Skip to main content

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

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.

Introduction

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

$$ \frac{2}{e^{t}+1}=\sum_{n=0}^{\infty}E_{n} \frac{t^{n}}{n!}\quad (\text{see [1--9]} ). $$
(1.1)

Carlitz [2] considered the degenerate Euler numbers defined by the generating function

$$ \frac{2}{ (1+\lambda t )^{\frac{1}{\lambda}}+1}=\sum_{n=0}^{\infty} \mathcal{E}_{n,\lambda}\frac{t^{n}}{n!}. $$
(1.2)

In [7], the modified degenerate Euler numbers, which are slightly different from Carlitz’s degenerate Euler numbers, are defined by

$$ \frac{2}{ (1+\lambda )^{\frac{t}{\lambda}}+1}=\sum_{n=0}^{\infty}\tilde{ \mathcal{E}}_{n,\lambda}\frac{t^{n}}{n!}. $$
(1.3)

Note that \(\lim_{\lambda\rightarrow0}\tilde{\mathcal{E}}_{n,\lambda }=\lim_{\lambda\rightarrow0}\mathcal{E}_{n,\lambda}=E_{n}\) (\(n\ge0\)). Recently, Kim and Kim [6] studied nonlinear differential equations given by

$$ \biggl(\frac{d}{dt} \biggr)^{N} \biggl(\frac{1}{ (1+\lambda t )^{\frac{1}{\lambda}}+1} \biggr) =\frac{ (-1 )^{N}}{ (1+\lambda t )^{N}}\sum_{i=1}^{N+1}a_{i} (N,\lambda )F^{i}, $$
(1.4)

where \(F=\frac{1}{ (1+\lambda t )^{\frac{1}{\lambda}}+1}\).

Let α, a, b be nonzero real numbers. Then we consider the general modified degenerate Euler numbers as follows:

$$ \frac{2}{\alpha (1+\lambda )^{\frac{at}{\lambda}}+b}=\sum_{n=0}^{\infty}\tilde{ \mathcal{E}}_{n,\lambda} (\alpha\mid a,b )\frac{t^{n}}{n!}. $$
(1.5)

From (1.5) we note that

$$\begin{aligned} \lim_{\lambda\rightarrow0}\frac{2}{\alpha (1+\lambda )^{\frac{at}{\lambda}}+b} =&\frac{2}{\alpha e^{at}+b} \\ =&\frac{1}{b}\frac{2}{\frac{\alpha}{b}e^{at}+1} \\ =&\frac{1}{b}\sum_{n=0}^{\infty}E_{n,\frac{\alpha}{b}}a^{n} \frac {t^{n}}{n!}, \end{aligned}$$
(1.6)

where \(E_{n,q}\) (\(n\ge0\)) are the Apostol-Euler numbers given by the generating function

$$ \frac{2}{qe^{t}+1}=\sum_{n=0}^{\infty}E_{n,q} \frac{t^{n}}{n!}\quad (\text{see [1, 3]} ). $$
(1.7)

Thus, by (1.5) and (1.6) we get

$$\frac{a^{n}}{b}E_{n,\frac{\alpha}{b}}=\lim_{\lambda\rightarrow0}\tilde { \mathcal{E}}_{n,\lambda} (\alpha\mid a,b )\quad (n\ge 0 ). $$

Bayad and Kim [1] studied the following nonlinear differential equations related to Apostol-Euler numbers:

$$ F_{q}^{N}=\frac{1}{ (N-1 )!}\sum _{k=0}^{N}a_{k} (N )F_{q}^{ (k-1 )} \quad (N\in\mathbb{N} ), $$
(1.8)

where \(F_{q}^{ (k )}= (\frac{d}{dt} )^{k}F_{q} (t )\), \(F_{q} (t )=\frac{1}{qe^{t}+1}\).

In this paper, we study the ordinary differential equations associated with the generating 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.

