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

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.