Degenerate Bernstein polynomials

Abstract

Here we consider the degenerate Bernstein polynomials as a degenerate version of Bernstein polynomials, which are motivated by Simsek's recent work 'Generating functions for unification of the multidimensional Bernstein polynomials and their applications' ([15,16]) and Carlitz's degenerate Bernoulli polynomials. We derived thier generating function, symmetric identities, recurrence relations, and some connections with generalized falling factorial polynomials, higher-order degenerate Bernoulli polynomials and degenerate Stirling numbers of the second kind.2010 Mathematics Subject Classification. 11B83.

Some identities of degenerate Euler polynomials associated with degenerate Bernstein polynomials

“…In [1,2], Carlitz initiated the study of degenerate versions of some special polynomials and numbers, namely the degenerate Bernoulli and Euler polynomials and numbers. Here we would like to draw the attention of the reader to the fact that Kim et al have introduced various degenerate polynomials and numbers and investigating their properties, some identities related to them and their applications by means of generating functions, combinatorial methods, umbral calculus, p-adic analysis and differential equations (see [8,9] and the references therein). It is amusing that this line of study led them even to the introduction of degenerate gamma functions and degenerate Laplace transforms (see [7]).…”

“…Recently, Kim-Kim introduced the degenerate Bernstein polynomials given by (x) k,λ k! t k (1 + λt) [8][9][10]).…”

“…Let f be a continuous function on Z p . Then the degenerate Bernstein operator of order n is given by [8,9,[12][13][14][15][17][18][19]).…”

“…As a continuation of this initiative of Carlitz, Kim and his colleagues have been introducing various degenerate special polynomials and numbers and investigating their properties, some identities related to them and their applications. This research has been carried out by means of generating functions, combinatorial methods, umbral calculus, padic analysis and differential equations (see [8,9] and the references therein). Here, along the same line and by virtue of fermionic p-adic integrals on Z p and generating functions, we investigate some properties and identities for degenerate Euler polynomials related to degenerate Bernstein polynomials.…”

Symmetric Properties of Carlitz’s Type q-Changhee Polynomials

“…Symmetric identities of special polynomials are important and interesting in number theory, pure and applied mathematics. Symmetric identities of many different polynomials were investigated in [5,10,14,16,[32][33][34][35][36][37][38][39]. In particular, C. Cesarano [40] presented some techniques regarding the generating functions used, and these identities can be applicable to the theory of porous materials [41].…”