Some properties of higher order Daehee polynomials of the second kind arising from umbral calculus

R E S E A R C H

Open Access

Some properties of higher-order Daehee

polynomials of the second kind arising from

umbral calculus

Dae San Kim

_{and Taekyun Kim}

*_{Correspondence: [email protected]} 2_{Department of Mathematics,}

Kwangwoon University, Seoul, 139-701, Republic of Korea Full list of author information is available at the end of the article

Abstract

In this paper, we study the higher-order Daehee polynomials of the second kind from the umbral calculus viewpoint and give various identities of the higher-order Daehee polynomials of the second kind arising from umbral calculus.

1 Introduction

Letk∈Z≥. The Daehee polynomials of the second kind of orderkare defined by the

generating function to be

( +t)log( +t) t

k

( +t)x= ∞

n=

ˆ

D(k)_n (x)t

n

n! ()

(see []).

Whenx= ,Dˆ(k)n =Dˆ(k)n () are called the Daehee numbers of the second kind of orderk.

The Stirling number of the first kind is defined by the falling factorial to be

(x)n=x(x– )· · ·(x–n+ ) = n

l=

S(n,l)xl. ()

Thus, by (), we get

log( +t)m=m! ∞

l=m

S(l,m)

tl

l! ()

(see [–]), wherem∈Z≥.

Forλ∈Cwithλ= , the Frobenius-Euler polynomials of orders(∈N) are given by

 –λ

et_–λ

s

ext= ∞

n=

H_n(s)(x|λ)t

n

n! ()

(see [–]).

Whenx= ,Hn(s)(λ) =Hn(s)(λ|) are called the Frobenius-Euler numbers of orders.

As is well known, the Bernoulli polynomials of orderk(∈N) are defined by the gener-ating function to be

t et_{– }

k

ext= ∞

n=

B(k)_n (x)t

n

n! ()

(see [–]).

Whenx= ,B(k)n =B(k)n () are called the Bernoulli numbers of orderk.

In this paper, we study the higher-order Daehee polynomials of the second kind with umbral calculus viewpoint and give various identities of the higher-order Daehee polyno-mials of the second kind arising from umbral calculus.

2 Umbral calculus

LetCbe the complex number field and letFbe the set of all formal power series

F=

f(t) = ∞

k=

ak

tk

k!

ak∈C .

LetP=C[x], and letP∗be the vector space of all linear functionals onP.L|p(x) indi-cates the action of the linear functionalLon the polynomialp(x). Then the vector space operations onP∗are given byL+M|p(x)=L|p(x)+M|p(x), andcL|p(x)=cL|p(x), wherecis a complex constant inC. Forf(t)∈F, the linear functional onPis defined by

f(t)|xn_=a

n. Then, in particular, we have

tk|xn=n!δn,k (n,k≥) ()

(see [, ]), whereδn,kis the Kronecker symbol.

LetfL(t) =

∞

k= L|xk

k! tk. By (), we getfL(t)|xn=L|xn. That is,L=fL(t). The map

L →fL(t) is a vector space isomorphism fromP∗ onto F. Henceforth,F denotes both

the algebra of the formal power series intand the vector space of all linear functionals onP, and so an elementf(t) ofFwill be thought of as both a formal power series and a linear functional. We callFthe umbral algebra and the umbral calculus is the study of the umbral algebra. The ordero(f(t)) of the power seriesf(t) (= ) is the smallest integer for which the coefficient oftk_{does not vanish. Ifo(f(t)) = , thenf(t) is called an invertible}

series; ifo(f(t)) = , thenf(t) is called a delta series.

Letf(t),g(t)∈F witho(f(t)) =  ando(g(t)) = . Then there exists a unique sequence sn(x) (degsn(x) =n) such thatg(t)f(t)k|sn(x)=n!δn,k, for n,k≥. The sequence sn(x)

is called the Sheffer sequence for (g(t),f(t)) which is denoted bysn(x)∼(g(t),f(t)). 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). ()

From (), we note that

f(t) = ∞

k=

f(t)|xkt

k

k!, p(x) =

∞

k=

tk|p(x)x

k

and, by (), we get

tkp(x) =p(k)(x) =d

k_p(x)

dxk and e

yt_{p(x) =p(x+y) ()}

(see [, ]).

