Novel Formulas for B -Splines, Bernstein Basis Functions, and Special Numbers: Approach to Derivative and Functional Equations of Generating Functions

Yilmaz Simsek ()
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Yilmaz Simsek: Department of Mathematics, Faculty of Science, University of Akdeniz, Antalya TR-07058, Turkey

Mathematics, 2023, vol. 12, issue 1, 1-20

Abstract: The purpose of this article is to give relations among the uniform B -splines, the Bernstein basis functions, and certain families of special numbers and polynomials with the aid of the generating functions method. We derive a relation between generating functions for the uniform B -splines and generating functions for the Bernstein basis functions. We derive some functional equations for these generating functions. Using the higher-order partial derivative equations of these generating functions, we derive both the generalized de Boor recursion relation and the higher-order derivative formula of uniform B -splines in terms of Bernstein basis functions. Using the functional equations of these generating functions, we derive the relations among the Bernstein basis functions, the uniform B -splines, the Apostol-Bernoulli numbers and polynomials, the Aposto–Euler numbers and polynomials, the Eulerian numbers and polynomials, and the Stirling numbers. Applying the p -adic integrals to these polynomials, we derive many novel formulas. Furthermore, by applying the Laplace transformation to these generating functions, we derive infinite series representations for the uniform B -splines and the Bernstein basis functions.

Keywords: generating functions; uniform B -splines; Bernstein basis functions; Apostol–Bernoulli numbers and polynomials; Apostol–Euler numbers and polynomials; Eulerian numbers and polynomials; Laplace transforms; p -adic integral (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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