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Sheffer polynomials can be characterized using different Stieltjes integrals. These families of polynomials have been recently extended to the Dunkl context. In this way some classical operators as the derivative operator or the difference operator a
Externí odkaz:
http://arxiv.org/abs/2501.01364
Autor:
Kim, Taekyun, Kim, Dae san
The aim of this paper is to study degenerate Eulerian polynomials and degenerate Eulerian numbers, respectively as degenerate versions of the Eulerian polynomials and the Eulerian numbers, and to derive some of their properties. Specifically, we deri
Externí odkaz:
http://arxiv.org/abs/2412.03119
Autor:
Wang, Jiaqi, Ma, Rong
Let $p$ be an odd prime, Jianqiang Zhao has established a curious congruence $$ \sum_{i+j+k=p \atop i,j,k > 0} \frac{1}{ijk} \equiv -2B_{p-3}\pmod p , $$ where $B_{n}$ denotes the $n-$th Bernoulli numbers. In this paper, we will generalize this probl
Externí odkaz:
http://arxiv.org/abs/2411.03148
Autor:
Samsonadze, Eteri
The problem of finding the sum of a polynomial's values is considered. In particular, for any $n\geq 3$, the explicit formula for the sum of the $n$th powers of natural numbers $S_n=\sum_{x=1}^{m}x^{n}$ is proved: $$\sum_{x=1}^{m}x^{n}=(-1)^{n}m(m+1)
Externí odkaz:
http://arxiv.org/abs/2411.11859
Autor:
Kim, Taekyun, Kim, Dae San
The aim of this paper is to derive a summation formula for the alternating infinite series and an expression for zeta function by using hyperbolic secant random variables. These identities involve Euler numbers and are obtained by computing the momen
Externí odkaz:
http://arxiv.org/abs/2410.05895
Autor:
Murakami, Yuya
We introduce $ L $-functions attached to negative definite plumbed manifolds as the Mellin transforms of homological blocks. We prove that they are entire functions and their values at $ s=0 $ are equal to the Witten--Reshetikhin--Turaev invariants b
Externí odkaz:
http://arxiv.org/abs/2410.05611
Autor:
Farhi, Bakir
This paper is devoted to establishing several new formulas relating Bernoulli and Stirling numbers of both kinds.
Comment: 15 pages
Comment: 15 pages
Externí odkaz:
http://arxiv.org/abs/2410.05207
In recent years, both degenerate versions and probabilistic extensions of many special numbers and polynomials have been explored. For instance, degenerate Bernstein polynomials and probabilistic Bernstein polynomials were investigated earlier. Assum
Externí odkaz:
http://arxiv.org/abs/2410.04464
Autor:
Ikeda, Karin, Kadono, Yuta
The central binomial series is a subject that has been extensively studied, for example in the context of the irrationality of Riemann zeta values. In this paper, the Hurwitz version of the central binomial series is defined by adding one real parame
Externí odkaz:
http://arxiv.org/abs/2409.16026
Autor:
Kim, Taekyun, Kim, Dae san
In this paper, we consider sums of values of degenerate falling factorials and give a probabilistic proof of a recurrence relation for them. This may be viewed as a degenerate version of the recent probabilistic proofs on sums of powers of integers.<
Externí odkaz:
http://arxiv.org/abs/2409.07742