Kim et al. Advances in Difference Equations (2020)2020:444 https://doi.org/10.1186/s13662-020-02901-9

R E S E A R C H Open Access Degenerate poly-Bernoulli polynomials arising from degenerate polylogarithm

Taekyun Kim1,2,DansanKim3, Han-Young Kim2, Hyunseok Lee2 and Lee-Chae Jang4*

*Correspondence: [email protected] Abstract 4Graduate School of Education, Konkuk University, Seoul 05029, Recently, degenerate polylogarithm functions were introduced by Kim and Kim. In Republic of Korea this paper, we introduce degenerate poly-Bernoulli polynomials by means of the Full list of author information is degenerate polylogarithm functions and investigate some their properties. In more available at the end of the article detail, we find certain explicit expressions for those polynomials in terms of the Carlitz degenerate Bernoulli polynomials and the degenerate Stirling numbers of the second kind. Furthermore, we obtain some expressions for differences of the degenerate poly-Bernoulli polynomials. MSC: 11B68; 11B73; 11B83; 05A19 Keywords: Degenerate poly-Bernoulli polynomials; Degenerate polylogarithm

1 Introduction Carlitz [5, 6] initiated a study of degenerate versions of some special numbers and polyno- mials, the degenerate Bernoulli and Euler polynomials. In recent years, the idea of studying degenerate versions of many special polynomials and numbers regained interests of some mathematicians, and many interesting results were found (see [10–13, 16, 18, 19]). They have been explored by employing several different tools such as combinatorial methods, generating functions, p-adic analysis, umbral calculus techniques, differential equations, and probability theory. The aim of this paper is to introduce the degenerate poly-Bernoulli polynomials by means of the degenerate polylogarithm functions and to study their properties including their explicit expressions and differences. Here we note that those polynomials are slight modifications of the previously studied ones under the same name. The outline of this paper is as follows. In Sect. 1, as a preparation to the next section, we recall the Carlitz degenerate Bernoulli polynomials, the degenerate exponential func- tions, the degenerate polylogarithms, and the degenerate Stirling numbers of the second kind. In Sect. 2, we introduce the degenerate poly-Bernoulli polynomials by means of the degenerate polylogarithm functions; note that they reduce to the Carlitz degenerate Bernoulli polynomials when k = 1. We express the generating function of the degenerate poly-Bernoulli polynomials as an iterated integral, from which we find an explicit expres- sion for these polynomials when k = 2. Also, we find explicit expressions for the degenerate poly-Bernoulli polynomials in terms of the Carlitz degenerate Bernoulli polynomials and

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the degenerate Stirling numbers of the second kind. Finally, we find certain expressions for certain differences of the degenerate poly-Bernoulli polynomials. For 0 = λ ∈ R, Carlitz introduced the degenerate Bernoulli polynomials given by

∞  n t x t (1 + λt) λ = β λ(x) (see [5, 6]). (1) 1 n, n! (1 + λt) λ –1 n=0

In the case x =0,βn,λ = βn,λ(0) are called the degenerate Bernoulli numbers.

Note that limλ→0 βn,λ(x)=Bn(x)(n ≥ 0), where Bn(x) are the ordinary Bernoulli polyno- mials given by

∞ t  tn ext = B (x) (see [14, 15, 20, 23, 24]). (2) et –1 n n! n=0

For λ ∈ R, the degenerate exponential functions are defined by

∞  n x t 1 e (t)= (x) λ , eλ(t)=e (t)(see[10–13, 16, 18, 19]), (3) λ n, n! λ n=0

x λ ··· λ ≥ → where (x)0,λ =1,(x)n,λ = x(x – ) (x –(n –1) )(n 1). Note that limλ 0 eλ(t)= ∞ n tn xt n=0 x n! = e . For k ∈ Z, the polylogarithm is defined by

∞  xn Li (x)= (see [1–4, 9, 10, 12, 21]). (4) k nk n=1  ∞ xn Note that Li1(x)= n=1 n =–log(1 – x). Kaneko [7] considered the poly-Bernoulli numbers arising from the polylogarithm and defined by

∞ 1    tn Li 1–e–t = B(k) (see [1–4, 8, 17, 22]). (5) 1–e–t k n n! n=0

(1) Note that Bn = Bn(1) (n ≥ 0). Recently, Kim and Kim introduced the degenerate polylogarithm functions defined by

∞  n–1 (–λ) (1)n,1/λ n Li λ(x)= x (k ∈ Z)(see[10, 19]). (6) k, (n –1)!nk n=1

Note that limλ→0 Lik,λ(x)=Lik(x).

