + ··· 2k 2k−1 Poly-Bernoulli polynomials arising from umbral calculus 737 is a delta series.