scholarly journals Some Identities of the Degenerate Multi-Poly-Bernoulli Polynomials of Complex Variable

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
G. Muhiuddin ◽  
W. A. Khan ◽  
U. Duran ◽  
D. Al-Kadi
Keyword(s):  
Complex Variable ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Complex Variables ◽  
Whitney Numbers

In this paper, we introduce degenerate multi-poly-Bernoulli polynomials and derive some identities of these polynomials. We give some relationship between degenerate multi-poly-Bernoulli polynomials degenerate Whitney numbers and Stirling numbers of the first kind. Moreover, we define degenerate multi-poly-Bernoulli polynomials of complex variables, and then, we derive several properties and relations.

scholarly journals Construction on the Degenerate Poly-Frobenius-Euler Polynomials of Complex Variable

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Ghulam Muhiuddin ◽  
Waseem Ahmad Khan ◽  
Deena Al-Kadi
Keyword(s):  
Complex Variable ◽  
Stirling Numbers ◽  
Complex Variables ◽  
Euler Polynomials ◽  
Whitney Numbers

In this paper, we introduce degenerate poly-Frobenius-Euler polynomials and derive some identities of these polynomials. We give some relationships between degenerate poly-Frobenius-Euler polynomials and degenerate Whitney numbers and Stirling numbers of the first kind. Moreover, we define degenerate poly-Frobenius-Euler polynomials of complex variables and then we derive several properties and relations.

scholarly journals Note on the Type 2 Degenerate Multi-Poly-Euler Polynomials

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1691
Author(s):  
Waseem Ahmad Khan ◽  
Mehmet Acikgoz ◽  
Ugur Duran
Keyword(s):  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Euler Polynomials ◽  
Addition Formula ◽  
Whitney Numbers ◽  
Derivative Formula ◽  
New Type ◽  
Special Case ◽  
Math Phys

Kim and Kim (Russ. J. Math. Phys. 26, 2019, 40-49) introduced polyexponential function as an inverse to the polylogarithm function and by this, constructed a new type poly-Bernoulli polynomials. Recently, by using the polyexponential function, a number of generalizations of some polynomials and numbers have been presented and investigated. Motivated by these researches, in this paper, multi-poly-Euler polynomials are considered utilizing the degenerate multiple polyexponential functions and then, their properties and relations are investigated and studied. That the type 2 degenerate multi-poly-Euler polynomials equal a linear combination of the degenerate Euler polynomials of higher order and the degenerate Stirling numbers of the first kind is proved. Moreover, an addition formula and a derivative formula are derived. Furthermore, in a special case, a correlation between the type 2 degenerate multi-poly-Euler polynomials and degenerate Whitney numbers is shown.

scholarly journals A Note on Type 2 Degenerate Multi-Poly-Bernoulli Polynomials and Numbers

2020 ◽  
Author(s):  
Waseem A Khan ◽  
Aysha Khan ◽  
Ugur Duran
Keyword(s):  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Addition Formula ◽  
Genocchi Polynomials ◽  
Whitney Numbers ◽  
Derivative Formula ◽  
Bernoulli Polynomials And Numbers ◽  
Definition Of ◽  
Special Case

Inspired by the definition of degenerate multi-poly-Genocchi polynomials given by using the degenerate multi-polyexponential functions. In this paper, we consider a class of new generating function for the degenerate multi-poly-Bernoulli polynomials, called the type 2 degenerate multi-poly-Bernoulli polynomials by means of the degenerate multiple polyexponential functions. Then, we investigate their some properties and relations. We show that the type 2 degenerate multi-poly-Bernoulli polynomials equals a linear combination of the weighted degenerate Bernoulli polynomials and Stirling numbers of the first kind. Moreover, we provide an addition formula and a derivative formula. Furthermore, in a special case, we acquire a correlation between the type 2 degenerate multi-poly-Bernoulli numbers and degenerate Whitney numbers.

New family of Whitney numbers

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 309-320 ◽  
Author(s):  
B.S. El-Desouky ◽  
Nenad Cakic ◽  
F.A. Shiha
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Stirling Numbers ◽  
Explicit Formulas ◽  
Basic Properties ◽  
New Family ◽  
Whitney Numbers ◽  
Special Cases

In this paper we give a new family of numbers, called ??-Whitney numbers, which gives generalization of many types of Whitney numbers and Stirling numbers. Some basic properties of these numbers such as recurrence relations, explicit formulas and generating functions are given. Finally many interesting special cases are derived.

scholarly journals Two-Variable Type 2 Poly-Fubini Polynomials

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 281
Author(s):  
Ghulam Muhiuddin ◽  
Waseem Ahmad Khan ◽  
Ugur Duran
Keyword(s):  
Integral Representation ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Higher Order ◽  
Variable Type ◽  
Summation Formulas

