Bernoulli polynomials of the second kind and their identities arising from umbral calculus

Research Article

Bernoulli polynomials of the second kind and their identities arising from umbral calculus

Taekyun Kim^a,b, Dae San Kim^c,∗, Dmitry V. Dolgy^d, Jong-Jin Seo^e

aDepartment of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 300387, China.

bDepartment of Mathematics, Kwangwoon University, Seoul 139-701, S. Korea.

cDepartment of Mathematics, Sogang University, Seoul 121-742, Republic of Korea.

dSchool of Natural Sciences, Far Eastern Federal University, 690950 Vladivostok, Russia.

eDepartment of Applied Mathematics, Pukyong National University, Pusan 608-739, S. Korea.

Communicated by S.-H. Rim

Abstract

In this paper, we study the Bernoulli polynomials of the second kind with umbral calculus viewpoint and derive various identities involving those polynomials by using umbral calculus. c2016 All rights reserved.

Keywords: Bernoulli polynomial of the second kind, umbral calculus.

2010 MSC: 05A40, 11B68, 11B83.

1. Introduction and preliminaries

As is well known, the ordinary Bernoulli polynomials are defined by the generating function to be t

e^t−1e^xt=

∞

X

n=0

Bn(x)tⁿ

n!, (see [2, 10, 15, 17]). (1.1)

When x= 0, B_n =B_n(0) are called the Bernoulli numbers. The Bernoulli polynomials of the second kind are given by the generating function to be

t

log (1 +t)(1 +t)^x=

∞

X

n=0

bn(x)tⁿ

n!, (see [19, 21, 22]). (1.2)

∗Corresponding author

Email addresses: [email protected], [email protected](Taekyun Kim),[email protected](Dae San Kim), [email protected](Dmitry V. Dolgy ),[email protected](Jong-Jin Seo)

Received 2015-10-08

When x= 0, b_n=b_n(0) are called the Bernoulli numbers of the second kind.

The first few Bernoulli numbers bnof the second kind are b₀= 1, b₁= 1

2, b₂=− 1

12, b₃ = 1

24, b₄ =− 19

720, b₅= 3 160,· · · . By (1.2), we easily get

b_n(x) =

n

X

l=0

n l

b_l(x)_n−l, (1.3)

where (x)_n=x(x−1)· · ·(x−n+ 1), (n≥0), and

b_n(x) =B_n⁽ⁿ⁾(x+ 1), (see [21, 22]), (1.4) whereB^(α)n (x) are the Bernoulli polynomials of orderα.

The stirling number of the second kind is given by xⁿ=

n

X

l=0

S2(n, l) (x)_l, (n≥0). (1.5)

The Stirling number of the first kind is defined by (x)_n=x(x−1)· · ·(x−n+ 1) =

n

X

l=0

S1(n, l)x^l, (n≥0). (1.6) LetCbe the complex number field and letF be the set of all formal power series in the variablet:

F= (

f(t) =

∞

X

k=0

a_kt^k k!

a_k ∈C )

. (1.7)

Let us assume thatPis the algebra of polynomials in the variablexoverCandP^∗ is the vector space of all linear functionals on P. hL|p(x)i denotes the action of the linear functional L on a polynomial p(x). For f(t)∈ F, we define the continuous linear functional f(t) onP by

hf(t)|xⁿi=a_n, (n≥0), (see [21]). (1.8) Thus, by (1.7) and (1.8), we get

D t^k

xⁿE

=n!δ_n,k, (n, k≥0), (see [1–22]), (1.9) whereδ_n,k is the Kronecker’s symbol.

For fL(t) =

∞

P

k=0

h^L|^xki

k! t^k, we have hf_L(t)|xⁿi =hL|xⁿi,(n≥0). Thus, we see that fL(t) = L. The mapL7→f_L(t) is a vector space isomorphism fromP^∗ ontoF. Henceforth, F is thought of as both a formal power series and a linear functional. We call F the umbral algebra. The umbral calculus is the study of umbral algebra. The order o(f(t)) of the non-zero power series f(t) is the smallest integer k for which the coefficient of t^k does not vanish (see [8, 21]). If o(f(t)) = 1, then f(t) is called a delta series and if o(f(t)) = 0, thenf(t) is called an invertible series. Forf(t), g(t)∈ F with o(f(t)) = 1 ando(g(t)) = 0, there exists a unique sequencesn(x) of polynomials such that

D

g(t)f(t)^k sn(x)

E

=n!δ_n,k, wheren, k≥0.

