Kim et al. Advances in Difference Equations (2016) 2016:129 DOI 10.1186/s13662-016-0858-7
RESEARCH
Open Access
Differential equations arising from the generating function of general modified degenerate Euler numbers Tae Kyun Kim1,2* , Dae San Kim3 , Hyuck In Kwon2 and Jong Jin Seo4 *
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 ∞
tn = En et + n= n!
(see [–]).
(.)
Carlitz [] considered the degenerate Euler numbers defined by the generating function
=
λ
( + λt) +
∞
En,λ
n=
tn . n!
(.)
In [], the modified degenerate Euler numbers, which are slightly different from Carlitz’s degenerate Euler numbers, are defined by t λ
( + λ) +
=
∞
n
t E˜n,λ . n! n=
(.)
Note that limλ→ E˜n,λ = limλ→ En,λ = En (n ≥ ). Recently, Kim and Kim [] studied nonlinear differential equations given by
d dt
N
where F =
λ
( + λt) +
=
N+ (–)N ai (N, λ)F i , ( + λt)N i=
(.)
.
(+λt) λ +
© 2016 Kim et al. 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.
Kim et al. Advances in Difference Equations (2016) 2016:129
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Let α, a, b be nonzero real numbers. Then we consider the general modified degenerate Euler numbers as follows: at
α( + λ) λ + b
=
∞
n
t E˜n,λ (α | a, b) . n! n=
(.)
From (.) we note that
lim
λ→
α( + λ)
at λ
+b
=
αeat + b
=
b αb eat +
=
tn En, αb an , b n= n!
∞
(.)
where En,q (n ≥ ) are the Apostol-Euler numbers given by the generating function ∞
tn = E n,q qet + n= n!
(see [, ]).
(.)
Thus, by (.) and (.) we get an En, αb = lim E˜n,λ (α | a, b) (n ≥ ). λ→ b Bayad and Kim [] studied the following nonlinear differential equations related to Apostol-Euler numbers: ak (N)Fq(k–) (N – )! N
FqN =
(N ∈ N),
(.)
k=
where Fq(k) = ( dtd )k Fq (t), Fq (t) = qet+ . 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.
2 Generalized modified degenerate Euler numbers For nonzero real numbers α, a, b, let F = F(t) =
at
α( + λ) λ + b
.
Then by (.) we get dF (t) dt (–) λa log( + λ) αt = α( + λ) λ at (α( + λ) λ + b)
F () =
(.)
Kim et al. Advances in Difference Equations (2016) 2016:129
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(–) λa log( + λ) at α( + λ) λ + b – b at (α( + λ) λ + b) a = (–) log( + λ) F – bF . λ
=
(.)
Thus, from (.) we have F () =
a log( + λ) bF – F . λ
(.)
From (.) we derive the following equation: d () F dt a = log( + λ) bFF () – F () λ a = log( + λ)(bF – )F () λ a = log( + λ) (bF – ) bF – F λ a log( + λ) b F – bF + F . = λ
F () =
(.)
Continuing this process, we set F
(N)
= =
d dt
N F(t)
a log( + λ) λ
N N+
ak (N)bk– F k .
(.)
k=
By taking the derivative of (.) with respect to t we have dF (N) dt N N+ a log( + λ) ak (N)bk– kF k– F () = λ
F (N+) =
k=
= =
a log( + λ) λ a log( + λ) λ
N+ N+
kak (N)bk F k+ – ak (N)bk– kF k
k=
N+ N+
(k – )ak– (N)b
k– k
k=
F –
N+
kak (N)b
k– k
F
.
(.)
k=
Replacing N by N + in (.), we get F (N+) =
a log( + λ) λ
N+ N+
ak (N + )bk– F k .
(.)
k=
Comparing the coefficients on both sides of (.) and (.), we obtain a (N + ) = –a (N).
(.)
Kim et al. Advances in Difference Equations (2016) 2016:129
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Thus, by (.) we get a (N + ) = –a (N) = (–) a (N – ) = · · · = (–)N a ().
(.)
From (.) we have a a log( + λ) bF – F = log( + λ) a ()F + a ()bF . λ λ
(.)
