Umbral Calculus Associated with Bernoulli Polynomials ✩

JID:YJNTH AID:4725 /FLA [m1L; v 1.118; Prn:9/12/2013; 13:11] P.1 (1-12) Journal of Number Theory ••• (••••) •••–•••

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Journal of Number Theory

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Umbral calculus associated with Bernoulli polynomials ✩

Dae San Kim a, Taekyun Kim b,∗ a Department of Mathematics, Sogang University, Seoul 121-741, Republic of Korea b Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea article info abstract

Article history: Text. Recently, R. Dere and Y. Simsek have studied Received 6 November 2012 applications of umbral algebra to generating functions for Received in revised form 10 the Hermite type Genocchi polynomials and numbers [6].In September 2013 this paper, we investigate some interesting properties arising Accepted 17 September 2013 Available online xxxx from umbral calculus. These properties are useful in deriving Communicated by James Cogdell, some identities of Bernoulli polynomials. Herve Jacquet, Dihua Jiang, and Video. For a video summary of this paper, please click here Steve Kudla or visit http://youtu.be/r7_Bbf0BDbs. © 2013 The Authors. Published by Elsevier Inc. Keywords: All rights reserved. Bernoulli polynomials Umbral calculus Linear functional

1. Introduction

As is well known, the Bernoulli polynomials are deﬁned by the generating function to be ∞ text tn = eB(x)t = B (x) , (1) et − 1 n n! n=0

✩ This is an open-access article distributed under the terms of the Creative Commons Attribution- NonCommercial-ShareAlike License, which permits non-commercial use, distribution, and reproduction in any medium, provided the original author and source are credited. * Corresponding author. E-mail addresses: [email protected] (D.S. Kim), [email protected] (T. Kim).

0022-314X/$ – see front matter © 2013 The Authors. Published by Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jnt.2013.09.013 JID:YJNTH AID:4725 /FLA [m1L; v 1.118; Prn:9/12/2013; 13:11] P.2 (1-12) 2 D.S. Kim, T. Kim / Journal of Number Theory ••• (••••) •••–•••

n with the usual convention about replacing B (x)byBn(x). In the special case, x =0, Bn(0) = Bn are called the n-th Bernoulli numbers. From (1),wenotethat n n B0 =1, (B +1) − B = Bn(1) − Bn = δ1,n see [2–4] , where δm,k is the Kronecker symbol. In particular, by (1),weset n n l B (x)= B − x see [1–4,9,14] . (2) n l n l l=0

By (2),weseethatBn(x) is a monic polynomial of degree n. We recall the Euler polynomials are deﬁned by the generating function to be

∞ 2ext tn = eE(x)t = E (x) see [1–4,6,9–24] , (3) et +1 n n! n=0

n with the usual convention about replacing E (x)byEn(x). In the special case, x =0, En(0) = En are called the n-th Euler numbers. From (3), we can derive the following equation: n n l E (x)= E − x see [11,14] . (4) n l n l l=0

Thus by (4),weseethatEn(x) is also a monic polynomial of degree n.By(4),weget

n n E0 =1, (E +1) + E = En(1) + En =2δ0,n. (5)

Let C be the complex number ﬁeld and let F be the set of all formal power series in the variable t over C with ∞ a F = f(t)= k tk: a ∈ C . k! k k=0

We use the notation P = C[x]andP∗ denotes the vector space of all linear functional on P. Let L | p(x) be the action of a linear functional L on a polynomial p(x), and we remind that the vector space operations on P∗ are deﬁned by

L + M p(x) = L p(x) + M p(x) , cL p(x) = c L p(x) see [6,18] , where c is any constant in C. JID:YJNTH AID:4725 /FLA [m1L; v 1.118; Prn:9/12/2013; 13:11] P.3 (1-12) D.S. Kim, T. Kim / Journal of Number Theory ••• (••••) •••–••• 3

The formal power series

∞ a f(t)= k tk ∈F (6) k! k=0 deﬁnes a linear functional on P by setting

n f(t) x = an for all n 0. (7)