Forsn(x)∼(g(t),f(t)), we have

dsn(x)

dx =

n–

l=

n l

f(t)|xn–lsl(x), ()

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

 g(f(t))e

xf(t)₌

∞

n=

sn(x)

tn

n!, for allx∈C, ()

f(t)sn(x) =nsn–(x) (n≥), sn(x) = n

j=

 j!

gf(t)–f(t)j|xnxj, ()

sn(x+y) = n

j=

n

j

sj(x)pn–j(y), ()

wherepn(x) =g(t)sn(x).

f(t)|xp(x)=∂tf(t)|p(x)

, ()

with∂tf(t) =df_dt(t), and

sn+(x) =

x–g

_(t)

g(t)



f(t)sn(x) (n≥) ()

(see [, ]).

Let us assume thatsn(x)∼(g(t),f(t)) andrn(x)∼(h(t),l(t)). Then we see that

sn(x) = n

m=

Cn,mrm(x) (n≥), ()

where

Cn,m=

 m!

h(f(t)) g(f(t))l

f(t)mxn

()

(see [, ]).

3 Higher-order Daehee polynomials of the second kind

By (), we see that

ˆ

D(k)_n (x)∼

et_{– }

tet

k

,et– 

From (), we have

et– 

tet

k

ˆ

D(k)_n (x)∼,et–  and (x)n∼

,et– . ()

By (), we get

ˆ

D(k)_n (x) =

tet

et_{– }

k

(x)n

=

n

m=

S(n,m)

tet

et_{– }

k

xm

=

n

m=

S(n,m)ektB(k)n (x)

=

n

m=

S(n,m)B(k)m(x+k). ()

From () and (), we have

ˆ

D(k)_n (x) =

n

j=

 j!

( +t)log( +t) t

k

log( +t)jxn

xj, ()

where

( +t)log( +t) t

k

log( +t)jxn

=

log( +t) t

k+j

( +t)ktjxn

= (n)j

log( +t) t

k+j

min{k,n–j}

m=

k m

tmxn–j

= (n)j n–j

m=

k m

(n–j)m

∞

l=

(k+j)!

(l+k+j)!S(l+k+j,k+j)

tl|xn–j–m

= (n)j n–j

m=

k m

(n–j)m

(k+j)!

(n+k–m)!S(n+k–m,k+j)(n–j–m)!

= (n)j n–j

m=

k m

(n–j)m

S(n+k–m,k+j)

_n+k–m

k+j

. ()

Therefore, by () and (), we obtain the following theorem.

Theorem  For n∈Z≥and k≥,we have

ˆ

D(k)_n (x) =

n

j=

n

j n–j

m=

k m

(n–j)m

S(n+k–m,k+j)

_n+k–m

k+j

By () and (), we get

ˆ

D(k)_n (y) =

_∞

l=

ˆ

D(k)_l (y)t

l

l! xn

=

log( +t) t

k

( +t)y( +t)kxn

=

≤r≤min{k,n}

k r

(n)r

log( +t) t

k

( +t)yxn–r

=

≤r≤min{k,n}

k r

(n)r

≤m≤n–r

y m

(n–r)m

×

≤l≤n–r–m

k!S(l+k,k)

(l+k)!

tl|xn–r–m

=

≤r≤n

≤m≤n–r

(n)r

_k

r

_n–r

m

_n–r–m+k

k

S(n–r–m+k,k)(y)m. ()

Therefore, by (), we obtain the following theorem.

Theorem  For n≥,we have

ˆ

D(k)_n (x)

=

≤m≤n ≤r≤n–m

(n)r

_k

r

_n–r

m

_n–r–m+k

k

S(n–r–m+k,k)

(x)m

=

≤m≤n ≤r≤n–m

(n)r

_k

r

_n–r

n–m

_m–r+k

k

S(m–r+k,k)

(x)n–m.