Generalized modified degenerate Euler numbers

For nonzero real numbers α, a, b, let

$$ F=F (t )=\frac{1}{\alpha (1+\lambda )^{\frac {at}{\lambda}}+b}. $$
(2.1)

Then by (2.1) we get

$$\begin{aligned} F^{ (1 )} & =\frac{dF}{dt} (t ) \\ & =\frac{ (-1 )\frac{a}{\lambda}\log (1+\lambda )}{ (\alpha (1+\lambda )^{\frac{at}{\lambda}}+b )^{2}} \bigl(\alpha (1+\lambda )^{\frac{\alpha t}{\lambda }} \bigr) \\ & =\frac{ (-1 )\frac{a}{\lambda}\log (1+\lambda )}{ (\alpha (1+\lambda )^{\frac{at}{\lambda}}+b )^{2}} \bigl\{ \alpha (1+\lambda )^{\frac{at}{\lambda }}+b-b \bigr\} \\ & = (-1 )\frac{a}{\lambda}\log (1+\lambda ) \bigl(F-bF^{2} \bigr). \end{aligned}$$
(2.2)

Thus, from (2.2) we have

$$ F^{ (1 )}=\frac{a}{\lambda}\log (1+\lambda ) \bigl(bF^{2}-F \bigr). $$
(2.3)

From (2.3) we derive the following equation:

$$\begin{aligned} F^{ (2 )} & =\frac{d}{dt}F^{ (1 )} \\ & =\frac{a}{\lambda}\log (1+\lambda ) \bigl\{ 2bFF^{ (1 )}-F^{ (1 )} \bigr\} \\ & =\frac{a}{\lambda}\log (1+\lambda ) (2bF-1 )F^{ (1 )} \\ & = \biggl(\frac{a}{\lambda}\log (1+\lambda ) \biggr)^{2} (2bF-1 ) \bigl(bF^{2}-F \bigr) \\ & = \biggl(\frac{a}{\lambda}\log (1+\lambda ) \biggr)^{2} \bigl(2b^{2}F^{3}-3bF^{2}+F \bigr). \end{aligned}$$
(2.4)

Continuing this process, we set

$$\begin{aligned} F^{ (N )} & = \biggl(\frac{d}{dt} \biggr)^{N}F (t ) \\ & = \biggl(\frac{a}{\lambda}\log (1+\lambda ) \biggr)^{N}\sum _{k=1}^{N+1}a_{k} (N )b^{k-1}F^{k}. \end{aligned}$$
(2.5)

By taking the derivative of (2.5) with respect to t we have

$$\begin{aligned} F^{ (N+1 )} & =\frac{dF^{ (N )}}{dt} \\ & = \biggl(\frac{a}{\lambda}\log (1+\lambda ) \biggr)^{N}\sum _{k=1}^{N+1}a_{k} (N )b^{k-1}kF^{k-1}F^{ (1 )} \\ & = \biggl(\frac{a}{\lambda}\log (1+\lambda ) \biggr)^{N+1}\sum _{k=1}^{N+1} \bigl(ka_{k} (N )b^{k}F^{k+1}-a_{k} (N )b^{k-1}kF^{k} \bigr) \\ & = \biggl(\frac{a}{\lambda}\log (1+\lambda ) \biggr)^{N+1} \Biggl\{ \sum_{k=2}^{N+2} (k-1 )a_{k-1} (N ) b^{k-1}F^{k}-\sum_{k=1}^{N+1}ka_{k} (N )b^{k-1}F^{k} \Biggr\} . \end{aligned}$$
(2.6)

Replacing N by \(N+1\) in (2.5), we get

$$ F^{ (N+1 )}= \biggl(\frac{a}{\lambda}\log (1+\lambda ) \biggr)^{N+1}\sum_{k=1}^{N+2}a_{k} (N+1 )b^{k-1}F^{k}. $$
(2.7)

Comparing the coefficients on both sides of (2.6) and (2.7), we obtain

$$ a_{1} (N+1 )=-a_{1} (N ). $$
(2.8)

Thus, by (2.8) we get

$$ a_{1} (N+1 )=-a_{1} (N )= (-1 )^{2}a_{1} (N-1 )=\cdots= (-1 )^{N}a_{1} (1 ). $$
(2.9)

From (2.5) we have

$$ \frac{a}{\lambda}\log (1+\lambda ) \bigl\{ bF^{2}-F \bigr\} = \frac{a}{\lambda}\log (1+\lambda ) \bigl\{ a_{1} (1 )F+a_{2} (1 )bF^{2} \bigr\} . $$
(2.10)