Thus, by (6) and (7),wehave k n t x = n!δn,k see [6,18] . (8)

k ∞ L|x k | n | n Let fL(t)= k=0 k! t .Thenweseethat fL(t) x = L x and so as linear functionals L = fL(t)(see[6,18]). As is known in [18],themapL → fL(t) is a vector space isomorphism from P∗ onto F. Henceforth, F will denote both the algebra of formal poser series in t and the vector space of all linear functionals on P, and so an element f(t)ofF will be thought of as both a formal power series and a linear functional. We shall call F the umbral algebra. The umbral calculus is the study of umbral algebra and modern classical umbral calculus can be described as a systematic study of the class of Sheﬀer sequences (see [18]). The order ord(f(t)) of a nonzero power series f(t) is the smallest integer k for which the coeﬃcient of tk does not vanish. If a series f(t)iswithord(f(t)) = 1, then f(t)is called a delta series. If a series f(t)iswithord(f(t)) = 0, then f(t) is called an invertible series (see [6,18]). For f(t),g(t) ∈F,wehave f(t)g(t) p(x) = f(t) g(t)p(x) see [6,18] . (9)

Let us assume that Sn(x) denotes a polynomial of degree n.Iff(t) is a delta series and k g(t) is an invertible series, then there exists a unique sequence Sn(x) such that g(t)f(t) | Sn(x) = n!δn.k, n, k 0(see[6]). The sequence Sn(x) is called the Sheﬀer sequence for (g(t),f(t)), denoted by Sn(x) ∼ (g(t),f(t)). If Sn(x) ∼ (1,f(t)), then Sn(x) is called the associated sequence for f(t)orSn(x) is associated to f(t). If Sn(x) ∼ (g(t),t), then Sn(x) is called the Appell sequence for g(t)orSn(x)isAppellforg(t)(see[6,18]). For p(x) ∈ P,itisknown(see[6,18])that

y yt e − 1 p(x) = p(u) du, (10) t 0 f(t) xp(x) = ∂tf(t) p(x) = f (t) p(x) , (11) and eyt − 1 p(x) = p(y) − p(0) see [6,18] . (12) JID:YJNTH AID:4725 /FLA [m1L; v 1.118; Prn:9/12/2013; 13:11] P.4 (1-12) 4 D.S. Kim, T. Kim / Journal of Number Theory ••• (••••) •••–•••

Let us assume that Sn(x) ∼ (g(t),f(t)). Then we have the following equations (13)–(16):

∞ h(t) | S (x) h(t)= k g(t)f(t)k,h(t) ∈F, (13) k! k=0 ∞ g(t)f(t)k | p(x) p(t)= S (x),p(t) ∈ P, (14) k! k k=0 f(t)Sn(x)=nSn−1(x) n ∈ Z+ = N ∪{0} , (15) and ∞ 1 ¯ S (y) eyf(t) = k tk, for all y ∈ C, (16) g(f¯(t)) k! k=0 where f¯(t) is the compositional inverse of f(t)(see[18]).

Let f1(t),...,fm(t) ∈F.Thenasiswellknown,wehave n n i1 im f1(t)f2(t) ···fm(t) x = f1(t) x ··· fm(t) x , (17) i1,...,im where the sum is over all nonnegative integers i1,...,im such that i1 + ···+ im = n (see [6,18]). In [6], R. Dere and Y. Simsek have studied applications of umbral algebra to generating functions for the Hermite type Genocchi polynomials and numbers. In this paper, we derive some interesting properties of Bernoulli polynomials arising from umbral calculus. These properties will be used in studying identities on the Bernoulli polynomials

2. Umbral calculus and Bernoulli polynomials

Let Pn = {p(x) ∈ C[x]: deg p(x) n} and let Sn(x) ∼ (g(t),t). From (16),wehave

1 xn = S (x) ⇔ xn = g(t)S (x)(n 0). (18) g(t) n n

1 t − ∈F Let us take g(t)= t (e 1) .Theng(t) is invertible series. By (1),weget

∞ B (x) 1 k tk = ext. (19) k! g(t) k=0

Thus by (19),wehave

1 xn = B (x)(n 0), (20) g(t) n JID:YJNTH AID:4725 /FLA [m1L; v 1.118; Prn:9/12/2013; 13:11] P.5 (1-12) D.S. Kim, T. Kim / Journal of Number Theory ••• (••••) •••–••• 5 and

tBn(x)=Bn(x)=nBn−1(x). (21)

1 t − From (20) and (21),wenotethatBn(x) is an Appell sequence for t (e 1). By (2),we get

x+y 1 B (u) du = B (x + y) − B (x) n n +1 n+1 n+1 x ∞ yk eyt − 1 = tk−1B (x)= B (x). (22) k! n t n k=1

