Degenerate q Euler polynomials

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R E S E A R C H

Open Access

Degenerate

q-Euler polynomials

Taekyun Kim

1*

_{, Dae San Kim}

2

_{and Dmitry V Dolgy}

3

*_{Correspondence:}

[email protected]

1_{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

Recently, some identities of degenerate Euler polynomials arising fromp-adic fermionic integrals onZpwere introduced in Kim and Kim (Integral Transforms Spec. Funct. 26(4):295-302, 2015). In this paper, we study degenerateq-Euler polynomials which are derived fromp-adicq-integrals onZp.

MSC: 11B68; 11S80

Keywords: degenerate Euler polynomials;p-adicq-fermionic integral

1 Introduction

Letpbe a ﬁxed odd prime number. Throughout this paper,Zp,Qp andCpwill denote the ring ofp-adic integers, the ﬁeld ofp-adic rational numbers and the completion of algebraic closure ofQp, respectively. Letνpbe the normalized exponential valuation inCp with|p|p=p–νp(p)=_p.

Letqbe an indeterminate inCpsuch that| –q|p<p–



p–_{. The}_q_{-extension of}_x_{is deﬁned}

as [x]q=–q_–qx. Note thatlimq→[x]q=x. Forf ∈C(Zp) ={f |,f is aCp-valued continuous

function onZp}, the fermionicp-adicq-integral onZpis deﬁned by Kim to be

I–q(f) =

Zp

f(x)dμ–q(x) = lim N→∞

 [pN_]–q

pN_–

x=

f(x)(–q)x (see [, ]), (.)

where [x]–q=–(–q)_+qx. By (.), we easily get

qI–q(f) +I–q(f) = []qf()

f(x) =f(x+ )

, (.)

and

qnI–q(fn) + (–)n–I–q(f) = []q n–

l=

(–)n––lqlf(l) (n∈N), (.)

wherefn(x) =f(x+n) (see [–]).

The ordinary fermionicp-adic integral onZpis deﬁned as

lim

q→I–q(f) =I–(f) =

Zp

f(x)dμ–(x) = lim N→∞

pN– x=

f(x)(–)x (see []). (.)

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The degenerate Euler polynomials of orderr(∈N) are deﬁned by the generating function to be



( +λt)λ+ 

r

( +λt)xλ₌

∞

n= E(r)

n (x|λ)

tn

n! (see [, , ]), (.)

whereλ,t∈Zpsuch that|λt|p<p–



p–_.

From (.), we have

∞

n= lim

λ→E (r) n (x|λ)

tn

n!

=lim

λ→



( +λt)λ+ 

r

( +λt)xλ

=



et_{+ } r

ext

=

∞

n=

E(r)_n (x)t n

n!, (.)

whereE(r)n (x) are the higher-order Euler polynomials. Thus, by (.), we get

lim

λ→E (r)

n (x|λ) =En(r)(x) (n≥). (.)

Whenx= ,En(r)(λ) =En(r)(|λ) are called the higher-order degenerate Euler numbers, whilelimλ→En(r)(λ) =En(r)are called the higher-order Euler numbers.

In [], it was shown that

E(r) n (x|λ) =

Zp

· · ·

Zp

(x+x+· · ·+xr+x|λ)ndμ–(x)· · ·dμ–(xr), (.)

where (x)n=x(x– )· · ·(x–n+ ) andn∈Z≥.

In this paper, we studyq-extensions of the degenerate Euler polynomials and give some formulae and identities of those polynomials which are derived from the fermionicp-adic

q-integrals onZp.

2 Some identities ofq-analogues of higher-order degenerate Euler polynomials

In this section, we assume thatλ,t∈Zpwith|λt|p<p–



p–_{. From (.), we have}

Zp

· · ·

Zp

( +λt)(x+···+xr+x)/λ

dμ–q(x)· · ·dμ–q(xr)

=

[]q

q( +λt)/λ_{+ }

r

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Now, we deﬁne aq-analogue of degenerate Euler polynomials of orderras follows:

whereE(r)n,q(x) are called the higher-orderq-Euler polynomials (see [–]). Thus, by (.), we get generating function to be

Now, we deﬁne the degenerate Frobenius-Euler polynomials of orderras follows:

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Thus, by (.), we get

lim

λ→h (r)

n(x,u|λ) =Hn(x|u) (n≥).

