scholarly journals A Note on the Degenerate Poly-Cauchy Polynomials and Numbers of the Second Kind

Symmetry ◽  
2020 ◽  
Vol 12 (7) ◽  
pp. 1066
Author(s):  
Hye Kyung Kim ◽  
Lee-Chae Jang
Keyword(s):  
Cauchy Numbers

In this paper, we consider the degenerate Cauchy numbers of the second kind were defined by Kim (2015). By using modified polyexponential functions, first introduced by Kim-Kim (2019), we define the degenerate poly-Cauchy polynomials and numbers of the second kind and investigate some identities and relationship between various polynomials and the degenerate poly-Cauchy polynomials of the second kind. Using this as a basis of further research, we define the degenerate unipoly-Cauchy polynomials of the second kind and illustrate their important identities.

scholarly journals Two types of hypergeometric degenerate Cauchy numbers

2020 ◽  
Vol 18 (1) ◽  
pp. 417-433
Author(s):  
Takao Komatsu
Keyword(s):  
Zeta Functions ◽  
Bernoulli Polynomials ◽  
Cauchy Numbers

Abstract In 1985, Howard introduced degenerate Cauchy polynomials together with degenerate Bernoulli polynomials. His degenerate Bernoulli polynomials have been studied by many authors, but his degenerate Cauchy polynomials have been forgotten. In this paper, we introduce some kinds of hypergeometric degenerate Cauchy numbers and polynomials from the different viewpoints. By studying the properties of the first one, we give their expressions and determine the coefficients. Concerning the second one, called H-degenerate Cauchy polynomials, we show several identities and study zeta functions interpolating these polynomials.

scholarly journals Generalizations of poly-Bernoulli and poly-Cauchy numbers

2015 ◽  
Vol 1 (4) ◽  
pp. 799-828 ◽  
Author(s):  
Mehmet Cenkci ◽  
Paul Thomas Young
Keyword(s):  
Cauchy Numbers

A generalized class of restricted Stirling and Lah numbers

2018 ◽  
Vol 68 (4) ◽  
pp. 727-740 ◽  
Author(s):  
Toufik Mansour ◽  
Mark Shattuck
Keyword(s):  
Generating Function ◽  
Stirling Numbers ◽  
Combinatorial Properties ◽  
Exponential Generating Function ◽  
Cauchy Numbers ◽  
Restricted Stirling Numbers

Abstract In this paper, we consider a polynomial generalization, denoted by $\begin{array}{} u_m^{a,b} \end{array}$ (n, k), of the restricted Stirling numbers of the first and second kind, which reduces to these numbers when a = 1 and b = 0 or when a = 0 and b = 1, respectively. If a = b = 1, then $\begin{array}{} u_m^{a,b} \end{array}$ (n, k) gives the cardinality of the set of Lah distributions on n distinct objects in which no block has cardinality exceeding m with k blocks altogether. We derive several combinatorial properties satisfied by $\begin{array}{} u_m^{a,b} \end{array}$ (n, k) and some additional properties in the case when a = b = 1. Our results not only generalize previous formulas found for the restricted Stirling numbers of both kinds but also yield apparently new formulas for these numbers in several cases. Finally, an exponential generating function formula is derived for $\begin{array}{} u_m^{a,b} \end{array}$ (n, k) as well as for the associated Cauchy numbers.

scholarly journals A new class of identities involving Cauchy numbers, harmonic numbers and zeta values

2012 ◽  
Vol 27 (3) ◽  
pp. 305-328 ◽  
Author(s):  
Bernard Candelpergher ◽  
Marc-Antoine Coppo
Keyword(s):  
Harmonic Numbers ◽  
New Class ◽  
Cauchy Numbers ◽  
Zeta Values

scholarly journals POLY-CAUCHY NUMBERS

2013 ◽  
Vol 67 (1) ◽  
pp. 143-153 ◽  
Author(s):  
Takao KOMATSU
Keyword(s):  
Cauchy Numbers

scholarly journals Some Determinants Involving Incomplete Fubini Numbers

2018 ◽  
Vol 26 (3) ◽  
pp. 143-170 ◽  
Author(s):  
Takao Komatsu ◽  
José L. Ramírez
Keyword(s):  
Natural Extensions ◽  
Cauchy Numbers

AbstractWe study some properties of restricted and associated Fubini numbers. In particular, they have the natural extensions of the original Fubini numbers in the sense of determinants. We also introduce modified Bernoulli and Cauchy numbers and study characteristic properties.

Generalized Stirling numbers with poly-Bernoulli and poly-Cauchy numbers

2018 ◽  
Vol 14 (05) ◽  
pp. 1211-1222 ◽  
Author(s):  
Takao Komatsu ◽  
Paul Thomas Young
Keyword(s):  
Stirling Numbers ◽  
Combinatorial Properties ◽  
Cauchy Numbers ◽  
Generalized Stirling Numbers

By using the generalized Stirling numbers studied by Hsu and Shiue, we define a new kind of generalized poly-Bernoulli and poly-Cauchy numbers. By using the formulae of the generalized Stirling numbers, we give their characteristic and combinatorial properties.

scholarly journals Hurwitz-Lerch Type Multi-Poly-Cauchy Numbers

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 335
Author(s):  
Noel Lacpao ◽  
Roberto Corcino ◽  
Mary Vega
Keyword(s):  
Explicit Formula ◽  
Stirling Numbers ◽  
Cauchy Numbers ◽  
Factorial Function

In this paper, we define Hurwitz–Lerch multi-poly-Cauchy numbers using the multiple polylogarithm factorial function. Furthermore, we establish properties of these types of numbers and obtain two different forms of the explicit formula using Stirling numbers of the first kind.

Shifted poly-Cauchy numbers

2014 ◽  
Vol 54 (2) ◽  
pp. 166-181 ◽  
Author(s):  
Takao Komatsu ◽  
László Szalay
Keyword(s):  
Cauchy Numbers
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