In particular, for y =1,wehave

x+1 t t B (x)= B (u) du = xn. (23) n et − 1 n et − 1 x

By (15), we easily get 1 B (x)=t B (x) . (24) n n +1 n+1

From (24), we can derive the following equation:

y yt − e 1 yt 1 Bn(x) = e − 1 Bn+1(x) = Bn(u) du. (25) t n +1 0

For r ∈ N,then-th Bernoulli polynomials of order r are deﬁned by the generating function to be

∞ t r tn ext = B(r)(x) see [2,3] . (26) et − 1 n n! n=0

(r) (r) In the special case, x =0,Bn (0) = Bn are called the n-th Bernoulli numbers of order r.By(26),weget n n B(r)(x)= B(r) xr see [2,3] . (27) n l n−l l=0

Note that JID:YJNTH AID:4725 /FLA [m1L; v 1.118; Prn:9/12/2013; 13:11] P.6 (1-12) 6 D.S. Kim, T. Kim / Journal of Number Theory ••• (••••) •••–•••

(r) n ··· Bn = Bl1 Blr . (28) l1,...,lr l1+···+lr =n

(r) From (27) and (28),wenotethatBn (x) is a monic polynomial with coeﬃcients in Q. By (27),weget

x+y 1 B(r)(u) du = B(r) (x + y) − B(r) (x) n n +1 n+1 n+1 x ∞ yk eyt − 1 = tk−1B (x)= B(r)(x), (29) k! n t n k=1 and

(r) − (r) (r−1) Bn (x +1) Bn (x)=nBn−1 (x). (30)

From (29) and (30),wenotethat

x+1 et − 1 B(r)(x)= B(r)(u) du = B(r−1)(x). (31) t n n n x

By (31),weget

− t t r 1 t r B(r)(x)= B(r−1)(x)= B (x)= xn, (32) n et − 1 n et − 1 n et − 1 and t r tB(r)(x)=n xn−1 = nB(r) (x). (33) n et − 1 n−1

et−1 r F It is easy to show that ( t ) is an invertible series in . Therefore, by (32) and (33), we obtain the following lemma.

(r) et−1 r Lemma 1. Bn (x) is the Appell sequence for ( t ) .

By (33),weget 1 B(r)(x)=t B(r) (x) (n 0). (34) n n +1 n+1 JID:YJNTH AID:4725 /FLA [m1L; v 1.118; Prn:9/12/2013; 13:11] P.7 (1-12) D.S. Kim, T. Kim / Journal of Number Theory ••• (••••) •••–••• 7

Thus, from (34),wehave

y yt − e 1 (r) yt 1 (r) (r) B (x) = e − 1 B (x) = B (u) du. (35) t n n +1 n+1 n 0

In the special case, y =1,wehave

1 t − e 1 (r) (r) (r−1) B (x) = B (u) du = B . t n n n 0

By (17),weget t r n t t n i1 ··· ir t x = t x t x (36) e − 1 i1,...,ir e − 1 e − 1 n=i1+···+ir and r t n t n (r) x = Bn, x = B . (37) et − 1 et − 1 n

Thus, from (36) and (37),wehave

n ··· (r) Bi1 Bir = Bn . i!,...,ir n=i1+···+ir

Let us take p(x) ∈ Pn with

n p(x)= bkBk(x). (38) k=0

∼ et−1 From (20) and (21),wenotethatBn(x) ( t ,t). By the deﬁnition of Appell sequences, we get t − e 1 k t Bn(x) = n!δn,k (n, k 0), (39) t and, from (38),wehave t − n t − n e 1 k e 1 k t p(x) = bl t Bl(x) = bll!δl,k = k!bk. (40) t t l=0 l=0 JID:YJNTH AID:4725 /FLA [m1L; v 1.118; Prn:9/12/2013; 13:11] P.8 (1-12) 8 D.S. Kim, T. Kim / Journal of Number Theory ••• (••••) •••–•••

Thus, by (25) and (40),weget

1 t − t − 1 e 1 k 1 e 1 (k) 1 (k) bk = t p(x) = p (x) = p (u) du, (41) k! t k! t k! 0

(k) dk where p (u)= duk p(u). Therefore, by (38) and (41), we obtain the following theorem.

∈ P n Theorem 2. Let p(x) n with p(x)= k=0 bkBk(x). Then we have

1 t − 1 e 1 (k) 1 (k) bk = p (x) = p (u) du, k! t k! 0

(k) dk where p (u)= duk p(u).

(r) ∈ P n Let p(x)=Bn (x) n with p(x)= k=0 bkBk(x). Then we have n p(k)(x)=k! B(r) (x), (42) k n−k and t − t − 1 e 1 (k) n e 1 (r) bk = p (x) = B − (x) . (43) k! t k t n k

Therefore, by Theorem 2 and (43), we obtain the following corollary.