By (.) and (.), we get

E(r)

n,q(x|λ) =h(r)n

x, –q–|λ (n≥). (.)

From (.) and (.), we have

∞

n=

Zp

· · ·

Zp

x+· · ·+xr+x

λ

n

dμ–q(x)· · ·dμ–q(xr)λ n_tn

n!

=

∞

n= E(r)

n,q(x|λ)

tn

n!. (.)

Now, we deﬁne

(x|λ)n=x(x–λ)· · ·x– (n– )λ (n> ), (.)

(x|λ)= .

By (.) and (.), we get

Zp

· · ·

Zp

(x+x+· · ·+xr|λ)ndμ–q(x)· · ·dμ–q(xr) =En,q(r)(x|λ) (u≥). (.)

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

Theorem . For n≥,we have

E(r) n,q(x|λ) =

Zp

· · ·

Zp

(x+· · ·+xr+x|λ)ndμ–q(x)· · ·dμ–q(xr)

=h(r)_nx, –q–|λ (n≥),

where h(r)n (x,u|λ)are called the degenerate Frobenius-Euler polynomials of order r.

It is not diﬃcult to show that

(x+· · ·+xr+x|λ)n

= (x+· · ·+xr+x)(x+· · ·+xr+x–λ)· · ·

x+· · ·+xr+x– (n– )λ

=λn

x+· · ·+xr+x

λ

n

=λn

n

l=

S(n,l)

x+· · ·+xr+x

λ

l

= n

l=

λn–lS(n,l)(x+· · ·+xr+x)l, (.)

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We observe that

Zp

· · ·

Zp

e(x+···+xr+x)t_dμ

–q(x)· · ·dμ–q(xr) =

[]q

qet_{+ } r

ext. (.)

Thus, by (.), we get

∞

n=

Zp

· · ·

Zp

(x+· · ·+xr+x)ndμ–q(x)· · ·dμ–q(xr)

tn

n!

=

[]q

qet_{+ } r

ext=

∞

n=

E(r)_n,q(x)t n

n!. (.)

By comparing the coeﬃcients on both sides of (.), we get

E(r)_n,q(x) =

Zp

· · ·

Zp

(x+· · ·+xr+x)ndμ–q(x)· · ·dμ–q(xr). (.)

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

h(r)_n x, –q–|λ=

Zp

· · ·

Zp

(x+· · ·+xr+x|λ)ndμ–q(x)· · ·dμ–q(xr)

= n

l=

λn–lS(n,l)

Zp

· · ·

Zp

(x+· · ·+xr+x)ldμ–q(x)· · ·dμ–q(xr)

= n

l=

λn–lS(n,l)El,q(r)(x)

= n

l=

λn–lS(n,l)Hl(r)

x|–q–. (.)

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

Theorem . For n≥,we have

h(r)_n x, –q–|λ= n

l=

λn–lS(n,l)H_l(r)x|–q–.

In particular,

E(r) n,q(x|λ) =

n

l=

λn–lS(n,l)E_l,q(r)(x).

By replacingtby (eλt_{– )/}_λ_{in (.), we get}

[]q

qet_{+ } r

ext

=

∞

n= E(r)

n,q(x|λ) 

n! 

λn

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=

∞

n= E(r)

n,q(x|λ) 

λn

∞

m=n

S(m,n)λ m

m!t m

=

∞

m= _m

n= E(r)

n,q(x|λ)λm–nS(m,n)

tm

m!, (.)

whereS(m,n) is the Stirling number of the second kind. Thus, by (.), we obtain the following theorem.

Theorem . For m≥,we have

H_m(r)x|–q–= m

n=

h(r)_n x, –q–|λλm–nS(m,n).

In particular,

E(r)_m,q(x) = m

n= E(r)

n,q(x|λ)λm–nS(m,n).

Whenr= , En,q(x|λ) =En,q(() x|λ) are called the degenerateq-Euler polynomials. In particular,x= ,En,q(λ) =En,q(|λ) are called the degenerateq-Euler numbers.hn(x,u| λ) =h()n (x,u|λ) are called the degenerate Frobenius-Euler polynomials. When x= ,

hn(u|λ) =hn(,u|λ) are called the degenerate Frobenius-Euler numbers. From (.), we have

Zp

( +λt)x+λx_dμ_–q(_x_)

=

[]q

q( +λt)λ+ 

( +λt)xλ

=

 +q–

( +λt)λ+q–

( +λt)xλ

=

∞

n=

hn

x, –q–|λt

n

n!. (.)

