Degenerate poly-Bernoulli polynomials arising from degenerate polylogarithm

Kim, Taekyun; Kim, Dansan; Kim, Hanyoung; Lee, Hyunseok; Jang, Lee-Chae

doi:10.1186/s13662-020-02901-9

Cited by 13 publications

(9 citation statements)

References 14 publications

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“…These ways of investigating special polynomials and numbers can be also applied to degenerate versions of such polynomials and numbers. Indeed, in recent years, many mathematicians have drawn their attention to studies of degenerate versions of many special polynomials and numbers by using the aforementioned means ( [9,10,14] and references therein). The incomplete and complete Bell polynomials arise in many different contexts as we stated in the Introduction.…”

Section: Resultsmentioning

confidence: 99%

Complete and incomplete Bell polynomials associated with Lah–Bell numbers and polynomials

et al. 2021

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The nth r-extended Lah–Bell number is defined as the number of ways a set with $n+r$ n + r elements can be partitioned into ordered blocks such that r distinguished elements have to be in distinct ordered blocks. The aim of this paper is to introduce incomplete r-extended Lah–Bell polynomials and complete r-extended Lah–Bell polynomials respectively as multivariate versions of r-Lah numbers and the r-extended Lah–Bell numbers and to investigate some properties and identities for these polynomials. From these investigations we obtain some expressions for the r-Lah numbers and the r-extended Lah–Bell numbers as finite sums.

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Section: Resultsmentioning

confidence: 99%

Complete and incomplete Bell polynomials associated with Lah–Bell numbers and polynomials

et al. 2021

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“…To prove ( 4) and ( 5) we recall that Li k (−i) = −2 −k η(k) − iβ(k) and then use series representation (7).…”

Section: Resultsmentioning

confidence: 99%

“…Remark 3. Some relevant work highlighting the applications of the polylogarithm and related functions are [5,6,7,8].…”

Section: Remark 2 the Integral Relationmentioning

confidence: 99%

Some nested sums in terms of the Riemann zeta function and the Dirichlet beta function

Jha¹

2019

Preprint

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Let $k\geq 2$ be an integer, and $\zeta(k)=\sum_{n=1}^{\infty}n^{-k}$. Then we prove the following$$\zeta(k+1)=\frac{1}{k}\sum_{1\leq i_{1}\leq i_{2}\cdots \leq i_{k}}\frac{1}{i_{1}i_{2}\cdots i_{k}}\cdot\frac{1}{i_{k}},$$$$\zeta(k)=\frac{1}{k-1}\sum_{1\leq i_{1}\leq i_{2}\cdots \leq i_{k}}\frac{1}{i_{1}i_{2}\cdots i_{k}}\cdot\frac{1}{(i_{k}+1)},$$$$\zeta(k)=\frac{1}{2^{k}-2}\sum_{1\leq i_{1}\leq i_{2}\cdots \leq i_{k}}\frac{1}{i_{1}i_{2}\cdots i_{k}}\cdot \frac{\binom{2i_{k}}{i_{k}+1}}{i_{k}\cdot 2^{2i_{k}+1}}.,$$where the summation is over all $k$-tuples $(i_{1},i_{2},\cdots,i_{k})$ of natural numbers satisfying $1\leq i_{1}\leq i_{2}\leq \cdots\leq i_{k}$ for fixed $k$. Additionally, we also derive$$\eta(k)=\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^{k}}=2^{k}\sum_{1\leq i_{1}\leq i_{2}\cdots \leq i_{k}}\frac{1}{i_{1}i_{2}\cdots i_{k}}\cdot \frac{1}{2^{(i_{k}+1)/{2}}}\cos\left(\frac{\pi(i_k+1)}{4}\right),$$$$\beta (k)=\sum _{{n=0}}^{\infty}{\frac {(-1)^{n}}{(2n+1)^{k}}}=\sum_{1\leq i_{1}\leq i_{2}\cdots \leq i_{k}}\frac{1}{i_{1}i_{2}\cdots i_{k}}\cdot \frac{1}{2^{(i_{k}+1)/{2}}}\sin\left(\frac{\pi(i_k+1)}{4}\right).$$

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“…Carlitz is the first one who initiated the study of degenerate versions of some special numbers and polynomials, namely the degenerate Bernoulli and Euler polynomials and numbers (see [1]). In recent years, studying degenerate versions of some special numbers and polynomials regained interests of some mathematicians with their interests not only in combinatorial and arithmetic properties but also in applications to differential equations, identities of symmetry and probability theory (see [5,6,9,10,13,15,16] and the references therein). It is noteworthy that studying degenerate versions is not only limited to polynomials but also can be extended to transcendental functions like gamma functions (see [8]).…”

Section: Introductionmentioning

confidence: 99%

“…As is well-known, poly-Bernoulli polynomials are defined in terms of polylogarithm functions. Recently, as degenerate versions of such functions and polynomials, degenerate polylogarithm functions were introduced and degenerate poly-Bernoulli polynomials were defined by means of the degenerate polylogarithm functions, and some properties of the degenerate poly-Bernoulli polynomials were investigated (see [13]).…”

Section: Introductionmentioning

confidence: 99%

Representations of degenerate poly-Bernoulli polynomials

Kim

Ким

Kwon

et al. 2020

Preprint

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As is well-known, poly-Bernoulli polynomials are defined in terms of polylogarithm functions. Recently, as degenerate versions of such functions and polynomials, degenerate polylogarithm functions were introduced and degenerate poly-Bernoulli polynomials were defined by means of the degenerate polylogarithm functions, and some of their properties were investigated. The aim of this paper is to further study some properties of the degenerate poly-Bernoulli polynomials by using three formulas coming from the recently developed 'λ -umbral calculus.' In more detail, among other things, we represent the degenerate poly-Bernoulli polynomials by higher-order degenerate Bernoulli polynomials and by higher-order degenerate derangement polynomials.

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Degenerate poly-Bernoulli polynomials arising from degenerate polylogarithm

Cited by 13 publications

References 14 publications

Complete and incomplete Bell polynomials associated with Lah–Bell numbers and polynomials

Complete and incomplete Bell polynomials associated with Lah–Bell numbers and polynomials

Some nested sums in terms of the Riemann zeta function and the Dirichlet beta function

Representations of degenerate poly-Bernoulli polynomials

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