scholarly journals A New Generating Function of (q-) Bernstein-Type Polynomials and Their Interpolation Function

2010 ◽  
Vol 2010 ◽  
pp. 1-12 ◽  
Author(s):  
Yilmaz Simsek ◽  
Mehmet Acikgoz
Keyword(s):  
Recurrence Relation ◽  
Generating Function ◽  
Hermite Polynomials ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Higher Order ◽  
Interpolation Function ◽  
Bernstein Type ◽  
Mellin Transformation

The main object of this paper is to construct a new generating function of the (q-) Bernstein-type polynomials. We establish elementary properties of this function. By using this generating function, we derive recurrence relation and derivative of the (q-) Bernstein-type polynomials. We also give relations between the (q-) Bernstein-type polynomials, Hermite polynomials, Bernoulli polynomials of higher order, and the second-kind Stirling numbers. By applying Mellin transformation to this generating function, we define interpolation of the (q-) Bernstein-type polynomials. Moreover, we give some applications and questions on approximations of (q-) Bernstein-type polynomials, moments of some distributions in Statistics.

scholarly journals A New Class of Higher-Order Hypergeometric Bernoulli Polynomials Associated with Lagrange–Hermite Polynomials

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 648
Author(s):  
Ghulam Muhiuddin ◽  
Waseem Ahmad Khan ◽  
Ugur Duran ◽  
Deena Al-Kadi
Keyword(s):  
Generating Function ◽  
Functional Equations ◽  
Laguerre Polynomials ◽  
Hermite Polynomials ◽  
Bernoulli Polynomials ◽  
Higher Order ◽  
New Class

The purpose of this paper is to construct a unified generating function involving the families of the higher-order hypergeometric Bernoulli polynomials and Lagrange–Hermite polynomials. Using the generating function and their functional equations, we investigate some properties of these polynomials. Moreover, we derive several connected formulas and relations including the Miller–Lee polynomials, the Laguerre polynomials, and the Lagrange Hermite–Miller–Lee polynomials.

scholarly journals New type of degenerate Daehee polynomials of the second kind

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Sunil Kumar Sharma ◽  
Waseem A. Khan ◽  
Serkan Araci ◽  
Sameh S. Ahmed
Keyword(s):  
Generating Function ◽  
Special Class ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Higher Order ◽  
Bernoulli Numbers And Polynomials ◽  
New Type ◽  
Order Degenerate ◽  
Math Phys

Abstract Recently, Kim and Kim (Russ. J. Math. Phys. 27(2):227–235, 2020) have studied new type degenerate Bernoulli numbers and polynomials by making use of degenerate logarithm. Motivated by (Kim and Kim in Russ. J. Math. Phys. 27(2):227–235, 2020), we consider a special class of polynomials, which we call a new type of degenerate Daehee numbers and polynomials of the second kind. By using their generating function, we derive some new relations including the degenerate Stirling numbers of the first and second kinds. Moreover, we introduce a new type of higher-order degenerate Daehee polynomials of the second kind. We also derive some new identities and properties of this type of polynomials.

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 A Newton Interpolation Approach to Generalized Stirling Numbers

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Aimin Xu
Keyword(s):  
Explicit Expression ◽  
Recurrence Relation ◽  
Generating Function ◽  
Stirling Numbers ◽  
Divided Differences ◽  
Newton Interpolation ◽  
Basic Properties ◽  
Generalized Stirling Numbers

We employ the generalized factorials to define a Stirling-type pair{s(n,k;α,β,r),S(n,k;α,β,r)}which unifies various Stirling-type numbers investigated by previous authors. We make use of the Newton interpolation and divided differences to obtain some basic properties of the generalized Stirling numbers including the recurrence relation, explicit expression, and generating function. The generalizations of the well-known Dobinski's formula are further investigated.