By (.) we get a () = – and
a () = .
(.)
Thus, from (.) and (.) we have a (N + ) = (–)N a () = (–)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). Let k = 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 – ) .. .
(.)
Kim et al. Advances in Difference Equations (2016) 2016:129
=
N–
Page 5 of 7
(–)k a (N – k)k + (–)N N a ()
k=
=
N
(–)k a (N – k)k .
(.)
k=
For k = 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–
a (N – k)(–)k k + (–)N– N– a ()
k=
=
N–
a (N – k)(–)k k .
(.)
k=
Continuing this process, we deduce
N–j+
aj (N + ) = (j – )
aj– (N – k)(–)k jk ,
(.)
k=
where ≤ j ≤ N + . Now we give an explicit expression for aj (N + ) in (.). From (.) and (.) we can derive the following equation:
a (N + ) =
N (–)k a (N – k)k k=
=
N N (–)k (–)N–k k = (–)N k . k=
(.)
k=
By (.) we get
a (N + ) =
N–
a (N – k )(–)k k
k =
=
N–
(–)N–k –
k =
= (–)N–
N–k –
k (–)k k
k = N– N––k k = k =
k k .
(.)
Kim et al. Advances in Difference Equations (2016) 2016:129
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Continuing this process, we deduce that, for ≤ j ≤ N + , aj (N + ) = (j – )!(–)N–j+ N–j+ N–j+–kj– N–j+–kj– –kj–
kj– =
kj– =
kj– =
×
N–j+–kj– –···–k
···
jkj– (j – )kj– · · · k k .
(.)
k =
Therefore, by (.) and (.) we obtain the following theorem. Theorem Let α, a, b be nonzero real numbers. The family of nonlinear differential equations F
(N)
=
a log( + λ) λ
N N+
ak (N)bk– F k
k=
has a solution F = F(t) =
at
α(+λ) λ +b
, where a (N) = (–)N , and
aj (N) = (j – )!(–)N–j+ N–j+–kj– –···–k
N–j+ N–j+–kj–
kj– =
kj– =
×
···
jkj– (j – )kj– · · · k k
k =
for ≤ j ≤ N + . Now we define the general modified degenerate Euler numbers given by the generating function α( + λ)
at λ
+b
=
∞
n
t E˜n,λ (α; a, b) . n! n=
(.)
Note that E˜n,λ (; , ) are the modified degenerate Euler numbers given by t
( + λ) λ +
=
∞
n
t E˜n,λ . n! n=
Now we observe that d N at dt α( + λ) λ + b ∞ E˜n,λ (α; a, b) d N n t = n= n! dt
F (N) =
∞
=
˜ tn En+N,λ (α; a, b) . n= n!
(.)
Kim et al. Advances in Difference Equations (2016) 2016:129
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For r ∈ N, the higher-order general modified degenerate Euler numbers are defined by the generating function
r
α( + λ)
at λ
+b
=
∞
n
t (r) E˜n,λ (α; a, b) . n! n=
(.)
Therefore, by Theorem , (.), and (.) we obtain the following theorem. Theorem Let α, a, b be nonzero real numbers. For n ≥ , we have
E˜n+N (α; a, b) =
a log( + λ) λ
N N+
(k) ak (N)bk– –k E˜n,λ (α; a, b),
k=
where a (N) = (–)N , and, for ≤ j ≤ N + , aj (N) = (j – )!(–)N–j+ N–j+–kj– –···–k
N–j+ N–j+–kj–
×
kj– =
kj– =
···
jkj– (j – )kj– · · · k 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 References 1. Bayad, A, Kim, T: Higher recurrences for Apostol-Bernoulli-Euler numbers. Russ. J. Math. Phys. 19(1), 1-10 (2012). MR2892600 2. Carlitz, L: Degenerate Stirling, Bernoulli and Eulerian numbers. Util. Math. 15, 51-88 (1979). MR531621 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 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) 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 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) 7. Kwon, HI, Kim, T, Seo, JJ: Modified degenerate Euler polynomials. Adv. Stud. Contemp. Math. (Kyungshang) 26(1), 203-209 (2016) 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 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