The sequences_n(x) is called the Sheffer sequence for (g(t), f(t)) which is denoted bys_n(x)∼(g(t), f(t)).

For p(x) ∈P, we have e^yt

p(x)

=p(y) and e^ytp(x) = p(x+y). Let f(t)∈ F and p(x) ∈P. Then we see that

f(t) =

∞

X

k=0

f(t) x^k

k! t^k, p(x) =

∞

X

k=0

t^k p(x)

k! x^k. (1.10)

Thus, by (1.10), we get

p^(k)(0) = D

t^k p(x)

E ,

D

1|p^(k)(x) E

=p^(k)(0). (1.11)

From (1.11), we have

t^kp(x) =p^(k)(x) = d^kp(x)

dx^k , (k≥0). Forsn(x)∼(g(t), f(t)), we have

ds_n(x) dx =

n−1

X

l=0

n l

D f(t)

x^n−lE

s_l(x), (n≥1), (1.12)

wheref(t) is the compositional inverse off(t) withf(f(t)) =f f(t)

=t, 1

g f(t) =e^xf(t)=

∞

X

n=0

sn(x)tⁿ

n!, for allx∈C, (1.13)

f(t)sn(x) =nsn−1(x), (n≥1), sn(x+y) =

n

X

j=0

n j

sj(x)pn−j(y), (1.14) wherep_n(x) =g(t)s_n(x),

hf(t)|xp(x)i=h∂_tf(t)|p(x)i, (1.15) and

sn+1(x) =

x−g⁰(t) g(t)

1

f⁰(t)sn(x), (n≥0), (see [1, 13, 16, 21]). (1.16) Assume thatpn(x)∼ 1, f(t)

and qn(x)∼ 1, g(t)

. Then the transfer formula is given by q_n(x) =x

f(t) g(t)

n

x⁻¹p_n(x) (n≥1). (1.17)

Forsn(x)∼(g(t), f(t)), rn(x)∼(h(t), l(t)), we have sn(x) =

n

X

m=0

Cn,mrm(x), (n≥0), (1.18)

where

Cn,m = 1 m!

*h f(t)

g f(t) l f(t)m

x^m +

, (see [12, 21]). (1.19)

In this paper, we study the Bernoulli polynomials of the second kind with umbral calculus viewpoint and derive various identities involving those polynomials by using umbral calculus.

2. Bernoulli polynomials of the second kind

For α∈N, the Bernoulli polynomials of the second kind with orderα are defined by t

log (1 +t) α

(1 +t)^x =

∞

X

n=0

b^(α)_n (x)tⁿ

n!. (2.1)

Note that b_n(x) =b⁽¹⁾n (x). When x= 0,b^(α)n =b^(α)n (0) are called the Bernoulli numbers of the second kind with orderα. Indeed, we note that

b^(α)_n (x) =B_n^(n−α+1)(x+ 1). Let us consider the following two sheffer sequences :

qn(x)∼

1,

log (1 +t) t

α

e^t−1

and

(x)_n∼ 1, e^t−1 . Thus, by (1.17), we get

q_n(x) =x

t log (1 +t)

αn

x⁻¹(x)_n

=x

t log (1 +t)

αn

(x−1)_n−1

=xb^(αn)_n−1 (x−1), (n≥1). That is,

xb^(αn)_n−1 (x−1)∼

1,

log (1 +t) t

α

e^t−1

. From (1.2) and (1.13), we have

b_n(x)∼ t

e^t−1, e^t−1

. (2.2)

By (2.2), we get

t

e^t−1b_n(x)∼ 1, e^t−1

, (x)_n∼ 1, e^t−1

. (2.3)

Thus, we see that

b_n(x) = e^t−1

t (x)_n= e^t−1 t

n

X

l=0

S₁(n, l)x^l (2.4)

= e^t−1

n

X

l=0

S₁(n, l) l+ 1 x^l+1

=

n

X

l=0

S1(n, l) l+ 1

(x+ 1)^l+1−x^l+1 .

