A note on generating functions for the unification of the Bernstein type basis functions

In [3], Simsek unified generating function of the Bernstein basis functions. In this paper, by using knot sequence, we rewrite generating functions for the unification of the Bernstein type basis functions. By using these generating functions, we also find generating function for the Hermite type numbers. We investigate some properties of this functions and these basis. Finally, we simulate these polynomials with their plots for some selected numerical values.

Download Full-text

Deriving Novel Formulas and Identities for the Bernstein Basis Functions and Their Generating Functions

Mathematical Methods for Curves and Surfaces - Lecture Notes in Computer Science ◽

10.1007/978-3-642-54382-1_27 ◽

2014 ◽

pp. 471-490 ◽

Cited By ~ 1

Author(s):

Yilmaz Simsek

Keyword(s):

Generating Functions ◽

Basis Functions ◽

Bernstein Basis ◽

Bernstein Basis Functions

Download Full-text

Matrix representations for a certain class of combinatorial numbers associated with Bernstein basis functions and cyclic derangements and their probabilistic and asymptotic analyses

Applicable Analysis and Discrete Mathematics ◽

10.2298/aadm201017009k ◽

2021 ◽

pp. 9-9

Author(s):

Irem Kucukoglu ◽

Yilmaz Simsek

Keyword(s):

Generating Functions ◽

Hermite Polynomials ◽

Series Representation ◽

Basis Functions ◽

Standard Normal Distribution ◽

Bernstein Basis ◽

Matrix Representations ◽

New Family ◽

Standard Normal ◽

Bernstein Basis Functions

In this paper, we mainly concerned with an alternate form of the generating functions for a certain class of combinatorial numbers and polynomials. We give matrix representations for these numbers and polynomials with their applications. We also derive various identities such as Rodrigues-type formula, recurrence relation and derivative formula for the aforementioned combinatorial numbers. Besides, we present some plots of the generating functions for these numbers. Furthermore, we give relationships of these combinatorial numbers and polynomials with not only Bernstein basis functions, but the two-variable Hermite polynomials and the number of cyclic derangements. We also present some applications of these relationships. By applying Laplace transform and Mellin transform respectively to the aforementioned functions, we give not only an infinite series representation, but also an interpolation function of these combinatorial numbers. We also provide a contour integral representation of these combinatorial numbers. In addition, we construct exponential generating functions for a new family of numbers arising from the linear combination of the numbers of cyclic derangements in the wreath product of the finite cyclic group and the symmetric group of permutations of a set. Finally, we analyse the aforementioned functions in probabilistic and asymptotic manners, and we give some of their relationships with not only the Laplace distribution, but also the standard normal distribution. Then, we provide an asymptotic power series representation of the aforementioned exponential generating functions.

Download Full-text

Functional equations from generating functions: a novel approach to deriving identities for the Bernstein basis functions

Fixed Point Theory and Applications ◽

10.1186/1687-1812-2013-80 ◽

2013 ◽

Vol 2013 (1) ◽

Cited By ~ 9

Author(s):

Yilmaz Simsek

Keyword(s):

Generating Functions ◽

Functional Equations ◽

Basis Functions ◽

Bernstein Basis ◽

Novel Approach ◽

Bernstein Basis Functions

Download Full-text

Combinatorial identities associated with Bernstein type basis functions

Filomat ◽

10.2298/fil1607683s ◽

2016 ◽

Vol 30 (7) ◽

pp. 1683-1689 ◽

Cited By ~ 3

Author(s):

Yilmaz Simsek

Keyword(s):

Catalan Numbers ◽

Combinatorial Identities ◽

Basis Functions ◽

Binomial Coefficients ◽

Bernstein Basis ◽

Bernstein Type ◽

Combinatorial Sums ◽

Bernstein Basis Functions

In this paper, we give some identities and relations for the Bernstein basis functions and the beta type polynomials. Integrating these identities, we derive many identities and formulas, some old and some new, for combinatorial sums involving binomial coefficients and the Catalan numbers. We also give remarks and comments on these identities.

Download Full-text