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 Closed Forms for Derangement Numbers in Terms of the Hessenberg Determinants

2016 ◽  
Author(s):  
Feng Qi ◽  
Jiao-Lian Zhao ◽  
Bai-Ni Guo
Keyword(s):  
Recurrence Relation ◽  
Generating Function ◽  
Higher Order ◽  
Order Derivative ◽  
Exponential Generating Function ◽  
Higher Order Derivative

In the paper, the authors find closed forms for derangement numbers in terms of the Hessenberg determinants, discover a recurrence relation of derangement numbers, present a formula for any higher order derivative of the exponential generating function of derangement numbers, and compute some related Hessenberg and tridiagonal determinants.

scholarly journals Relationships between Mahler Expansion and Higher Order q-Daehee polynomials

2019 ◽  
Author(s):  
Ugur Duran ◽  
Mehmet Acikgoz
Keyword(s):  
Gamma Function ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Higher Order

In this paper, we derive multifarious relationships among the two types of higher order q-Daehee polynomials and p-adic gamma function via Mahler theorem. Also, we compute some weighted p-adic q-integrals of the derivative of p-adic gamma function related to the Stirling numbers of the both kinds and the q-Bernoulli polynomials of order k.

scholarly journals Some Identities Relating to Eulerian Polynomials and Involving Stirling Numbers

2017 ◽  
Author(s):  
Feng Qi ◽  
Dongkyu Lim ◽  
Bai-Ni Guo
Keyword(s):  
Differential Equations ◽  
Generating Function ◽  
Nonlinear Differential Equations ◽  
Stirling Numbers ◽  
Higher Order ◽  
Open Problems ◽  
Eulerian Polynomials ◽  
Finite Sums

In the paper, the authors establish two identities, which can be regarded as nonlinear differential equations, for the generating function of Eulerian polynomials, find two identities for the Stirling numbers of the second kind, and present two identities for Eulerian polynomials and higher order Eulerian polynomials, pose two open problems about summability of two finite sums involving the Stirling numbers of the second kind. Some of these conclusions meaningfully and significantly simplify several known results.

scholarly journals A New Class of Symmetric Beta Type Distributions Constructed by Means of Symmetric Bernstein Type Basis Functions

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 779 ◽  
Author(s):  
Fusun Yalcin ◽  
Yilmaz Simsek
Keyword(s):  
Bernoulli Polynomials ◽  
Random Variable ◽  
Stirling Numbers ◽  
Moment Generating Function ◽  
Basis Functions ◽  
Integral Representations ◽  
Harmonic Numbers ◽  
Bernstein Type ◽  
Digamma Function ◽  
New Class

The main aim of this paper is to define and investigate a new class of symmetric beta type distributions with the help of the symmetric Bernstein-type basis functions. We give symmetry property of these distributions and the Bernstein-type basis functions. Using the Bernstein-type basis functions and binomial series, we give some series and integral representations including moment generating function for these distributions. Using generating functions and their functional equations, we also give many new identities related to the moments, the polygamma function, the digamma function, the harmonic numbers, the Stirling numbers, generalized harmonic numbers, the Lah numbers, the Bernstein-type basis functions, the array polynomials, and the Apostol–Bernoulli polynomials. Moreover, some numerical values of the expected values for the logarithm of random variable are given.

scholarly journals Three closed forms for convolved Fibonacci numbers

2020 ◽  
Author(s):  
Feng Qi
Keyword(s):  
Generating Function ◽  
Golden Ratio ◽  
Fibonacci Numbers ◽  
Stirling Numbers ◽  
Higher Order ◽  
Bell Polynomials ◽  
Faà Di Bruno Formula ◽  
Derivatives Of

In the paper, by virtue of the Faa di Bruno formula and several properties of the Bell polynomials of the second kind, the author computes higher order derivatives of the generating function of convolved Fibonacci numbers and, consequently, derives three closed forms for convolved Fibonacci numbers in terms of the falling and rising factorials, the Lah numbers, the signed Stirling numbers of the first kind, and the golden ratio.

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