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Generating functions for unification of the multidimensional Bernstein polynomials and their applications
Author(s) -
Simsek Yilmaz
Publication year - 2018
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.4746
Subject(s) - mathematics , bernstein polynomial , unification , basis (linear algebra) , stirling numbers of the second kind , euler's formula , pure mathematics , operator (biology) , bernoulli number , algebra over a field , mathematical analysis , biochemistry , chemistry , geometry , repressor , computer science , transcription factor , gene , programming language
The aim of this paper is to construct generating functions for m ‐dimensional unification of the Bernstein basis functions. We give some properties of these functions. We also give derivative formulas and a recurrence relation of the m ‐dimensional unification of the Bernstein basis functions with help of their generating functions. By combining the m ‐dimensional unification of the Bernstein basis functions with m variable functions on simplex and cube, we give m ‐dimensional unification of the Bernstein operator. Furthermore, by applying integrals method including the Riemann integral, the q ‐integral, and the p ‐adic integral to some identities for the ( q ‐) Bernstein basis functions, we derive some combinatorial sums including the Bernoulli numbers and Euler numbers and also the Stirling numbers and the Cauchy numbers (the Bernoulli numbers of the second kind).