Corollary 3. For n 0, we have n t − (r) n e 1 (r) B (x)= B − (x) Bk(x). n k t n k k=0

In other words, n n − B(r)(x)= B(r 1)B (x), where n − k 0. n k n−k k k=0

From the deﬁnition of Appell sequences, we note that t − r e 1 k (r) t B (x) = n!δn,k (n, k 0). (44) t n JID:YJNTH AID:4725 /FLA [m1L; v 1.118; Prn:9/12/2013; 13:11] P.9 (1-12) D.S. Kim, T. Kim / Journal of Number Theory ••• (••••) •••–••• 9

∈ P n (r) (r) Let p(x) n with p(x)= k=0 bk Bk (x). By (44),weget t − r n t − r e 1 k (r) e 1 k (r) t p(x) = b t B (x) t l t l l=0 n (r) (r) = bl l!δl,k = k!bk . (45) l=0

Thus, by (45),wehave t − r (r) 1 e 1 k b = t p(x) . (46) k k! t

Therefore, by (46), we obtain the following theorem.

∈ P n (r) (r) Theorem 4. Let p(x) n with p(x)= k=0 bk Bk (x). Then we have t − r (r) 1 e 1 k b = t p(x) . k k! t

Let us consider p(x)=Bn(x)with

n (r) (r) Bn(x)=p(x)= bk Bk (x). (47) k=0

By Theorem 4 and (47),weget t − r t − r (r) 1 e 1 k 1 e 1 k b = t p(x) = t Bn(x) . (48) k k! t k! t

For k

Let k r.Thenby(48),weget (r) 1 t − r k−r 1 t − r k−r bk = e 1 t Bn(x) = e 1 t Bn(x) k! k! 1 n t r = (k − r)! e − 1 B − (x) k! k − r n+r k n r k−r r r−j jt = (−1) e B − (x) k j n+r k r! r j=0 n r k−r r r−j = (−1) B − (j). (50) k j n+r k r! r j=0

Therefore, by (47), (49) and (50), we obtain the following theorem.

Theorem 5. For n ∈ Z+ and r ∈ N, we have r−1 r r k r r−j (r) B (x)= (−1) B − (j)B (x) n n+r−k j n+r k k k=0 r! r−k j=0 n n r k−r r r−j (r) + (−1) B − (j)B (x). k j n+r k k k=r r! r j=0

3. Further remarks

For n, m ∈ Z+ with n − m 0, we have n n−m n − n − m − B(r)(x)B(r) (x)= B(r 1)B (x) B(r 1) B (x) n n−m l n−l l p n−m−p p l=0 p=0 2n−m k n − m n (r−1) (r−1) = B B B (x)B − (x), (51) p k − p n−m−p n−k+p p k p k=0 p=0 where n − m − p 0.

Let us consider p(x)=En(x)with

n En(x)=p(x)= bkBk(x). (52) k=0

Then we have (k) n p (x)=k! E − (x), (53) k n k and JID:YJNTH AID:4725 /FLA [m1L; v 1.118; Prn:9/12/2013; 13:11] P.11 (1-12) D.S. Kim, T. Kim / Journal of Number Theory ••• (••••) •••–••• 11 t − t − 1 e 1 (k) n e 1 bk = p (x) = En−k(x) k! t k t n E − (1) − E − n E − = n k+1 n k+1 = −2 n k+1 . (54) k n − k +1 k n − k +1

By (52) and (54),weget n n E − E (x)=−2 n k+1 B (x). (55) n k n − k +1 k k=0

From (55), we can derive the following equation.

En(x)En−m(x) n n−m n E − n − m E − − =4 n l+1 B (x) n m p+1 B (x) l n − l +1 l p n − m − p +1 p l=0 p=0 2n−m k n − m n En−m−l+1En−k+l+1 =4 B (x)B − (x), l k − l (n − m − l +1)(n − k + l +1) l k l k=0 l=0 where n, m ∈ Z+ with n − m 0.

Remark. Recently, R. Dere, Y. Simsek and H.M. Srivastava have studied special polynomials with viewpoint of umbral calculus (see [5,7,8]).

Acknowledgments

The authors would like to express their gratitude for the valuable comments and suggestions of referees. This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MOE) (No. 2012R1A1A2003786).

Appendix A. Supplementary material

The online version of this article contains additional supplementary material. Please visit http://dx.doi.org/10.1016/j.jnt.2013.09.013.

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