When x= 0, we have

bn=

n

X

l=0

S1(n, l) l+ 1 . By (1.12), we get

d

dxb_n(x) =

n−1

X

l=0

n l

D

log (1 +t)|x^n−lE

b_l(x) (2.5)

=

n−1

X

l=0

n l

* _∞ X

m=1

(−1)^m−1 m t^m

x^n−l +

bl(x)

=

n−1

X

l=0

n l

(n−l−1)! (−1)^n−l−1b_l(x)

=

n−1

X

l=0

n!

l!(n−l)(−1)^n−l−1bl(x), (n≥1). Therefore, by (2.5), we obtain the following lemma.

Lemma 1. Forn≥1, we have d

dxbn(x) =

n−1

X

l=0

n!

l!(n−l)(−1)^n−l−1b_l(x). From (1.9), we have

b_n(y) = t

log (1 +t)

(1 +t)^y

xⁿ

(2.6)

=

* t log (1 +t)

∞

X

m=0

(y)_m t^m m!xⁿ

+

=

n

X

m=0

(y)_m n

m

t log (1 +t)

x^n−m

=

n

X

m=0

(y)_m n

m

bn−m.

Therefore, by (2.6), we obtain the following proposition.

Proposition 2. Forn≥0, we have

b_n(x) =

n

X

m=0

n m

bn−m(x)_m

=

n

X

m=0

m!

n m

x m

bn−m.

By (1.2), we get

bn(x) = t

log (1 +t)(x)_n=

n

X

l=0

S1(n, l)

t log (1 +t)

x^l (2.7)

=

n

X

l=0

S1(n, l)

l

X

m=0

bm

m!t^mx^l

=

n

X

l=0

S1(n, l)

l

X

m=0

l m

bmx^l−m

=

n

X

l=0 l

X

m=0

S1(n, l) l

m

bl−mx^m.

By (1.14), we get

b_n(x+y) =

n

X

j=0

n j

b_j(x) (y)_n−j. (2.8)

Let

Pn={p(x)∈C[x]|degp(x)≤n}, (n≥0).

Then it is an (n+ 1)-dimensional vector space overC. Now, we consider the polynomial p(x) in Pn which is given by

p(x) =

n

X

m=0

Cmbm(x). (2.9)

Thus, by (2.9), we get

t

e^t−1 e^t−1m

p(x)

=

n

X

l=0

C_l t

e^t−1 e^t−1m

b_l(x)

(2.10)

=

n

X

l=0

C_ll!δ_m,l=m!Cm. From (2.10), we have

Cm= 1 m!

t e^t−1

e^t−1m

p(x)

. (2.11)

Therefore, by (2.11), we obtain the following theorem.

Theorem 3. Let p(x)∈Pn with

p(x) =

n

X

m=0

Cmbm(x). Then, we have

C_m= 1 m!

t e^t−1

e^t−1m

p(x)

.

For example, let us takep(x) =Bn(x)∈Pn. Then, we have Bn(x) =

n

X

m=0

Cmbm(x), (2.12)

where

Cm = 1 m!

t e^t−1

e^t−1m

Bn(x)

(2.13)

=

n

X

l=m

S₂(l, m) n

l

t e^t−1

B_n−l(x)

=

n

X

l=m

S2(l, m) n

l ^n−l

X

k=0

Bn−l−k

n−l k

t e^t−1

x^k

=

n

X

l=m n−l

X

k=0

S2(l, m) n

l

n−l k

Bn−l−kBk.

Therefore, by (2.12) and (2.13), we obtain the following theorem.

Theorem 4. Forn≥0, we have B_n(x) =

n

X

m=0

( _n X

l=m n−l

X

k=0

n l

n−l k

S₂(l, m)Bn−l−kB_k )

b_m(x). Remark. From (2.13), for m≥1, we have

C_m = 1 m!

t e^t−1

e^t−1m

B_n(x)

(2.14)

= 1 m!

D

e^t−1m−1

tB_n(x)E

= n m!

D

e^t−1m−1

Bn−1(x)E

= n

m!(m−1)!

n−1

X

l=m−1

S2(l, m−1)1 l!

D t^l

Bn−1(x) E

= n m

n−1

X

l=m−1

S₂(l, m−1)

n−1 l

Bn−1−l.

Therefore, by (2.12) and (2.14), we get B_n(x) =

n

X

m=1

(n m

n−1

X

l=m−1

S₂(l, m−1)

n−1 l

Bn−1−l

)

b_m(x) +

n

X

k=0

n k

Bn−kB_k.

