scholarly journals A Parametric Kind of the Degenerate Fubini Numbers and Polynomials

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 405 ◽  
Author(s):  
Sunil Kumar Sharma ◽  
Waseem A. Khan ◽  
Cheon Seoung Ryoo
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Stirling Numbers ◽  
Trigonometric Functions ◽  
Summation Formulas ◽  
Degenerate Type

In this article, we introduce the parametric kinds of degenerate type Fubini polynomials and numbers. We derive recurrence relations, identities and summation formulas of these polynomials with the aid of generating functions and trigonometric functions. Further, we show that the parametric kind of the degenerate type Fubini polynomials are represented in terms of the Stirling 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 Multifarious Results for q-Hermite Based Frobenius Type Eulerian Polynomials

2019 ◽  
Author(s):  
Waseem Khan ◽  
Idrees Ahmad Khan ◽  
Mehmet Acikgoz ◽  
Ugur Duran
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Series Representation ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Eulerian Polynomials ◽  
New Class ◽  
Genocchi Polynomials

In this paper, a new class of q-Hermite based Frobenius type Eulerian polynomials is introduced by means of generating function and series representation. Several fundamental formulas and recurrence relations for these polynomials are derived via different generating methods. Furthermore, diverse correlations including the q-Apostol-Bernoulli polynomials, the q-Apostol-Euler poynoomials, the q-Apostol-Genocchi polynomials and the q-Stirling numbers of the second kind are also established by means of the their generating functions.

scholarly journals The multiparameter r-Whitney numbers

Filomat ◽  
2019 ◽  
Vol 33 (3) ◽  
pp. 931-943 ◽  
Author(s):  
B. El-Desouky ◽  
F.A. Shiha ◽  
Ethar Shokr
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Matrix Representation ◽  
Stirling Numbers ◽  
Combinatorial Identities ◽  
Harmonic Numbers ◽  
Explicit Formulas ◽  
Whitney Numbers ◽  
Special Cases ◽  
Generalized Stirling Numbers

In this paper, we define the multiparameter r-Whitney numbers of the first and second kind. The recurrence relations, generating functions , explicit formulas of these numbers and some combinatorial identities are derived. Some relations between these numbers and generalized Stirling numbers of the first and second kind, Lah numbers, C-numbers and harmonic numbers are deduced. Furthermore, some interesting special cases are given. Finally matrix representation for these relations are given.

scholarly journals A note on degenerate r-Stirling numbers

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim ◽  
Hyunseok Lee ◽  
Jin-Woo Park
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Stirling Numbers ◽  
Explicit Expressions

Abstract The aim of this paper is to study the unsigned degenerate r-Stirling numbers of the first kind as degenerate versions of the r-Stirling numbers of the first kind and the degenerate r-Stirling numbers of the second kind as those of the r-Stirling numbers of the second kind. For the degenerate r-Stirling numbers of both kinds, we derive recurrence relations, generating functions, explicit expressions, and some identities involving them.

scholarly journals New classes of recurrence relations involving hyperbolic functions, special numbers and polynomials

2021 ◽  
pp. 15-15
Author(s):  
Yilmaz Simsek
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Stirling Numbers ◽  
Umbral Calculus ◽  
Discrete Mathematics ◽  
New Class ◽  
Integral Formulas ◽  
Euler Numbers And Polynomials ◽  
Derivative Formulas ◽  
Special Numbers And Polynomials

By using the calculus of finite differences methods and the umbral calculus, we construct recurrence relations for a new class of special numbers. Using this recurrence relation, we define generating functions for this class of special numbers and also new classes of special polynomials. We investigate some properties of these generating functions. By using these generating functions with their functional equations, we obtain many new and interesting identities and relations related to these classes of special numbers and polynomials, the Bernoulli numbers and polynomials, the Euler numbers and polynomials, the Stirling numbers. Finally, some derivative formulas and integral formulas for these classes of special numbers and polynomials are given. In general, this article includes results that have the potential to be used in areas such as discrete mathematics, combinatorics analysis and their applications.

scholarly journals Hyperbolic Jacobsthal Numbers

2019 ◽  
pp. 1-9
Author(s):  
Can Murat Dikmen
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Inner Product ◽  
Summation Formulas