From () and (), we have

et– Dˆ(k)_n (x) =nDˆ(k)_n–(x) () and

et– Dˆ(k)_n (x) =Dˆ(k)_n (x+ ) –Dˆ(k)_n (x). Thus, by (), we get

ˆ

D(k)_n (x+ ) –Dˆ(k)_n (x) =nDˆ(k)_n–(x) (n≥). () From () and (), we derive the following equation:

ˆ

D(k)_n+(x) =

x+ke

t_{–  –t}

t(et_{– )}

e–tDˆ(k)_n (x)

=xDˆ(k)_n (x– ) +ke–te

t_{–  –t}

t(et_{– )}Dˆ (k)

where

e–te

t_{–  –t}

t(et_{– )}Dˆ (k) n (x)

=e–te

t_{–  –t}

t(et_{– )}

≤j≤n

n

j

≤m≤n–j

m!_mkn–j_{m n+k–m}

k+j

×S(n+k–m,k+j)

xj

=

≤j≤n

n

j

≤m≤n–j

m!_mkn–j_{m n+k–m}

k+j

×S(n+k–m,k+j)e–t

et_{–  –t}

t(et_{– )}x j

=

≤j≤n

n

j

≤m≤n–j

m!m+k_m n–j_{m n+k–m}

k+j

×S(n+k–m,k+j)e–t

et_{–  –t}

et_{– }

xj+

j+ 

=

≤j≤n

n

j

≤m≤n–j

m!m+k_m n–j_{m n+k–m}

k+j

×S(n+k–m,k+j)

j+  e

–t_xj+_–B j+(x)

=

≤j≤n

n

j

≤m≤n–j

m!m+k_m n–j_{m n+k–m}

k+j

×S(n+k–m,k+j)

j+  e

–t_{(x– )}j+_–B

j+(x– )

. ()

Therefore, from () and (), we obtain the following theorem.

Theorem  For n≥,k≥,we have

ˆ

D(k)_n+(x)

=xDˆ(k)_n (x– ) +k

≤j≤n

n

j

≤m≤n–j

m!m+k_m n–j_{m n+k–m}

k+j

× S(n+k–m,k+j)

j+ 

(x– )j+–Bj+(x– )

.

Now, we observe that

e–te

t_{–  –t}

t(et_{– )}Dˆ (k) n (x)

=

n

j=

n j

_n+k

j+k

S(n+k,j+k)e–t

et_{–  –t}

= n j= _n j _n+k j+k

S(n+k,j+k)e(k–)t

et_{–  –t}

t(et_{– )}x j = n j= _n j _n+k j+k

S(n+k,j+k)

j+  e

(k–)t_xj+_–B j+(x) = n j= _n j _n+k j+k

S(n+k,j+k)

j+ 

(x+k– )j+–Bj+(x+k– )

. ()

Thus, by (), we get

ˆ

D(k)_n+(x) =xDˆ(k)_n (x– ) +k

n j= _n j _n+k j+k

S(n+k,j+k)

j+ 

(x+k– )j+–Bj+(x+k– )

.

From () and (), we note that

d dxDˆ

(k) n (x) =n!

n–

l=

(–)n–l–

l!(n–l)Dˆ

(k)

l (x). ()

By () and (), we see that

ˆ

D(k)_n (y) =

_∞

l=

ˆ

D(k)_l (y)t

l

l! xn

(n≥)

=

( +t)log( +t) t

k

( +t)yxn

=

∂t

( +t)log( +t) t

k

( +t)y xn– = ∂t

( +t)log( +t) t

k

( +t)yxn–

+y

( +t)log( +t) t

k

( +t)y–xn–

=yDˆ(k)_n–(y– ) +k

( +t)log( +t) t

k–

( +t)y

log( +t) +  –( +t)log( +t) t

xn

n

=yDˆ(k)_n–(y– ) + k n

( +t)log( +t) t

k–

( +t)ylog( +t)xn

+k n

( +t)log( +t) t

k–

( +t)yxn

–k n

( +t)log( +t) t

k

( +t)yxn

=yDˆ(k)_n–(y– ) + k nDˆ

(k–) n (y) –

k nDˆ

(k) n (y)

+k n

≤l≤n

(–)l–_(n) l

l

( +t)log( +t) t

k–

( +t)y_xn–l

Thus, by (), we get

ˆ

D(k)_n (x) = n n+kxDˆ

(k)

n–(x– ) +

k n+kDˆ

(k–) n (x)

+ k n+k

≤l≤n

(–)l–

n l

(l– )!Dˆ(k–)_n–l (x). ()

Therefore, by (), we obtain the following theorem.

