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 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 of the Type 2 Degenerate Multi-Poly-Euler Polynomials and Numbers

2020 ◽  
Author(s):  
Waseem Ahmad Khan ◽  
Mehmet Acikgoz ◽  
Ugur Duran
Keyword(s):  
Linear Combination ◽  
Stirling Numbers ◽  
Euler Polynomials ◽  
Addition Formula ◽  
New Class ◽  
Whitney Numbers ◽  
Derivative Formula ◽  
Euler Polynomials And Numbers ◽  
Special Case

In this paper, we consider a new class of polynomials which is called the multi-poly-Euler polynomials. Then, we investigate their some properties and relations. We provide that the type 2 degenerate multi-poly-Euler polynomials equals a linear combination of the degenerate Euler polynomials of higher order and the degenerate 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-Euler polynomials and degenerate Whitney numbers.

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.

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 Notes on the Poly-Korobov polynomials and related polynomials

Filomat ◽  
2020 ◽  
Vol 34 (2) ◽  
pp. 469-474
Author(s):  
Burak Kurt
Keyword(s):  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Euler Polynomials ◽  
Changhee Polynomials

In recent years, many mathematicians ([2], [7], [8], [9], [15], [16], [21]) introduced and investigated for the Korobov polynomials. They gave some identities and relations for the Korobov type polynomials. In this work, we give some relations for the first kind Korobov polynomials and Korobov type Changhee polynomials. Further, wegive two relations between the poly-Changhee polynomials and the poly-Korobov polynomials. Also, we give a relation among the poly-Korobov type Changhee polynomials, the Stirling numbers of the second kind, the Euler polynomials and the Bernoulli numbers.

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.

scholarly journals On Degenerate Truncated Special Polynomials

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 144 ◽  
Author(s):  
Ugur Duran ◽  
Mehmet Acikgoz
Keyword(s):  
Bernstein Polynomials ◽  
Stirling Numbers ◽  
Bell Polynomials ◽  
Euler Polynomials ◽  
Exponential Polynomials ◽  
Special Polynomials ◽  
Summation Formulas ◽  
Special Cases ◽  
Proof Techniques

The main aim of this paper is to introduce the degenerate truncated forms of multifarious special polynomials and numbers and is to investigate their various properties and relationships by using the series manipulation method and diverse special proof techniques. The degenerate truncated exponential polynomials are first considered and their several properties are given. Then the degenerate truncated Stirling polynomials of the second kind are defined and their elementary properties and relations are proved. Also, the degenerate truncated forms of the bivariate Fubini and Bell polynomials and numbers are introduced and various relations and formulas for these polynomials and numbers, which cover several summation formulas, addition identities, recurrence relationships, derivative property and correlations with the degenerate truncated Stirling polynomials of the second kind, are acquired. Thereafter, the truncated degenerate Bernoulli and Euler polynomials are considered and multifarious correlations and formulas including summation formulas, derivation rules and correlations with the degenerate truncated Stirling numbers of the second are derived. In addition, regarding applications, by introducing the degenerate truncated forms of the classical Bernstein polynomials, we obtain diverse correlations and formulas. Some interesting surface plots of these polynomials in the special cases are provided.

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 [...]

scholarly journals On the generalized q-poly-Euler polynomials of the second kind

Filomat ◽  
2020 ◽  
Vol 34 (2) ◽  
pp. 475-482
Author(s):  
Veli Kurt
Keyword(s):  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Euler Polynomials ◽  
Euler Numbers ◽  
Basic Properties

In this work, we define the generalized q-poly-Euler numbers of the second kind of order ? and the generalized q-poly-Euler polynomials of the second kind of order ?. We investigate some basic properties for these polynomials and numbers. In addition, we obtain many identities, relations including the Roger-Sz?go polynomials, the Al-Salam Carlitz polynomials, q-analogue Stirling numbers of the second kind and two variable Bernoulli polynomials.

scholarly journals Analytical functions of generalized complex variables and some appllications

2014 ◽  
Vol 5 (1) ◽  
pp. 569-575
Author(s):  
Bimurat Sagindykov
Keyword(s):  
Fluid Dynamics ◽  
Elasticity Theory ◽  
Laplace Equation ◽  
Complex Variable ◽  
Complex Variables ◽  
Poisson Formula ◽  
Generalized Poisson ◽  
Analytical Functions ◽  
Generalized Complex ◽  
Generalized Laplace Equation

The object of this work is to use functions of a generalized complex variable to solve the problems of fluid dynamics and elasticity theory. In this paper, we obtain Cauchy-Riemann conditions, generalized Laplace equation and the generalized Poisson formula for such functions.

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