Frobenius Euler polynomials and umbral calculus in the p adic case

R E S E A R C H

Open Access

Frobenius-Euler polynomials and umbral

calculus in the

p-adic case

Dae San Kim

_{, Taekyun Kim}

_{, Sang-Hun Lee}

_{and Seog-Hoon Rim}

*_{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 somep-adic Frobenius-Euler measure related to umbral calculus in thep-adic case. Finally, we derive some identities of Frobenius-Euler polynomials from our study.

MSC: 05A10; 05A19

Keywords: Frobenius-Euler polynomials; umbral calculus;p-adic integral

1 Introduction

Letpbe a fixed prime number. Throughout this paperZp,QpandCpwill denote the ring ofp-adic integers, the field ofp-adic rational numbers and the completion of algebraic closure ofQp, respectively.

Forf∈Nwith (f,p) = , let

X=lim_← –

N

Zf pNZ;

a+fpNZp=

x∈X|x≡amodfpN, ≤a≤fpN– ,

X*=

<a<fp,(a,p)=

a+fpNZp

, N∈N(see [–]).

Note that the natural mapZfpN_Z→ZpN_Zinduces π:X→Zp.

Ifgis a function onZp, we denote by the samegthe functiong◦πonX. Namely, we can considergas a function onX.

Fork≥ andλ∈Cpwith| –λ|p> , the Frobenius-Euler measure onXis defined by

μλ

x+fpNZp

= λ fpN_–x

 –λfpN (see [, ]), (.)

where thep-adic absolute value onCpis normalized by|p|p=_p.

As is well known, the Frobenius-Euler polynomials are defined by the generating func-tion to be

 –λ

et_–λ

ext=eH(x|λ)t₌

∞

n= Hn(x|λ)

tn

n! (see [, , ]), (.)

with the usual convention about replacingHn_{(x|λ) byHn(x|λ). In the special case,x= ,}

Hn(|λ) =Hn(λ) are called thenth Frobenius-Euler numbers

Hn(x|λ) =H(λ) +xn=

∞

l=

n l

Hl(λ)xn–l (see [, , ]). (.)

Thus, by (.) and (.), we easily get

H(λ) + n–λHn(λ) = ( –λ)δ,n (see [–]), (.)

whereδn,kis the Kronecker symbol.

Forr∈N, the Frobenius-Euler polynomials of orderrare defined by the generating function

 –λ

et_–λ

r

ext=

 –λ

et_–λ

× · · · ×

 –λ

et_–λ

r-times

ext

=

∞

n=

H_n(r)(x|λ)t n

n! (see [, ]). (.)

In the special case,x= ,Hn(r)(|λ) =Hn(r)(λ) are called thenth Frobenius-Euler numbers of orderr. Thenth Frobenius-Euler polynomials can be represented by (.) as follows:

λHn(x|λ)  –λ =

X

(x+y)ndμλ(y) =

Zp

(x+y)ndμλ(y)

= lim

N→∞   –λpN

pN_–

y=

(x+y)n_λpN–y _{(see [, ]). (.)}

LetFbe the set of all formal power series in the variabletoverCpwith

F=

f(t) =

∞

k= ak

k!t k_a

k∈Cp

. (.)

LetP=Cp[x] andP*denote the vector space of all linear functionals onP. The formal power series

f(t) =

∞

k= ak

k!t

k_{∈F (see [, ]) (.)}

defines a linear functional onPby setting

f(t)|xn=an for alln≥. (.)

From (.) and (.), we have

Here,F denotes both the algebra of formal power series intand the vector space of all linear functionals onP, and so an elementf(t) ofF will be thought of as both a formal power series and a linear functional (see [, ]). We will callFthe umbral algebra. The umbral calculus is the study of umbral algebra (see [, ]).

The ordero(f(t)) of power seriesf(t) (= ) is the smallest integerkfor whichakdoes not vanish (see [, ]). A seriesf(t) for whicho(f(t)) =  is called a delta series. If a seriesf(t) haso(f(t)) = , thenf(t) is called an invertible series (see [, ]). Letf(t),g(t)∈F. Then we easily see thatf(t)g(t)|p(x)=f(t)|g(t)p(x)=g(t)|f(t)p(x). From (.), we note that

eyt|xn=yn, eyt|p(x)=p(y), (.)

f(t) =

∞

k=

f(t)|xk k! t

k_{, f(t)∈F, (.)}

and

p(x) =

∞

k=

tk_|p(x)

k! x

k_{, p(x)∈P(see []). (.)}

Forf(t),f(t), . . . ,fm(t)∈F, we have

f(t)f(t)· · ·fm(t)|xn

=

i+···+im=n

n i, . . . ,im

f(t)|xi

· · ·fm(t)|xim_{. (.)}

By (.), we get

p(k)(x) =d k_p(x)

dxk = n

l=k

tl|p(x)l

k

k!

l!x

l–k _(.)

and

p(k)() =tk|p(x)=|p(k)(x).

Thus, by (.), we get

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

dxk (see [, ]). (.)

By (.), we easily see that

eytp(x) =p(x+y) (see []). (.)

Let Sn(x) denote a polynomial of degreen. Suppose thatf(t),g(t)∈F witho(f(t)) =  and o(g(t)) = . Then there exists a unique sequence Sn(x) of polynomials satisfying g(t)f(t)k_|Sn(x)=n!δ

Forp(x)∈P, we have

f(t)|xp(x)=∂tf(t)|p(x)

=f(t)|p(x), (.)

eyt– |p(x)=p(y) –p(). (.)