Theorem  For n≥,k≥,we have

ˆ

D(k)_n (x) = n n+kxDˆ

(k)

n–(x– ) +

k n+kDˆ

(k–) n (x)

+ k n+k

≤l≤n

(–)l–

n

l

(l– )!Dˆ(k–)_n–l (x).

Now, we compute((+t)log_t (+t))k_{(log( +t))}m_|xn_{in two different ways:}

( +t)log( +t) t

k

log( +t)mxn

=

( +t)log( +t) t

k

log( +t)mxn

=

≤l≤n–m

m!

(l+m)!S(l+m,m)(n)l+m

( +t)log( +t) t

k

xn–l–m

=

≤l≤n–m

m!

n l+m

S(l+m,m)Dˆ(k)n–l–m

=

≤l≤n–m

m!

n l

S(n–l,m)Dˆ(k)l . ()

On the other hand,

( +t)log( +t) t

k

log( +t)mxn

=

∂t

( +t)log( +t) t

k

log( +t)m

xn–

=k

( +t)log( +t) t

k–_{log( +t) +  –}(+t)log(+t) t

t

log( +t)mxn–

+m

( +t)log( +t) t

k

( +t)–log( +t)m–xn–

=k n

( +t)log( +t) t

k–

log( +t)m+xn

+k n

( +t)log( +t) t

k–

–k n

( +t)log( +t) t

k

log( +t)mxn

+m

( +t)log( +t) t

k

( +t)–log( +t)m–xn–

. ()

Thus, by (), we get

n+k n

( +t)log( +t) t

k

log( +t)mxn

=k n

( +t)log( +t) t

k–

log( +t)m+xn

+k n

( +t)log( +t) t

k–

log( +t)mxn

+m

( +t)log( +t) t

k

( +t)–log( +t)m–xn–

. ()

From (), we derive the following equation:

n+k k

≤l≤n–m

m!

n l

S(n–l,m)Dˆ(k)_l

=k n

≤l≤n–m–

(m+ )!

n l

S(n–l,m+ )Dˆ(k–)l

+k n

≤l≤n–m

m!

n l

S(n–l,m)Dˆ(k–)_l

+m

≤l≤n–m

(m– )!

n–  l

S(n–l– ,m– )Dˆ(k)l (–). ()

Therefore, by (), we obtain the following theorem.

Theorem  For n– ≥m≥,we have

n–m

l=

n

l

S(n–l,m)Dˆ(k)l

=k(m+ ) n+k

≤l≤n–m–

n

l

S(n–l,m+ )Dˆ(k–)l

+ k n+k

≤l≤n–m

n l

S(n–l,m)Dˆ(k–)_l

+ n n+k

≤l≤n–m

n– 

l

S(n–l– ,m– )Dˆ(k)l (–).

ForDˆ(k)n (x)∼((e t_–

tet )k,et– ), and (x)n∼(,et– ), let us assume that

ˆ

D(k)_n (x) =

n

m=

Then, by () and (), we get

Cn,m=

 m!

( +t)log( +t) t

k

tmxn

=

n m

( +t)log( +t) t

k

xn–m

=

n m

ˆ

D(k)_n–m. ()

Therefore, by () and (), we obtain the following theorem.

Theorem  For n≥,we have

ˆ

D(k)_n (x) =

≤m≤n

n m

ˆ

D(k)_n–m(x)m

=

≤m≤n

m!

n m

ˆ

D(k)_n–m

x m

.

