On Degenerate Truncated Special Polynomials

Abstract

The main aim of this paper is to introduce the degenerate truncated forms of multifarious special polynomials and numbers and is to investigate their various properties and relationships by using the series manipulation method and diverse special proof techniques. The degenerate truncated exponential polynomials are first considered and their several properties are given. Then the degenerate truncated Stirling polynomials of the second kind are defined and their elementary properties and relations are proved. Also, the degenerate truncated forms of the bivariate Fubini and Bell polynomials and numbers are introduced and various relations and formulas for these polynomials and numbers, which cover several summation formulas, addition identities, recurrence relationships, derivative property and correlations with the degenerate truncated Stirling polynomials of the second kind, are acquired. Thereafter, the truncated degenerate Bernoulli and Euler polynomials are considered and multifarious correlations and formulas including summation formulas, derivation rules and correlations with the degenerate truncated Stirling numbers of the second are derived. In addition, regarding applications, by introducing the degenerate truncated forms of the classical Bernstein polynomials, we obtain diverse correlations and formulas. Some interesting surface plots of these polynomials in the special cases are provided.

Construction of the Type 2 Degenerate Multi-Poly-Euler Polynomials and Numbers

“…[3]), and the degenerate Bernoulli B n; (x) and Euler E n; (x) (cf. [1,6,8,[14][15][16][17][18][19][20][21]23]) polynomials are de…ned by the following generating functions:…”

“…One of the most important families of special functions is the family of special polynomials, cf. [1,[3][4][5][6][7][8]10,21,23,24]. Intense research activities in such an area as the family of special polynomials are principally motivated by their importance in both pure and applied mathematics and other disciplines.…”

“…Now, the degenerate forms of special polynomials are considered and studied intensively by several mathematicians, cf. [6][7][8][14][15][16][17][18][19][20][21]23]. For instance, Duran and Acikgoz [6] considered the degenerate truncated exponential polynomials and gave their several properties.…”

“…[6][7][8][14][15][16][17][18][19][20][21]23]. For instance, Duran and Acikgoz [6] considered the degenerate truncated exponential polynomials and gave their several properties. After that, degenerate truncated forms of various special polynomials including Genocchi, Bell, Bernstein, Fubini, Euler and Bernoulli polynomials were introduced via the degenerate truncated exponential polynomials and their various properties and relationships by using the series manipulation method and diverse special proof techniques were derived in [6].…”

“…For instance, Duran and Acikgoz [6] considered the degenerate truncated exponential polynomials and gave their several properties. After that, degenerate truncated forms of various special polynomials including Genocchi, Bell, Bernstein, Fubini, Euler and Bernoulli polynomials were introduced via the degenerate truncated exponential polynomials and their various properties and relationships by using the series manipulation method and diverse special proof techniques were derived in [6]. Duran and Sadjang [8] considered the fully degenerate Gould-Hopper polynomials with a q parameter and the Gould-Hopper-based fully degenerate poly-Bernoulli polynomials with a q parameter and then gave some of their properties including not only di¤erence rule, inversion formula and addition formula but also multifarious correlations and implicit summation formulas.…”