IfSn(x)∼(g(t),f(t)), then we have

h(t) =

∞

k=

h(t)|Sk(x) k! g(t)f(t)

k_{, h(t)∈F, (.)}

p(x) =

∞

k=

g(t)f(t)k|p(x)

k! Sk(x), p(x)∈P, (.)

f(t)Sn(x) =nSn–(x), (.)

and



g(f¯(t))e y¯f(t)₌

∞

k= Sk(y)

k! t

k _{for ally∈Cp, (.)}

wheref¯(t) is compositional inverse off(t) (see [, ]). In [], Kim and Kim have stud-ied some identities of Frobenius-Euler polynomials arising from umbral calculus. In this paper, we study somep-adic Frobenius-Euler integral onZprelated to umbral calculus in thep-adic case. Finally, we derive some new and interesting identities of Frobenius-Euler polynomials from our study.

2 Frobenius-Euler polynomials associated with umbral calculus Let

g(t;λ) =e t_–λ

 –λ ∈F. (.)

Then we see thatg(t;λ) is an invertible series. From (.), we have

∞

k= Hk(x|λ)

tk

k! = 

g(t;λ)e

xt_{. (.)}

Hence, by (.), we get

 –λ

et_–λ

xn= 

g(t;λ)x

n_{=Hn(x|λ). (.)}

By (.) and (.), we get

Hn(x|λ)∼g(t;λ),t.

From (.), we have

Zp

e(x+y)tdμλ(y) = λ

et_–λe

and

Zp

e(x+y+)tdμλ(y) –λ

Zp

e(x+y)tdμλ(y) =λext. (.)

By (.), we get

Zp

(x+y+ )ndμλ(y) –λ

Zp

(x+y)ndμλ(y) =λxn. (.)

From (.) and (.), we have

λ

 –λHn(x+ |λ) – λ

 –λHn(x|λ) =λx

n_{. (.)}

From (.), we can easily derive

Hn+(x|λ) =

x–g

_(t;λ)

g(t;λ)

Hn(x|λ). (.)

By (.), we get

g(t;λ)Hn+(x|λ) =g(t;λ)xHn(x|λ) –g(t;λ)Hn(x|λ). (.)

Thus, from (.), we have

et–λHn+(x|λ) =

et–λxHn(x|λ) –etHn(x|λ). (.)

By (.), we get

Hn+(x+ |λ) –λHn+(x|λ) =x

Hn(x+ |λ) –λHn(x|λ)

. (.)

From (.), we note that

Hn(x+ |λ) –λHn(x|λ) =xHn–(x+ |λ) –λHn–(x|λ)

=xHn–(x+ |λ) –λHn–(x|λ)

=· · ·

=xnH(x+ |λ) –λH(x|λ)

=xn( –λ). (.)

Let us consider the linear functionalf(t) such that

f(t)|p(x)=

Zp

p(u)dμλ(u) (.)

for all polynomialsp(x) can be determined from (.) to be

f(t) =

∞

k=

f(t)|xk

k! t k₌

∞

k=

Zp

ukdμλ(u)t k

k! =

Zp

By (.) and (.), we get

f(t) =

Zp

eutdμλ(u) = λ

et_–λ. (.)

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

Theorem . For p(x)∈P,we have

λ

et_–λ

p(x)

=

Zp

p(u)dμλ(u).

In particular,

λ

 –λHn(λ) =

Zp

eytdμλ(y)xn

.

From (.), we have

∞

n=

Zp

(x+y)ndμλ(y)t n

n! =

Zp

e(x+y)tdμλ(y) =

∞

n=

Zp

eytdμλ(y)xnt

n

n!. (.)

By (.), (.) and (.), we get

λ

 –λHn(x|λ) =

Zp

eytdμλ(y)xn= λ

et_–λx

n_{, forn≥. (.)}

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

Theorem . For p(x)∈P,we have

Zp

p(x+y)dμλ(y) =

Zp

eytdμλ(y)p(x) = λ

et_–λp(x).

In particular,

λ

 –λHn(x|λ) =

Zp

eytdμλ(y)xn= λ

et_–λx

n _(n≥).

By (.) and (.), we get

λ

 –λHn(x|λ)∼

et_–λ

λ ,t

. (.)

From Appell identity and (.), we can derive the following identities:

Hn(x+y|λ) = n

k=

n k

Let

From (.), we can derive the following identity:

×

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

In particular,

for all polynomialsp(x) can be determined from (.) to be

f*(t) =

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

Theorem . For p(x)∈P,we have

In particular,

Indeed, thenth Frobenius-Euler number of orderris given by

Remark From (.) and (.), we note that

Continuing this process, we obtain the following equation:

dk

wherekis a positive integer.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the manuscript and typed, read, and approved the final manuscript.

Author details

1_{Department of Mathematics, Sogang University, Seoul, 121-741, Republic of Korea.}2_{Department of Mathematics,}

Kwangwoon University, Seoul, 139-701, Republic of Korea.3_{Division of General Education, Kwangwoon University, Seoul,}

139-701, Republic of Korea.4_{Department of Mathematics Education, Kyungpook National University, Taegu, 702-701,}

Republic of Korea.

Acknowledgements

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology 2012R1A1A2003786.

Received: 21 November 2012 Accepted: 6 December 2012 Published: 20 December 2012

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doi:10.1186/1687-1847-2012-222