Operational Methods and Truncated Exponential-Based Mittag-Leffler Polynomials

Abstract

This article deals with the introduction of truncated exponential-based Mittag-Leffler polynomials and derivation of their properties. The operational correspondence between these polynomials and Mittag-Leffler polynomials is established. An integral representation for these polynomials is also derived.Mathematics Subject Classification. 33E20, 33B10, 33E30, 11T23.

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

“…n (x, y), For example, we have e (2) n (x, y) = [2]e n (x, y) e (1) n (x, 1) = [r]e n (x) e (2) n (x, 1) = [2]e n (x) and e (1) n (x, 1) = e n (x). (17) As it is shown in [6,7], the 2VTEP e (r)…”

“…We note the following representation: e n (x) = 1 n! ∞ 0 e −ξ (x + ξ) n dξ, (7) which follows readily from the classical gamma-function representation (see, for details, [3]). Consequently, we have the following generating function for the truncated-exponential polynomials e n (x) (see [4]):…”

A note on truncated degenerate Bell polynomials

“…The minimum variance unbiased estimation is discussed in [4] for the zero class truncated bivariate Poisson and logarithmic series distributions, the maximum likelihood estimation of the Poisson parameter λ is concerned in [5] when the zero class has been truncated and a new family of Hermite polynomials is constructed in [7] by using the truncated exponential with applications to flattened beams in optics. More recently, degenerate exponential truncated polynomials and numbers are studied in [9], the degenerate zero-truncated Poisson random variables are introduced in [14], the truncated-exponential-based Apostol-type polynomials are investigated in [23] and the truncated exponential-based Mittag-Leffler polynomials are considered in [25]. Further, in [12] an umbral calculus approach is given for Bernoulli-Padé polynomials of fixed order, which include the truncated Bernoulli polynomials as a special case and whose generating function is based on the Padé approximant of the exponential function.…”

Truncated Fubini Polynomials

“…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].…”