By (2.10) we get

$$ a_{1} (1 )=-1\quad \text{and}\quad a_{2} (1 )=1. $$
(2.11)

Thus, from (2.9) and (2.11) we have

$$ a_{1} (N+1 )= (-1 )^{N}a_{1} (1 )= (-1 )^{N+1}. $$
(2.12)

By (2.6) and (2.7) we see that

$$\begin{aligned} a_{N+2} (N+1 ) =& (N+1 )a_{N+1} (N ) \\ =& (N+1 )Na_{N} (N-1 ) \\ =& (N+1 )N (N-1 )a_{N-1} (N-2 ) \\ & \vdots \\ =& (N+1 )N (N-1 )\cdots2a_{2} (1 ) \\ =& (N+1 )!. \end{aligned}$$
(2.13)

Thus, by (2.13) we have

$$ a_{N+2} (N+1 )= (N+1 )!. $$
(2.14)

For \(2\le k\le N+1\), by comparing the coefficients on both sides of (2.6) and (2.7) we have

$$ a_{k} (N+1 )= (k-1 )a_{k-1} (N )-ka_{k} (N ). $$
(2.15)

Let \(k=2\) in (2.15). Then we have

$$\begin{aligned} a_{2} (N+1 ) =&a_{1} (N )-2a_{2} (N ) \\ =&a_{1} (N )-2 \bigl(a_{1} (N-1 )-2a_{2} (N-1 ) \bigr) \\ =&a_{1} (N )-2a_{1} (N-1 )+ (-1 )^{2}2^{2}a_{2} (N-1 ) \\ =&a_{1} (N )-2a_{1} (N-1 )+ (-1 )^{2}2^{2} \bigl\{ a_{1} (N-2 )-2a_{2} (N-2 ) \bigr\} \\ =&a_{1} (N )-2a_{1} (N-1 )+ (-1 )^{2}2^{2}a_{1} (N-2 )+ (-1 )^{3}2^{3}a_{2} (N-2 ) \\ & \vdots \\ =&\sum_{k=0}^{N-1} (-1 )^{k}a_{1} (N-k )2^{k}+ (-1 )^{N}2^{N}a_{2} (1 ) \\ =&\sum_{k=0}^{N} (-1 )^{k}a_{1} (N-k )2^{k}. \end{aligned}$$
(2.16)

For \(k=3\) in (2.15), we have

$$\begin{aligned} a_{3} (N+1 ) =&2a_{2} (N )-3a_{3} (N ) \\ =&2a_{2} (N )-3 \bigl\{ 2a_{2} (N-1 )-3a_{3} (N-1 ) \bigr\} \\ =&2a_{2} (N )-3\cdot2a_{2} (N-1 )+ (-1 )^{2}3^{2}a_{3} (N-1 ) \\ =&2a_{2} (N )-3\cdot2a_{2} (N-1 )+ (-1 )^{2}3^{2} \bigl\{ 2a_{2} (N-2 )-3a_{3} (N-2 ) \bigr\} \\ =&2a_{2} (N )-3\cdot2a_{2} (N-1 )+ (-1 )^{2}3^{2}2a_{2} (N-2 )+ (-1 )^{3}3^{3}a_{3} (N-2 ) \\ & \vdots \\ =&2\sum_{k=0}^{N-2}a_{2} (N-k ) (-1 )^{k}3^{k}+ (-1 )^{N-1}3^{N-1}a_{3} (2 ) \\ =&2\sum_{k=0}^{N-1}a_{2} (N-k ) (-1 )^{k}3^{k}. \end{aligned}$$
(2.17)

Continuing this process, we deduce

$$ a_{j} (N+1 )= (j-1 )\sum_{k=0}^{N-j+2}a_{j-1} (N-k ) (-1 )^{k}j^{k}, $$
(2.18)

where \(2\le j\le N+1\).