In this paper, we introduce the Hyperbolic Jacobsthal numbers and we present recurrence relations, Binet's formulas, generating functions and the summation formulas for these numbers. Moreover, we investgate Lorentzian inner product for the hyperbolic Jacobsthal vectors.

scholarly journals Truncated Fubini Polynomials

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 431 ◽  
Author(s):  
Ugur Duran ◽  
Mehmet Acikgoz
Keyword(s):  
Recurrence Relations ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Euler Polynomials ◽  
Summation Formulas

In this paper, we introduce the two-variable truncated Fubini polynomials and numbers and then investigate many relations and formulas for these polynomials and numbers, including summation formulas, recurrence relations, and the derivative property. We also give some formulas related to the truncated Stirling numbers of the second kind and Apostol-type Stirling numbers of the second kind. Moreover, we derive multifarious correlations associated with the truncated Euler polynomials and truncated Bernoulli polynomials.

scholarly journals Identities and Computation Formulas for Combinatorial Numbers Including Negative Order Changhee Polynomials

Symmetry ◽  
2019 ◽  
Vol 12 (1) ◽  
pp. 9 ◽  
Author(s):  
Daeyeoul Kim ◽  
Yilmaz Simsek ◽  
Ji Suk So
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Euler Numbers ◽  
Integral Formulas ◽  
Euler Numbers And Polynomials ◽  
Negative Order ◽  
Changhee Polynomials ◽  
Changhee Numbers

The purpose of this paper is to construct generating functions for negative order Changhee numbers and polynomials. Using these generating functions with their functional equation, we prove computation formulas for combinatorial numbers and polynomials. These formulas include Euler numbers and polynomials of higher order, Stirling numbers, and negative order Changhee numbers and polynomials. We also give some properties of these numbers and polynomials with their generating functions. Moreover, we give relations among Changhee numbers and polynomials of negative order, combinatorial numbers and polynomials and Bernoulli numbers of the second kind. Finally, we give a partial derivative of an equation for generating functions for Changhee numbers and polynomials of negative order. Using these differential equations, we derive recurrence relations, differential and integral formulas for these numbers and polynomials. We also give p-adic integrals representations for negative order Changhee polynomials including Changhee numbers and Deahee numbers.

scholarly journals New families of special numbers for computing negative order Euler numbers and related numbers and polynomials

2018 ◽  
Vol 12 (1) ◽  
pp. 1-35 ◽  
Author(s):  
Yilmaz Simsek
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Bernstein Polynomials ◽  
Fibonacci Numbers ◽  
Stirling Numbers ◽  
Euler Numbers ◽  
Open Questions ◽  
Negative Order ◽  
Central Factorial Numbers ◽  
Probability And Statistics

The main purpose of this paper is to construct new families of special numbers with their generating functions. These numbers are related to many well-known numbers, which are Bernoulli numbers, Fibonacci numbers, Lucas numbers, Stirling numbers of the second kind and central factorial numbers. Our other inspiration of this paper is related to the Golombek's problem [15] "Aufgabe 1088. El. Math., 49 (1994), 126-127". Our first numbers are not only related to the Golombek's problem, but also computation of the negative order Euler numbers. We compute a few values of the numbers which are given by tables. We give some applications in probability and statistics. That is, special values of mathematical expectation of the binomial distribution and the Bernstein polynomials give us the value of our numbers. Taking derivative of our generating functions, we give partial differential equations and also functional equations. By using these equations, we derive recurrence relations and some formulas of our numbers. Moreover, we come up with a conjecture with two open questions related to our new numbers. We give two algorithms for computation of our numbers. We also give some combinatorial applications, further remarks on our new numbers and their generating functions.

scholarly journals Identities and relations for special numbers and polynomials: An approach to trigonometric functions

Filomat ◽  
2020 ◽  
Vol 34 (2) ◽  
pp. 535-542
Author(s):  
Neslihan Kilar ◽  
Yilmaz Simsek
Keyword(s):  
Generating Functions ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Trigonometric Functions ◽  
Euler Numbers ◽  
Derived Relations ◽  
Special Numbers And Polynomials

In this paper, by using trigonometric functions and generating functions, identities and relations associated with special numbers and polynomials are derived. Relations among the combinatorial numbers, the Bernoulli polynomials, the Euler numbers, the Stirling numbers and others special numbers and polynomials are given.

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