Differential equations arising from the generating function of general modified degenerate Euler numbers

R E S E A R C H

Open Access

Differential equations arising from the

generating function of general modified

degenerate Euler numbers

Tae Kyun Kim

_{, Dae San Kim}

_{, Hyuck In Kwon}

_{and Jong Jin Seo}

*_{Correspondence:} [email protected] 1_{Department of Mathematics,} College of Science, Tianjin Polytechnic University, Tianjin, 300387, China

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 introduce the general modified degenerate Euler numbers and study ordinary differential equations arising from the generating function of these numbers. In addition, we give some new explicit identities for the general modified degenerate Euler numbers arising from our differential equations.

MSC: 05A19; 11B37; 11B83; 34A34

Keywords: general modified degenerate Euler numbers; differential equations

1 Introduction

As is known, the Euler numbers are defined by the generating function



et_{+ }= ∞

n=

En

tn

n! (see [–]). (.)

Carlitz [] considered the degenerate Euler numbers defined by the generating function



( +λt)λ + 

= ∞

n=

En,λ

tn

n!. (.)

In [], the modified degenerate Euler numbers, which are slightly different from Carlitz’s degenerate Euler numbers, are defined by



( +λ)λt + 

= ∞

n= ˜

En,λ

tn

n!. (.)

Note thatlim_λ→E˜n,λ=limλ→En,λ=En(n≥). Recently, Kim and Kim [] studied non-linear differential equations given by

d dt

N 

( +λt)λ+ 

= (–) N

( +λt)N N+

i=

ai(N,λ)Fi, (.)

whereF=  (+λt)λ+

.

Letα,a,bbe nonzero real numbers. Then we consider the general modified degenerate Euler numbers as follows:



α( +λ)atλ _+b

= ∞

n= ˜

En,λ(α|a,b)

tn

n!. (.)

From (.) we note that

lim λ→



α( +λ)atλ _+b

= 

αeat_+b

= 

b

 α beat+ 

= 

b

∞

n=

En,α_ban

tn

n!, (.)

whereEn,q(n≥) are the Apostol-Euler numbers given by the generating function



qet_{+ }= ∞

n=

En,q

tn

n! (see [, ]). (.)

Thus, by (.) and (.) we get

an

b En,αb =_λ→lim_E˜n,λ(α|a,b) (n≥).

Bayad and Kim [] studied the following nonlinear differential equations related to Apostol-Euler numbers:

F_qN=  (N– )!

N

k=

ak(N)Fq(k–) (N∈N), (.)

whereFq(k)= (_dtd)kFq(t),Fq(t) =_qet_+.

In this paper, we study the ordinary differential equations associated with the gener-ating function of general modified degenerate Euler numbers. In addition, we give some new and explicit formulas and identities for those numbers arising from our differential equations.

2 Generalized modified degenerate Euler numbers

For nonzero real numbersα,a,b, let

F=F(t) = 

α( +λ)atλ _+b

. (.)

Then by (.) we get

F()=dF

dt(t)

= (–) a

λlog( +λ) (α( +λ)atλ _+b)

= (–)

From (.) we derive the following equation:

F()= d

Continuing this process, we set

F(N)=

By taking the derivative of (.) with respect totwe have

F(N+)=dF

Comparing the coefficients on both sides of (.) and (.), we obtain

Thus, by (.) we get

a(N+ ) = –a(N) = (–)a(N– ) =· · ·= (–)Na(). (.)

From (.) we have

a

λlog( +λ)

bF–F=a

λlog( +λ)

a()F+a()bF

. (.)

By (.) we get

a() = – and a() = . (.)

Thus, from (.) and (.) we have

a(N+ ) = (–)Na() = (–)N+. (.)

By (.) and (.) we see that

aN+(N+ ) = (N+ )aN+(N)

= (N+ )NaN(N– )

= (N+ )N(N– )aN–(N– )

.. .

= (N+ )N(N– )· · ·a()

= (N+ )!. (.)

Thus, by (.) we have

aN+(N+ ) = (N+ )!. (.)

For ≤k≤N+ , by comparing the coefficients on both sides of (.) and (.) we have

ak(N+ ) = (k– )ak–(N) –kak(N). (.)

Letk=  in (.). Then we have

a(N+ ) =a(N) – a(N)

=a(N) – 

a(N– ) – a(N– )

=a(N) – a(N– ) + (–)a(N– )

=a(N) – a(N– ) + (–)

a(N– ) – a(N– )

=a(N) – a(N– ) + (–)a(N– ) + (–)a(N– )

= N–

k=

(–)ka(N–k)k+ (–)NNa()

= N

k=

(–)ka(N–k)k. (.)

Fork=  in (.), we have

a(N+ ) = a(N) – a(N)

= a(N) – 

a(N– ) – a(N– )

= a(N) – ·a(N– ) + (–)a(N– )

= a(N) – ·a(N– ) + (–)

a(N– ) – a(N– )

= a(N) – ·a(N– ) + (–)a(N– ) + (–)a(N– )

.. .

=  N–

k=

a(N–k)(–)kk+ (–)N–N–a()

=  N–

k=

a(N–k)(–)kk. (.)

Continuing this process, we deduce

aj(N+ ) = (j– ) N–j+

k=

aj–(N–k)(–)kjk, (.)

where ≤j≤N+ .

Now we give an explicit expression foraj(N+ ) in (.). From (.) and (.) we can derive the following equation:

a(N+ ) = N

k=

(–)ka(N–k)k

= N

k=

(–)k(–)N–kk= (–)N N

k=

k. (.)

By (.) we get

a(N+ ) =  N–

k=

a(N–k)(–)kk

=  N–

k=

(–)N–k– N–k–

k=

k_(–)k_k

= (–)N– N–

k=

N––k

k=

Continuing this process, we deduce that, for ≤j≤N+ ,

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

Theorem  Letα,a,b be nonzero real numbers.The family of nonlinear differential

equa-tions

Now we define the general modified degenerate Euler numbers given by the generating function

Note thatE˜n,λ(; , ) are the modified degenerate Euler numbers given by



Now we observe that

Forr∈N, the higher-order general modified degenerate Euler numbers are defined by

the generating function



α( +λ)atλ _+b

r =

∞

n= ˜

E(r) n,λ(α;a,b)

tn

n!. (.)

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

Theorem  Letα,a,b be nonzero real numbers.For n≥,we have

˜

En+N(α;a,b) =

a

λlog( +λ)

N N+

k=

ak(N)bk––kE˜_n(k_,λ)(α;a,b),

where a(N) = (–)N,and,for≤j≤N+ ,

aj(N) = (j– )!(–)N–j+

× N–j+

kj–=

N–j+–kj–

kj–=

· · ·

N–j+–kj––···–k

k=

jkj–_{(j– )}kj–· · ·_k_k_.

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, College of Science, Tianjin Polytechnic University, Tianjin, 300387, China.}2_{Department of} Mathematics, Kwangwoon University, Seoul, 139-701, Republic of Korea.3_{Department of Mathematics, Sogang} University, Seoul, 121-742, Republic of Korea. 4_{Department of Applied Mathematics, Pukyong National University, Busan,} Republic of Korea.

Acknowledgements

The first author is appointed as a chair professor at Tianjin Polytechnic University, Tianjin City, China, from August 2015 to August 2019. We would like to thank the referee for his detailed suggestions that helped to improve the original manuscript.

Received: 11 April 2016 Accepted: 9 May 2016

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