scholarly journals Certain Identities Associated with (p,q)-Binomial Coefficients and (p,q)-Stirling Polynomials of the Second Kind

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1436
Author(s):  
Talha Usman ◽  
Mohd Saif ◽  
Junesang Choi
Keyword(s):  
Bernstein Polynomials ◽  
Stirling Numbers ◽  
Binomial Coefficients ◽  
Research Subjects

The q-Stirling numbers (polynomials) of the second kind have been investigated and applied in a variety of research subjects including, even, the q-analogue of Bernstein polynomials. The (p,q)-Stirling numbers (polynomials) of the second kind have been studied, particularly, in relation to combinatorics. In this paper, we aim to introduce new (p,q)-Stirling polynomials of the second kind which are shown to be fit for the (p,q)-analogue of Bernstein polynomials. We also present some interesting identities involving the (p,q)-binomial coefficients. We further discuss certain vanishing identities associated with the q-and (p,q)-Stirling polynomials of the second kind.

scholarly journals A note on characteristic function for Bernstein polynomials involving special numbers and polynomials

Filomat ◽  
2020 ◽  
Vol 34 (2) ◽  
pp. 543-549
Author(s):  
Buket Simsek
Keyword(s):  
Characteristic Function ◽  
Binomial Distribution ◽  
Bernstein Polynomials ◽  
Random Variable ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Discrete Random Variable ◽  
Euler Numbers ◽  
Negative Order ◽  
Special Numbers And Polynomials

The aim of this present paper is to establish and study generating function associated with a characteristic function for the Bernstein polynomials. By this function, we derive many identities, relations and formulas relevant to moments of discrete random variable for the Bernstein polynomials (binomial distribution), Bernoulli numbers of negative order, Euler numbers of negative order and the Stirling numbers.

scholarly journals Onq-Euler Numbers Related to the Modifiedq-Bernstein Polynomials

2010 ◽  
Vol 2010 ◽  
pp. 1-15
Author(s):  
Min-Soo Kim ◽  
Daeyeoul Kim ◽  
Taekyun Kim
Keyword(s):  
Bernstein Polynomials ◽  
Stirling Numbers ◽  
Euler Numbers

We considerq-Euler numbers, polynomials, andq-Stirling numbers of first and second kinds. Finally, we investigate some interesting properties of the modifiedq-Bernstein polynomials related toq-Euler numbers andq-Stirling numbers by using fermionicp-adic integrals onℤp.

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 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 Generating Functions for New Families of Combinatorial Numbers and Polynomials: Approach to Poisson–Charlier Polynomials and Probability Distribution Function

Axioms ◽  
2019 ◽  
Vol 8 (4) ◽  
pp. 112 ◽  
Author(s):  
Irem Kucukoglu ◽  
Burcin Simsek ◽  
Yilmaz Simsek
Keyword(s):  
Probability Distribution ◽  
Generating Functions ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Distribution Functions ◽  
Binomial Coefficients ◽  
Bell Polynomials ◽  
Exponential Polynomials ◽  
Charlier Polynomials ◽  
Combinatorial Sums

The aim of this paper is to construct generating functions for new families of combinatorial numbers and polynomials. By using these generating functions with their functional and differential equations, we not only investigate properties of these new families, but also derive many new identities, relations, derivative formulas, and combinatorial sums with the inclusion of binomials coefficients, falling factorial, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), the Poisson–Charlier polynomials, combinatorial numbers and polynomials, the Bersntein basis functions, and the probability distribution functions. Furthermore, by applying the p-adic integrals and Riemann integral, we obtain some combinatorial sums including the binomial coefficients, falling factorial, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), and the Cauchy numbers (or the Bernoulli numbers of the second kind). Finally, we give some remarks and observations on our results related to some probability distributions such as the binomial distribution and the Poisson distribution.

scholarly journals Unimodality Problems of Multinomial Coefficients and Symmetric Functions

10.37236/560 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Xun-Tuan Su ◽  
Yi Wang ◽  
Yeong-Nan Yeh
Keyword(s):  
Symmetric Functions ◽  
Stirling Numbers ◽  
Binomial Coefficients ◽  
Early Results ◽  
Multinomial Coefficients

In this note we consider unimodality problems of sequences of multinomial coefficients and symmetric functions. The results presented here generalize our early results for binomial coefficients. We also give an answer to a question of Sagan about strong $q$-log-concavity of certain sequences of symmetric functions, which can unify many known results for $q$-binomial coefficients and $q$-Stirling numbers of two kinds.

scholarly journals Generalized Stirling Numbers and Hyper-Sums of Powers of Binomials Coefficients

10.37236/3787 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Claudio de J. Pita-Ruiz V.
Keyword(s):  
Stirling Numbers ◽  
Binomial Coefficients ◽  
Partial Sums ◽  
Linear Combinations ◽  
Normal Ordering ◽  
Sums Of Powers ◽  
Generalized Stirling Numbers ◽  
Natural Way

We work with a generalization of Stirling numbers of the second kind related to the boson normal ordering problem (P. Blasiak et al.). We show that these numbers appear as part of the coefficients of expressions in which certain sequences of products of binomials, together with their partial sums, are written as linear combinations of some other binomials. We show that the number arrays formed by these coefficients can be seen as natural generalizations of Pascal and Lucas triangles, since many of the known properties on rows, columns, falling diagonals and rising diagonals in Pascal and Lucas triangles, are also valid (some natural generalizations of them) in the arrays considered in this work. We also show that certain closed formulas for hyper-sums of powers of binomial coefficients appear in a natural way in these arrays.

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