Now we give an explicit expression for \(a_{j} (N+1 )\) in (2.18). From (2.12) and (2.16) we can derive the following equation:

$$\begin{aligned} a_{2} (N+1 ) & =\sum_{k=0}^{N} (-1 )^{k}a_{1} (N-k )2^{k} \\ & =\sum_{k=0}^{N} (-1 )^{k} (-1 )^{N-k}2^{k} = (-1 )^{N}\sum_{k=0}^{N}2^{k}. \end{aligned}$$
(2.19)

By (2.17) we get

$$\begin{aligned} a_{3} (N+1 ) & =2\sum_{k_{2}=0}^{N-1}a_{2} (N-k_{2} ) (-1 )^{k_{2}}3^{k_{2}} \\ & =2\sum_{k_{2}=0}^{N-1} (-1 )^{N-k_{2}-1} \sum_{k_{1}=0}^{N-k_{2}-1}2^{k_{1}} (-1 )^{k_{2}}3^{k_{2}} \\ & =2 (-1 )^{N-1}\sum_{k_{2}=0}^{N-1} \sum_{k_{1}=0}^{N-1-k_{2}}2^{k_{1}}3^{k_{2}}. \end{aligned}$$
(2.20)

Continuing this process, we deduce that, for \(2\le j\le N+1\),

$$\begin{aligned}& a_{j} (N+1 ) \\& \quad = (j-1 )! (-1 )^{N-j+2} \\& \qquad {} \times\sum _{k_{j-1}=0}^{N-j+2}\sum_{k_{j-2}=0}^{N-j+2-k_{j-1}}\sum_{k_{j-3}=0}^{N-j+2-k_{j-1}-k_{j-2}}\cdots\sum _{k_{1}=0}^{N-j+2-k_{j-1}-\cdots-k_{2}}j^{k_{j-1}} (j-1 )^{k_{j-2}}\cdots3^{k_{2}}2^{k_{1}}. \end{aligned}$$
(2.21)

Therefore, by (2.5) and (2.21) we obtain the following theorem.

Theorem 1

Let α, a, b be nonzero real numbers. The family of nonlinear differential equations

$$F^{ (N )}= \biggl(\frac{a}{\lambda}\log (1+\lambda ) \biggr)^{N}\sum_{k=1}^{N+1}a_{k} (N )b^{k-1}F^{k} $$

has a solution \(F=F (t )=\frac{1}{\alpha (1+\lambda )^{\frac{at}{\lambda}}+b}\), where \(a_{1} (N )= (-1 )^{N}\), and

$$\begin{aligned} a_{j} (N ) =& (j-1 )! (-1 )^{N-j+1} \\ &{}\times\sum _{k_{j-1}=0}^{N-j+1}\sum_{k_{j-2}=0}^{N-j+1-k_{j-1}}\cdots\sum _{k_{1}=0}^{N-j+1-k_{j-1}-\cdots-k_{2}}j^{k_{j-1}} (j-1 )^{k_{j-2}}\cdots3^{k_{2}}2^{k_{1}} \end{aligned}$$

for \(2\le j\le N+1\).

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

$$ \frac{2}{\alpha (1+\lambda )^{\frac{at}{\lambda}}+b}=\sum_{n=0}^{\infty}\tilde{ \mathcal{E}}_{n,\lambda} (\alpha;a,b )\frac{t^{n}}{n!}. $$
(2.22)

Note that \(\tilde{\mathcal{E}}_{n,\lambda} (1;1,1 )\) are the modified degenerate Euler numbers given by

$$\frac{2}{ (1+\lambda )^{\frac{t}{\lambda}}+1}=\sum_{n=0}^{\infty}\tilde{ \mathcal{E}}_{n,\lambda}\frac{t^{n}}{n!}. $$

Now we observe that

$$\begin{aligned} F^{ (N )} & =\frac{1}{2} \biggl(\frac{d}{dt} \biggr)^{N} \biggl(\frac{2}{\alpha (1+\lambda )^{\frac{at}{\lambda}}+b} \biggr) \\ & =\frac{1}{2}\sum_{n=0}^{\infty} \frac{\tilde{\mathcal{E}}_{n,\lambda } (\alpha;a,b )}{n!} \biggl(\frac{d}{dt} \biggr)^{N}t^{n} \\ & =\frac{1}{2}\sum_{n=0}^{\infty}\tilde{ \mathcal{E}}_{n+N,\lambda} (\alpha;a,b )\frac{t^{n}}{n!}. \end{aligned}$$
(2.23)

For \(r\in\mathbb{N}\), the higher-order general modified degenerate Euler numbers are defined by the generating function

$$ \biggl(\frac{2}{\alpha (1+\lambda )^{\frac{at}{\lambda }}+b} \biggr)^{r}=\sum _{n=0}^{\infty}\tilde{\mathcal{E}}_{n,\lambda }^{ (r )} (\alpha;a,b )\frac{t^{n}}{n!}. $$
(2.24)

Therefore, by Theorem 1, (2.23), and (2.21) we obtain the following theorem.