Let logλ(t) be the inverse function of eλ(t)suchthatlogλ(eλ(t)) = eλ(logλ(t)) = t.Then we have

∞  n–1   λ (1) λ 1 log (1 + t)= n,1/ tn = (1 + t)λ –1 (see [10, 13, 16, 18]). (7) λ n! λ n=1

Note that from (6)wehaveLi1,λ(x)=–logλ(1 – x). Kim et al. Advances in Difference Equations (2020)2020:444 Page 3 of 9

In [10] the degenerate poly-Bernoulli polynomials are defined by means of the degener- ate polylogarithm function as

∞ Li (1 – e (–t))  tn k,λ λ ex (–t)= β(k)(x) .(8) 1–e (–t) λ n,λ n! λ n=0

(1) n Note that βn,λ(x)=(–1) βn,λ(x), where βn,λ(x) are the Carlitz degenerate Bernoulli poly- nomials defined by (1). In [10, 11, 16] the degenerate Stirling numbers of the second kind are defined by

n (x)n,λ = S2,λ(n, l)(x)l (n ≥ 0). (9) l=0

From (9) we can easily show that the generating function of the degenerate Stirling num- bers of the second kind is given by

∞    n 1 k t eλ(t)–1 = S λ(n, k) (see [13, 16, 18]). (10) k! 2, n! n=k

2 Degenerate poly-Bernoulli polynomials We slightly modify the definition of degenerate poly-Bernoulli polynomials in (8)by

∞ Li (1 – e (–t))  tn k,λ λ ex (t)= B(k) (x) , (11) e (t)–1 λ n,λ n! λ n=0

which are again called the degenerate poly-Bernoulli polynomials. Note different defini- (k) (k) ≥ tions in (8)and(11). In the case x =0,Bn,λ = Bn,λ(0) (n 0) are called the degenerate poly-Bernoulli numbers. Note that by (11)

∞  tn Li (1 – e (–t)) B(1) (x) = 1,λ λ ex (t) n,λ n! e (t)–1 λ n=0 λ ∞  n t x t = e (t)= β λ(x) . (12) e (t)–1 λ n, n! λ n=0

(1) ≥ Thus by (12)wegetBn,λ(x)=βn,λ(x)(n 0). By (11)weget

∞  tn Li (1 – e (–t)) B(k) (x) = k,λ λ ex (t) n,λ n! e (t)–1 λ n=0 λ ∞ ∞  l  m (k) t t = B (x) λ l,λ l! m, m! l=0 m=0  ∞    n n n (k) t = B (x) λ . (13) l l,λ n–l, n! n=0 l=0 Kim et al. Advances in Difference Equations (2020)2020:444 Page 4 of 9

Thus we have   n (k) n (k) B (x)= B (x) λ (n ≥ 0). n,λ l l,λ n–l, l=0

It is not difficult to show that

Lik,λ(1 – eλ(–t)) x eλ(t) eλ(t)–1 ∞  tn = B(k) (x) n,λ n! n=0

ex (t) t e1–λ(–t) t e1–λ(–t) t e1–λ(–t) = λ λ λ ··· λ tdt··· dt. (14) eλ(t)–1 0 1–eλ(–t) 0 1– eλ(–t) 0 1–eλ(–t) (k–1)-times

By (14)weget

∞  tn ex (t) t e1–λ(–t) B(2) (x) = λ λ tdt n,λ n! e (t)–1 1–e (–t) n=0 λ 0 λ ∞ tex (t)  β (1 – λ) tl = λ l,λ (–1)l eλ(t)–1 l +1 l! l=0 ∞ ∞  m  l t l βl,λ(1 – λ) t = β λ(x) (–1) m, m! l +1 l! m=0 l=0  ∞    n n n βl,λ(1 – λ) l t = (–1) β λ(x) . (15) l l +1 n–l, n! n=0 l=0

Comparing the coefficients on both sides of (15), we obtain the following theorem.

Theorem 1 For n ≥ 0, we have   n (2) n n l βl,λ(1 – λ) B (x)=β λ(x)– (1 – 2λ)β λ(x)+ (–1) β λ(x). n,λ n, 4 n–1, l l +1 n–l, l=2

In general,   n (k) n (k) B (x)= B (x) λ. n,λ l l,λ n–l, l=0

Note that by (11)

∞  n t Li λ(1 – eλ(–t)) B(k) (x) = k, ex (t) n,λ n! e (t)–1 λ n=0 λ   t x · 1 = eλ(t) Lik,λ 1–eλ(–t) . (16) eλ(t)–1 t Kim et al. Advances in Difference Equations (2020)2020:444 Page 5 of 9

