A New Generating Function of (q-) Bernstein-Type Polynomials and Their Interpolation Function

Abstract

The main object of this paper is to construct a new generating function of the (q-) Bernstein type polynomials. We establish elementary properties of this function. By using this generating function, we derive recurrence relation and derivative of the (q-) Bernstein type polynomials. We also give relations between the (q-) Bernstein type polynomials, Hermite polynomials, Bernoulli polynomials of higher-order and the second kind Stirling numbers. By applying Mellin transformation to this generating function, we define interpolation of the (q-) Bernstein type polynomials. Moreover, we give some applications and questions on approximations of (q-) Bernstein type polynomials, moments of some distributions in Statistics.

Explicit Formulas for Meixner Polynomials

“…There are many authors who have studied polynomials and their properties (see [1][2][3][4][5][6][7][8][9][10]). The polynomials are applied in many areas of mathematics, for instance, continued fractions, operator theory, analytic functions, interpolation, approximation theory, numerical analysis, electrostatics, statistical quantum mechanics, special functions, number theory, combinatorics, stochastic processes, sorting, and data compression.…”

Explicit Formulas for Some Generalized Polynomials

“…Numerous investigations related to the generating functions for many polynomials can be found in many books and articles (see, for example, [7][8][9][10][11][12][13][14][15] The main purpose of this paper is to obtain explicit formulas for the Generalised Hermite polynomials, the Generalised Humbert polynomials, the Lerch polynomials, and the Mahler polynomials.…”

q‐Bernstein polynomials related to q‐Frobenius–Euler polynomials, l‐functions, and q‐Stirling numbers

“…One has the following limit cases: cf. .…”