A Note on the Truncated-Exponential Based Apostol-Type Polynomials

Abstract

In this paper, we propose to investigate the truncated-exponential-based Apostol-type polynomials and derive their various properties. In particular, we establish the operational correspondence between this new family of polynomials and the familiar Apostol-type polynomials. We also obtain some implicit summation formulas and symmetric identities by using their generating functions. The results, which we have derived here, provide generalizations of the corresponding known formulas including identities involving generalized Hermite-Bernoulli polynomials.

Truncated Fubini Polynomials

“…see [1][2][3][4][5][6][7][8][9][10] for details about the aforesaid polynomials. The Bernoulli numbers B n and Euler numbers E n are obtained by the special cases of the corresponding polynomials at x = 0, namely: B n (0) := B n and E n (0) := E n .…”

“…Recently, several mathematicians have studied truncated-type special polynomials such as truncated Bernoulli polynomials and truncated Euler polynomials; cf. [1,4,7,9,11,12].…”

“…where the notation (x) µ called the falling factorial equals [2][3][4][5][7][8][9][10]13], and see also the references cited therein. The Apostol-type Stirling numbers of the second kind is defined by (cf.…”

Truncated-exponential-based Frobenius–Euler polynomials

“…The resulting formulas are very important and potentially useful, because they include expansions for many transcendent expressions of mathematical physics in series of the classical orthogonal polynomials. The developments bear heavily upon the work of many researchers who have earlier studied the special polynomials with applications to p-adic analysis, q-analysis, umbral analysis, and so on (see, for example, the recent work [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] and [23]).…”

“…The two-variable families of the Appell polynomials were originated by Bretti et al [3] with the usage of an iterated isomorphism. The two-variable truncated-exponential, Hermite, Legendre and Laguerre polynomials along their extensions are investigated and examined in [2,[5][6][7]23] by several authors.…”