On the other hand,

∞    n–1   1 1 (–λ) (1)n,1/λ n Li λ 1–eλ(–t) = 1–eλ(–t) t k, t (n –1)!nk n=1 ∞ ∞  n–1  l 1 λ (1)n,1/λ l–1 t = S λ(l, n)(–1) t nk–1 2, l! n=1 l=n ∞  l n–1 l 1 λ (1)n,1/λ l–1 t = (–1) S λ(l, n) t nk–1 2, l! l=1 n=1  ∞  l+1 λn–1(1) S (l +1,n) tl = n,1/λ · 2,λ (–1)l . (17) nk–1 l +1 l! l=0 n=1

By (16)and(17)weget

∞ ∞ ∞  n  m  l+1 p–1 l (k) t t λ (1)p,1/λ l S2,λ(l +1,p) t B (x) = β λ(x) (–1) n,λ n! m, m! pk–1 l +1 l! n=0 m=0 l=0 p=1  ∞    n l+1 p–1 n l n λ S2,λ(l +1,p) t = (–1) (1) λ β λ(x) . (18) l pk–1 p,1/ l +1 n–l, n! n=0 l=0 p=1

Therefore by (18) we obtain the following theorem.

Theorem 2 For n ≥ 0, we have

   n l+1 p–1 (k) n l λ (1)p,1/λ S2,λ(l +1,p) B (x)= (–1) · β λ(x). n,λ l pk–1 l +1 n–l, l=0 p=1

Now we observe that

Lik,λ(1 – eλ(–t)) x+1 Lik,λ(1 – eλ(–t)) x eλ (t)– eλ(t) eλ(t)–1 eλ(t)–1   x = eλ(t) Lik,λ 1–eλ(–t) ∞ ∞  l  m–1   t (–λ) (1)m,1/λ m = (x) λ 1–eλ(–t) l, l! (m –1)!mk l=0 m=1 ∞ ∞ ∞  l  m–1  p t λ (1)m,1/λ p–1 t = (x) λ S λ(p, m)(–1) l, l! mk–1 2, p! l=0 m=1 p=m ∞ ∞  l  p m–1 p t λ (1)m,1/λ p–1 t = (x) λ (–1) S λ(p, m) l, l! mk–1 2, p! l=0 p=1 m=1  ∞    n p m–1 n n p–1 λ (1)m,1/λ t = (–1) S λ(p, m)(x) λ . (19) p mk–1 2, n–p, n! n=1 p=1 m=1 Kim et al. Advances in Difference Equations (2020)2020:444 Page 6 of 9

On the other hand,

Lik,λ(1 – eλ(–t)) x+1 Lik,λ(1 – eλ(–t)) x eλ (t)– eλ(t) eλ(t)–1 eλ(t)–1 ∞ ∞  tn  tn = B(k) (x +1) – B(k) (x) n,λ n! n,λ n! n=0 n=0 ∞  tn = B(k) (x +1)–B(k) (x) . (20) n,λ n,λ n! n=1

Therefore by (19)and(20) we obtain the following theorem.

Theorem 3 For n ∈ N, we have    n p m–1 (k) (k) n p–1 λ (1)m,1/λ B (x +1)–B (x)= (–1) S λ(p, m) (x) λ. n,λ n,λ p mk–1 2, n–p, p=1 m=1

Note that by (11)

Lik,λ(1 – eλ(–t)) x eλ(t) eλ(t)–1 d–1 Lik,λ(1 – eλ(–t)) a+x = d eλ (t) eλ(t)–1 a=0

Li (1 – e (–t)) 1 d–1 dt = k,λ λ ea+x(t) t d d λ a=0 eλ(t)–1

d–1 a+x Lik,λ(1 – eλ(–t)) 1 dt d = e λ (dt) t d e λ (dt)–1 d a=0 d ∞ ∞    p–1   d–1  m 1 (–λ) (1)p,1/λ p a + x m–1 t = 1–eλ(–t) β λ d t (p –1)!pk m, /d d m! p=1 a=0 m=0

∞ ∞ ∞    p–1  l  d–1 m 1 λ (1)p,1/λ l–1 t m–1 a + x t = S λ(l, p)(–1) d β λ t pk–1 2, l! m, /d d m! p=1 l=p m=0 a=0