Thus, by (.), we get

hn

x, –q–|λ

=

Zp

(x+x|λ)ndμ–q(x)

=λn

Zp

x+x

λ

n

dμ–q(x)

= n

l=

S(n,l)λn–l

Zp

(x+x)ldμ–q(x)

= n

l=

S(n,l)λn–lHl

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and

hn

–q–|λ

=

Zp

(x|λ)ndμ–q(x)

=λn

Zp

x

λ

n

dμ–q(x)

= n

l=

S(n,l)λn–lHl

–q–. (.)

Ford∈N, by (.), we get

qd

Zp

(x+d|λ)ndμ–q(x) + (–)d–

Zp

(x|λ)ndμ–q(x)

= []q d–

l=

(–)d––lql(l|λ)n. (.)

Letd≡ (mod). Then we have

[]q d–

l=

(–)lql(l|λ)n=qdhn

d, –q–|λ+hn

–q–|λ. (.)

Ford∈Nwithd≡ (mod), we get

[]q d–

l=

(–)l–ql(l|λ)n=qdhn

d, –q–|λ–hn

–q–|λ. (.)

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

Theorem . Let d∈Nand n≥.

(i) Ford≡ (mod),we have

qdhn

d, –q–|λ+hn

–q–|λ= []q d–

l=

(–)lql(l|λ)n.

(ii) Ford≡ (mod),we have

qdhn

d, –q–|λ–hn

–q–|λ= []q d–

l=

(–)l–ql(l|λ)n.

Corollary . Let d∈Nand n≥.

(i) Ford≡ (mod),we have

qdEn,q(d|λ) +En,q(λ) = []q d–

l=

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(ii) Ford≡ (mod),we have

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

Theorem . For n≥,d∈Nwith d≡ (mod),we have

Now, we consider the degenerateq-Euler polynomials of the second kind as follows:

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= ( +λt)–x/λ

Zp

( +λt)–x/λ_dμ_–q(_x_)

= []q

( +λt)/λ₊_q( +λt)

(–x)/λ

. (.)

Whenx= ,En,q(ˆ λ) =En,q(ˆ |λ) are called the degenerateq-Euler numbers of the second kind.

By (.), we get

ˆ En,q(x|λ)

=λn

Zp

–x+x

λ

n

dμ–q(x)

=λn

n

l=

S(n,l)(–) l

λl

Zp

(x+x)ldμ–q(x)

= n

l=

S(n,l)λn–l(–)lEl,q(x). (.)

Thus, from (.), we have

(–)nEn,q(ˆ x|λ)

= n

l=

(–)n–lS(n,l)λn–lEl,q(x)

= n

l=

S(n,l)λn–lEl,q(x). (.)

We observe that

∞

n=

E_n,q–( –x) tn

n!

=  +q –

q–_et_{+ }e

(–x)t₌  +q

qe–t_{+ }e –xt

= []q

qe–t_{+ }e –xt₌

∞

n=

(–)nEn,q(x)t n

n!. (.)

From (.), we have

En,q–( –x) = (–)nEn,q(x) (n≥). (.)

By replacingtby eλt–

λ in (.), we get

∞

n= ˆ

En,q(x|λ)

n! 

λn

eλt_{– }n

=  +q

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= []q–

On the other hand, we have

∞

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

Theorem . For n≥,we have

It is easy to show that

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=λn

n

l=

n– 

l– 



λl_l_!

Zp

(–x|λ)ldμ–q(x)

= n

l=

n– 

l– 

λn–l

l!El,q(ˆ λ) (.)

and

(–)n

n! En,q(ˆ λ) = n

l=

n– 

l– 

λn–l

l!El,q(λ). (.)

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, Kwangwoon University, Seoul, 139-701, Republic of Korea.}2_{Department of Mathematics,}

Sogang University, Seoul, 121-742, Republic of Korea. 3_{Institute of Natural Sciences, Far Eastern Federal University,}

Vladivostok, 690950, Russia.

Acknowledgements

This paper is supported by Grant No. 14-11-00022 of Russian Scientiﬁc Fund.

Received: 5 May 2015 Accepted: 2 July 2015

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