Poly-Bernoulli polynomials arising from umbral calculus by Dae San Kim, Taekyun Kim and Sang-Hun Lee Abstract In this paper, we give some recurrence formula and new and interesting identities for the poly-Bernoulli numbers and polynomials which are derived from umbral calculus. 1 Introduction The classical polylogarithmic function Lis(x) are ∞ xk Li (x)= , s ∈ Z, (see [3, 5]). (1) s ks k X=1 In [5], poly-Bernoulli polynomials are defined by the generating function to be −t ∞ n Li (1 − e ) (k) t k ext = eB (x)t = B(k)(x) , (see [3, 5]), (2) 1 − e−t n n! n=0 X (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 arXiv:1306.6697v1 [math.NT] 28 Jun 2013 generating function to be t r ∞ tn ext = B(r)(x) , (see [7, 9]). (3) et − 1 n n! n=0 X (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 1 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 n n B(k)(x)= B(k) xl = B(k)xn−l. (4) n l n−l l l l l X=0 X=0 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! ) k=0 X P P∗ and let = C[x] and denote the vector space of all linear functionals on P. hL|p(x)i 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 by hL + M|p(x)i = hL|p(x)i + hM|p(x)i, hcL|p(x)i = c hL|p(x)i, 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 f (t) xn = L xn . That is, f (t)= L. Additionally, the map L 7−→ f (t) is L L L a vector space isomorphism fromPP∗ 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) of F 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) =6 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 hf(t)g(t)|p(x)i = hf(t)|g(t)p(x)i = hg(t)|f(t)p(x)i . (7) Let f(t) ∈F and p(x) ∈ P. Then we have ∞ tk ∞ xk f(t)= f(t) xk , p(x)= tk p(x) , (see [6, 9]). (8) k! k! k=0 k=0 X X 2 From (8), we can easily derive ∞ dkp(x) tl p(x) p(k)(x)= = xl−k. (9) dxk (l − k)! l=k X Thus, by (8) and (9), we get p(k)(0) = tk p(x) = 1 p(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), eyt p(x) = p(y). (12) 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 s (x) is called a Sheffer sequence for (g(t), f(t)) which is denoted by n 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 ′ hf(t)|xp(x)i = h∂tf(t)|p(x)i = hf (t)|p(x)i , (see [9, 10]). (13) Let sn(x) ∼ (g(t), f(t)). Then the following equations are known: ∞ ∞ k h(t) sk(x) g(t)f(t) p(x) h(t)= g(t)f(t)k, p(x)= s (x), (14) k! k! k k=0 k=0 X X where h(t) ∈F, p(x) ∈ P, ∞ 1 tk eyf¯(t) = s (y) , for all y ∈ C, (15) g f¯ t k k! ( ) k X=0 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 ∞ m t m! et − 1 = m! S (l, m) = S (l + m, m)tl+m. (17) 2 l! (l + m)! 2 l=m l=0 X X 3 Let sn(x) ∼ (g(t), t). Then the Appell identity is given by n n n n s (x + y)= s (y)xn−k = s (y)xk, (see [6, 9]), (18) n k k k n−k k k X=0 X=0 and g′(t) s (x)= x − s (x), (see [6.9]). (19) n+1 g(t) n 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 X where 1 h f¯(t) m c l f¯ t xn , . n,m = ¯ ( ) (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 g t −t Let k( )= Lik(1−e ) . Then, by (2) and (15), we get (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 d Li (x)= Li (x), tB(k)(x)= B(k)(x)= nB(k) (x). (23) dx k x k−1 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 4 Let k ∈ Z and n ≥ 0. Then we have ∞ Li (1 − e−t) (1 − e−t)m−1 B(k)(x)= k xn = xn (25) n 1 − e−t mk m=1 ∞ X ∞ m 1 m 1 m = 1 − e−t xn = (−1)j e−jtxn (m + 1)k (m + 1)k j m=0 m=0 j=0 Xn m X X 1 m = (−1)j (x − j)n. (m + 1)k j m=0 j=0 X X By (17) and (25), we get n ∞ 1 m! B(k)(x)= (−1)a S (a + m, m)ta+mxn (26) n (m + 1)k (a + m)! 2 m=0 a=0 Xn nX−m 1 m! = (−1)a S (a + m, m)(n) xn−a−m (m + 1)k (a + m)! 2 a+m m=0 a=0 Xn n−l X (−1)n−m−l n = m!S (n − l, m) xl, (m + 1)k l 2 l (m=0 ) X=0 X 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 g′ (t) B(k) (x)= x − k B(k)(x), (27) n+1 g (t) n k 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 Li (1 − e ) − Li − (1 − e ) = k k 1 . et − 1 Li (1 − e−t) k 5 From (27) and (28), we have ′ (k) (k) gk(t) (k) Bn+1(x)= xBn (x) − Bn (x) (29) gk(t) −t −t t Li (1 − e ) − Li − (1 − e ) = xB(k)(x) − k k 1 xn. n et − 1 t (1 − e−t) Here, we note that −t −t ∞ Li (1 − e ) − Li − (1 − e ) 1 1 m−1 k k 1 = − 1 − e−t (30) 1 − e−t mk mk−1 m=2 X 1 1 = − t + ··· 2k 2k−1 is a delta series. For any delta series f(t), we observe that f(t) xn+1 xn = f(t) . (31) t n +1 By (29), (30) and (31), we get t 1 Li (1 − e−t) − Li (1 − e−t) B(k) (x)= xB(k)(x) − k k−1 xn+1 n+1 n et − 1 n +1 1 − e−t ∞ 1 B = xB(k)(x) − l tl B(k) (x) − B(k−1)(x) n n +1 l! n+1 n+1 l=0 n o Xn 1 +1 n +1 = xB(k)(x) − B B(k) (x) − B(k−1) (x) .