∞ ∞    l l–1 p–1 l  d–1 m 1 (–1) λ t m–1 a + x t = (1) λS λ(l, p) d β λ t pk–1 p,1/ 2, l! m, /d d m! l=1 p=1 m=0 a=0 ∞ ∞    l+1 p–1 l  d–1 m l λ (1)p,1/λ S2,λ(l +1,p) t m–1 a + x t = (–1) d β λ pk–1 l +1 l! m, /d d m! l=0 p=1 m=0 a=0 ∞      n l+1 p–1 d–1 n n l λ (1)p,1.λ S2,λ(l +1,p) n–l–1 a + x t = (–1) d β λ ) l pk–1 l +1 n–l, /d d n! n=0 l=0 p=1 a=0  ∞      d–1 n l+1 p–1 n n l n–l–1 λ (1)p,1/λ S2,λ(l +1,p) a + x t = (–1) d β λ . l pk–1 l +1 n–l, /d d n! n=0 a=0 l=0 p=1 (21) Kim et al. Advances in Difference Equations (2020)2020:444 Page 7 of 9

On the other hand,

∞  n Li λ(1 – eλ(–t)) t k, ex (t)= B(k) (x) . (22) e (t)–1 λ n,λ n! λ n=0

Therefore by (21)and(22) we obtain the following theorem.

Theorem 4 For n, k ∈ Z with n ≥ 0 and d ∈ N, we have     d–1 n l+1 p–1 (k) n l n–l–1 λ (1)p,1/λ S2,λ(l +1,p) a + x B (x)= (–1) d β λ . n,λ l pk–1 l +1 n–l, /d d a=0 l=0 p=1

From the definition of the degenerate poly-Bernoulli polynomials we have

∞    tn Li (1 – e (–t)) Li (1 – e (–t)) B(k) (x + y) = k,λ λ e(x+y)(t)= k,λ λ ex (t) ey (t) n,λ n! e (t)–1 λ e (t)–1 λ λ n=0 λ λ  ∞ ∞ ∞    l  m  n n (k) t t n (k) t = B (x) (y) λ = B (x)(y) λ . (23) l,λ l! m, m! l l,λ n–l, n! l=0 m=0 n=0 l=0

Comparing the coefficients on both sides of (23), we have   n (k) n (k) B (x + y)= B (x)(y) λ (n ≥ 0). (24) n,λ l l,λ n–l, l=0

Consider the following expression:   n   1 n (k) B (x + λ) λ –(x) λ . (25) λ l n–l,λ l, l, l=0

From (13)weseethat(25)isequalto

1   B(k) (x + λ)–B(k) (x) . (26) λ n,λ n,λ

On the other hand, we see that (25)isalsoequalto   n 1 n (k) B λl(x) λ λ l n–l,λ l–1, l=1   n–1 n –1 (k) = n B (x) λ l n–1–l,λ l, l=0 (k) = nBn–1,λ(x). (27)

Thus from (24), combining (26)with(27), we obtain the following result.

Theorem 5 For n ≥ 1, we have

1   nB(k) (x)= B(k) (x + λ)–B(k) (x) n–1,λ λ n,λ n,λ Kim et al. Advances in Difference Equations (2020)2020:444 Page 8 of 9

and   n (k) n (k) B (x + y)= B (x)(y) λ (n ≥ 0). n,λ l l,λ n–l, l=0

3Conclusion In this paper, we defined the degenerate poly-Bernoulli polynomials, which are slight mod- ifications of the previous ones. We represented the generating function of those polyno- mials as an iterated integral, from which we obtained an explicit expression of those poly- nomials for k = 2 in terms of the Carlitz degenerate Bernoulli polynomials. We also found two explicit expressions of the degenerate poly-Bernoulli polynomials involving the Car- litz degenerate Bernoulli polynomials and the degenerate Stirling numbers of the second kind in Theorems 2 and 4. In addition, we were able to find certain expressions for differ- ences of the degenerate poly-Bernoulli polynomials in Theorems 3 and 5. We refer the reader to [18] and the references therein for three possible immediate ap- plications of our results to probability, differential equations, and symmetry. As one of our future projects, we would like to continue studying degenerate versions of many special polynomials and numbers and their applications to physics, science, and engineering, and mathematics.

Acknowledgements The authors would like to thank the reviewers for their valuable comments and suggestions and Jangjeon Institute for Mathematical Science for the support of this research.

Funding Not applicable.

Availability of data and materials Not applicable.

Competing interests The authors declare that they have no conflicts of interest.

Authors’ contributions TK and DSK conceived of the framework and structured the whole paper; DSK and TK wrote the paper; LCJ, HL, and HYK checked the results of the paper; DSK and TK completed the revision of the paper. All authors have read and approved the final version of the manuscript.

Author details 1School of Science, Xi’an Technological University, Xi’an 710021, China. 2Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea. 3Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea. 4Graduate School of Education, Konkuk University, Seoul 05029, Republic of Korea.

Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 8 June 2020 Accepted: 16 August 2020

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