The classical polylogarithm function is given by Lik(x) =

∞

X

n=1

xⁿ

n^k, (k∈Z, x >0). (2.15) The poly-Bernoulli polynomials are defined by the generating function to be

Li_k 1−e^t e^t−1 e^xt=

∞

X

n=0

B_n^(k)(x)tⁿ

n!. (2.16)

Thus, by (2.16), we see that

B^(k)_n (x)∼

e^t−1 Lik(1−e^−t), t

. (2.17)

Let us take p(x) =Bn^(k)(x)∈Pn.Then we have B_n^(k)(x) =

n

X

m=0

Cmbm(x), (2.18)

where

Cm = 1 m!

t

e^t−1 e^t−1m

B_n^(k)(x)

(2.19)

=

n

X

l=m

S2(l, m) n

l

t e^t−1

B_n−l^(k) (x)

=

n

X

l=m

S2(l, m) n

l n−l

X

j=0

n−l j

B^(k)_n−l−j

t e^t−1

x^j

=

n

X

l=m n−l

X

j=0

n l

n−l j

S2(l, m)B_n−l−j^(k) Bj,

where B^(k)n = Bn^(k)(0) are the poly-Bernoulli numbers. Therefore, by (2.18) and (2.19), we obtain the following theorem.

Theorem 5. Forn≥0, we have B_n^(k)(x) =

n

X

m=0







n

X

l=m n−l

X

j=0

n l

n−l j

S2(l, m)B_n−j−l^(k) Bj







bm(x). Let us considerp(x) =xⁿ∈Pn. Then, we have

xⁿ=

n

X

m=0

C_mb_m(x), (2.20)

where

Cm= 1 m!

t e^t−1

e^t−1m

xⁿ

(2.21)

=

n

X

l=m

S₂(l, m) n

l

t e^t−1

x^n−l

=

n

X

l=m

S₂(l, m) n

l

Bn−l.

Thus, by (2.20) and (2.21), we get xⁿ=

n

X

m=0

( _n X

l=m

S₂(l, m) n

l

Bn−l

)

b_m(x). (2.22)

Let us consider the following two Sheffer sequences : b_n(x)∼

t

e^t−1, e^t−1

, (2.23)

and

B^(k)_n (x)∼

e^t−1 Li_k(1−e^−t), t

. Then, by (1.17) and (1.18), we get

B_n^(k)(x) =

n

X

m=0

Cn,mbm(x), (2.24)

where

C_n,m= 1 m!

*Li_k 1−e^−t e^t−1

t e^t−1

e^t−1m

xⁿ +

(2.25)

=

n

X

l=m

S₂(l, m) n

l *

Li_k 1−e^−t e^t−1

t e^t−1x^n−l

+

=

n

X

l=m

S₂(l, m) n

l ^n−l

X

j=0

n−l j

B_n−l−j

*Li_k 1−e^−t e^t−1

x^j +

=

n

X

l=m n−l

X

j=0

n l

n−l j

S₂(l, m)B_n−l−jB_j^(k). Therefore, by (2.24) and (2.25), we obtain the following theorem.

Theorem 6. Forn≥0, we have B_n^(k)(x) =

n

X

m=0







n

X

l=m n−l

X

j=0

n l

n−l j

S2(l, m)Bn−l−jB_j^(k)







bm(x). Let us consider the following Sheffer sequences:

bn(x)∼ t

e^t−1, e^t−1

, (2.26)

B_n(x)∼

e^t−1 t , t

. Then we have

b_n(x) =

n

X

m=0

C_n,mB_m(x), (2.27)

where

Cn,m = 1 m!

t log (1 +t)

t

log (1 +t)(log (1 +t))^m

xⁿ

(2.28)

=

n

X

l=m

n l

S1(l, m)

* t log (1 +t)

2

x^n−l +

=

n

X

l=m

n l

S1(l, m)b⁽²⁾_n−l,

whereb⁽²⁾n are di-Bernoulli numbers of the second kind.

Therefore, by (2.27) and (2.28), we get bn(x) =

n

X

m=0 n

X

l=m

n l

S1(l, m)b⁽²⁾_n−l

!

Bm(x). (2.29)

Acknowledgement

This paper is supported by grant No. 14-11-00022 of Russian Scientific fund.

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