In the present work, a new extension of the two-variable Fubini polynomials is introduced by means of the polyexponential function, which is called the two-variable type 2 poly-Fubini polynomials. Then, some useful relations including the Stirling numbers of the second and the first kinds, the usual Fubini polynomials, and the higher-order Bernoulli polynomials are derived. Also, some summation formulas and an integral representation for type 2 poly-Fubini polynomials are investigated. Moreover, two-variable unipoly-Fubini polynomials are introduced utilizing the unipoly function, and diverse properties involving integral and derivative properties are attained. Furthermore, some relationships covering the two-variable unipoly-Fubini polynomials, the Stirling numbers of the second and the first kinds, and the Daehee polynomials are acquired.

scholarly journals Sums of powers of integers and hyperharmonic numbers

2021 ◽  
Vol 27 (2) ◽  
pp. 101-110
Author(s):  
José Luis Cereceda
Keyword(s):  
Arithmetic Progression ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Explicit Representation ◽  
Sums Of Powers ◽  
Hyperharmonic Numbers ◽  
New Formula ◽  
Positive Integers

In this paper, we obtain a new formula for the sums of k-th powers of the first n positive integers, Sk(n), that involves the hyperharmonic numbers and the Stirling numbers of the second kind. Then, using an explicit representation for the hyperharmonic numbers, we generalize this formula to the sums of powers of an arbitrary arithmetic progression. Furthermore, we express the Bernoulli polynomials in terms of hyperharmonic polynomials and Stirling numbers of the second kind. Finally, we extend the obtained formula for Sk(n) to negative values of n.

Introduction to the axiomatic theory of elementary functions

2021 ◽  
Author(s):  
Andrey Shishkin
Keyword(s):  
Complex Variable ◽  
Real Variable ◽  
Complex Variables ◽  
State University ◽  
Axiomatic Theory ◽  
Elementary Functions ◽  
State Pedagogical ◽  
Pedagogical Institute ◽  
State Pedagogical Institute ◽  
Basic Concepts

Contains an exposition of the basic concepts and theorems of the axiomatic theory of the basic elementary functions of real and complex variables. The textbook is written on the basis of lectures given by the author for a number of years at the Armavir State Pedagogical University, at the Slavyansk-on-Kuban State Pedagogical Institute and at the branch of the Kuban State University in Slavyansk-on-Kuban. It is intended for students of natural-mathematical profiles of preparation of the direction "Pedagogical education". It can be used in the study of mathematical analysis, the theory of functions of a real variable, the theory of functions of a complex variable, etc.

CLOSED FORMS FOR DEGENERATE BERNOULLI POLYNOMIALS

2020 ◽  
Vol 101 (2) ◽  
pp. 207-217 ◽  
Author(s):  
LEI DAI ◽  
HAO PAN
Keyword(s):  
Number Theory ◽  
Closed Form ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Closed Form Expression ◽  
Form Expression

Qi and Chapman [‘Two closed forms for the Bernoulli polynomials’, J. Number Theory159 (2016), 89–100] gave a closed form expression for the Bernoulli polynomials as polynomials with coefficients involving Stirling numbers of the second kind. We extend the formula to the degenerate Bernoulli polynomials, replacing the Stirling numbers by degenerate Stirling numbers of the second kind.

scholarly journals New results on the q-generalized Bernoulli polynomials of level m

2019 ◽  
Vol 52 (1) ◽  
pp. 511-522
Author(s):  
Alejandro Urieles ◽  
María José Ortega ◽  
William Ramírez ◽  
Samuel Vega
Keyword(s):  
Gamma Function ◽  
Bernstein Polynomials ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Algebraic Properties ◽  
Generalized Bernoulli Polynomials ◽  
Polynomial Class

AbstractThis paper aims to show new algebraic properties from the q-generalized Bernoulli polynomials B_n^{[m - 1]}(x;q) of level m, as well as some others identities which connect this polynomial class with the q-generalized Bernoulli polynomials of level m, as well as the q-gamma function, and the q-Stirling numbers of the second kind and the q-Bernstein polynomials.

scholarly journals Corrections: Kim, T.; et al. Some Identities for Euler and Bernoulli Polynomials and Their Zeros. Axioms 2018, 7, 56

Axioms ◽  
2019 ◽  
Vol 8 (4) ◽  
pp. 107
Author(s):  
Taekyun Kim ◽  
Cheon Ryoo
Keyword(s):  
Complex Variable ◽  
Bernoulli Polynomials ◽  
Real Variable ◽  
Euler Polynomials

The authors, Kim and Ryoo in [1], studied Euler polynomials and Bernoulli polynomials withan extended variable to a complex variable, replacing real variable x by complex variable x + iy,and achieved several useful identities and properties [...]

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