Theorem 2

Let α, a, b be nonzero real numbers. For \(n\ge0\), we have

$$\tilde{\mathcal{E}}_{n+N} (\alpha;a,b )= \biggl(\frac{a}{\lambda }\log (1+\lambda ) \biggr)^{N}\sum_{k=1}^{N+1}a_{k} (N )b^{k-1}2^{1-k}\tilde{\mathcal{E}}_{n,\lambda}^{ (k )} (\alpha;a,b ), $$

where \(a_{1} (N )=(-1)^{N}\), and, for \(2\le j\le N+1\),

$$\begin{aligned} a_{j} (N ) =& (j-1 )! (-1 )^{N-j+1} \\ &{}\times\sum _{k_{j-1}=0}^{N-j+1}\sum_{k_{j-2}=0}^{N-j+1-k_{j-1}}\cdots\sum _{k_{1}=0}^{N-j+1-k_{j-1}-\cdots-k_{2}}j^{k_{j-1}} (j-1 )^{k_{j-2}}\cdots3^{k_{2}}2^{k_{1}}. \end{aligned}$$

References

  1. 1.

    Bayad, A, Kim, T: Higher recurrences for Apostol-Bernoulli-Euler numbers. Russ. J. Math. Phys. 19(1), 1-10 (2012). MR2892600

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Carlitz, L: Degenerate Stirling, Bernoulli and Eulerian numbers. Util. Math. 15, 51-88 (1979). MR531621

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Ding, D, Yang, J: Some identities related to the Apostol-Euler and Apostol-Bernoulli polynomials. Adv. Stud. Contemp. Math. (Kyungshang) 20(1), 7-21 (2010). MR2597988

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Dolgy, DV, Kim, T, Kwon, HI, Seo, JJ: On the modified degenerate Bernoulli polynomials. Adv. Stud. Contemp. Math. (Kyungshang) 26(1), 1-9 (2016)

    Google Scholar 

  5. 5.

    Gaboury, S, Tremblay, R, Fugère, B-J: Some explicit formulas for certain new classes of Bernoulli, Euler and Genocchi polynomials. Proc. Jangjeon Math. Soc. 17(1), 115-123 (2014). MR3184467

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Kim, T, Kim, DS: Identities involving degenerate Euler numbers and polynomials arising from non-linear differential equations. J. Nonlinear Sci. Appl. 9, 2086-2098 (2016)

    MATH  Google Scholar 

  7. 7.

    Kwon, HI, Kim, T, Seo, JJ: Modified degenerate Euler polynomials. Adv. Stud. Contemp. Math. (Kyungshang) 26(1), 203-209 (2016)

    Google Scholar 

  8. 8.

    Rim, S-H, Jeong, J: On the modified q-Euler numbers of higher order with weight. Adv. Stud. Contemp. Math. (Kyungshang) 22(1), 93-98 (2012). MR2931608

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Zhang, Z, Yang, H: Some closed formulas for generalized Bernoulli-Euler numbers and polynomials. Proc. Jangjeon Math. Soc. 11(2), 191-198 (2008). MR2482602

    MathSciNet  MATH  Google Scholar 

Download references

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.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Tae Kyun Kim.

Additional information

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.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kim, T.K., Kim, D.S., Kwon, H.I. et al. Differential equations arising from the generating function of general modified degenerate Euler numbers. Adv Differ Equ 2016, 129 (2016). https://doi.org/10.1186/s13662-016-0858-7

Download citation

MSC

  • 05A19
  • 11B37
  • 11B83
  • 34A34

Keywords

  • general modified degenerate Euler numbers
  • differential equations