Now, we consider the following two Sheffer sequences:

ˆ

D(k)_n (x)∼

et_{– }

tet

k

,et– 

()

and

H_n(s)(x|λ)∼

et_–λ

 –λ

s

,t

, s∈N,λ∈Cwithλ= .

Let

ˆ

D(k)_n (x) =

n

m=

Cn,mHm(s)(x|λ). ()

Here

Cn,m=

 m!( –λ)s

( +t)log( +t) t

k

log( +t)m( –λ+t)sxn

= 

m!( –λ)s n

j=

s j

( –λ)s–j(n)j

×

( +t)log( +t) t

k

log( +t)mxn–j

=

n–m

j=

s j

( –λ)–j(n)j n–m–j

l=

n–j l+m

S(l+m,m)Dˆ(k)_{n–j–l–m}

=

n–m

j= n–m–j

l=

s j

n–j

l

(n)j( –λ)–jS(n–j–l,m)Dˆ(k)l . ()

Theorem  For n≥,k≥andλ∈Cwithλ= ,we have

ˆ

D(k)_n (x) =

n

m=

_n–m

j= n–m–j

l=

s j

n–j

l

(n)j

×( –λ)–jS(n–j–l,m)Dˆ(k)l Hm(s)(x|λ).

We consider the following two Sheffer sequences:

ˆ

D(k)_n (x)∼

et_{– }

tet

k

,et– 

, B(s)_n (x)∼

et_{– }

t s

,t

.

Let

ˆ

D(k)_n (x) =

n

m=

Cn,mB(s)m(x). ()

Here

Cn,m=

 m!

₍ t

log(+t)) s

( t (+t)log(+t))k

log( +t)mxn

=  m!

( +t)s(

t (+t)log(+t))s

(_(+t)logt _(+t))k

log( +t)mxn

. ()

Case . Fors>k, we have

Cn,m=

 m!

t ( +t)log( +t)

s–k

log( +t)m( +t)sxn

=  m!

≤j≤n

s j

(n)j

t ( +t)log( +t)

s–k

log( +t)mxn–j

=

≤j≤n–m

s j

(n)j

m≤l≤n–j

S(l,m)

×

n–j

l

t ( +t)log( +t)

s–k

xn–j–l

=

≤j≤n–m

m≤l≤n–j

s j

n–j

l

(n)jS(l,m)Cˆ_n–j–l(s–k), ()

whereCˆ_i(s–k)is theith Cauchy number of the second kind of orders–k(see []). Case . Fors=k, we have

Cn,m=

 m!

log( +t)m|( +t)sxn

=  m!

log( +t)m

s

j=

s j

=

≤j≤n–m

s j (n)j

m≤l<∞ S(l,m)

l!

tl|xn–j

=

≤j≤n–m

s j

(n)jS(n–j,m). ()

Case . Fors<k, we have

Cn,m=

 m!

( +t)log( +t) t

k–s

log( +t)m( +t)sxn

=

≤j≤n–m

m≤l≤n–j

s j

n–j

l

(n)jS(l,m)Dˆ(k–s)n–j–l. ()

Therefore, by (), (), (), (), and (), we obtain the following theorem.

Theorem  Let n≥,we have:

(I) Fors>k,we have

ˆ

D(k)_n (x) =

≤m≤n ≤j≤n–m

m≤l≤n–j

s j

n–j

l

×(n)jS(l,m)Cˆ(s–k)n–j–l

B(s)_m(x).

(II) Fors=k,we have

ˆ

D(k)_n (x) =

≤m≤n ≤j≤n–m

s j

(n)jS(n–j,m)

B(s)_m(x).

(III) Fors<k,we have

ˆ

D(k)_n (x) =

≤m≤n ≤j≤n–m

m≤l≤n–j

s j

n–j

l

×(n)jS(l,m)Dˆ(k–s)_n–j–l

B(s)_m(x).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to this work. All authors read and approved the final manuscript.

Author details

1_{Department of Mathematics, Sogang University, Seoul, 121-742, Republic of Korea.}2_{Department of Mathematics,}

Kwangwoon University, Seoul, 139-701, Republic of Korea.

Acknowledgements

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

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10.1186/1029-242X-2014-195