The aim of this paper is to derive some new identities related to the Frobenius-Euler polynomials. We also give relation between the generalized Frobenius-Euler polynomials and the generalized Hurwitz-Lerch zeta function at negative integers. Furthermore, our results give generalized Carliz's results which are associated with Frobenius-Euler polynomials. MSC: 05A10; 11B65; 28B99; 11B68
The aim of this paper is to study generating function of the Hermite-Kampė de Fėriet based second kind Genocchi polynomials. We also give some identities related to these polynomials.
The main objective of this paper is to introduce and investigate two new classes of generalized Apostol- In particular, we obtain addition formula for the new class of the generalized Apostol-Bernoulli polynomials. We also give some recurrence relations and Raabe relations for these polynomials.
This is as follows. Motivated naturally by the extensively and widely investigated known generalizations of the classical Bernoulli polynomials and the classical Genocchi polynomials to include any general (real or complex) order α, Luo and Srivastava (see [29,30,32]) introduced and studied the corresponding generalizations of the Apostol-Bernoulli polynomials and the Apostol-Genocchi polynomials. However, just as in the aforementioned works of Luo and Srivastava (and in their many sequels such as [4,37,46,54,[57][58][59][60] that are cited in this paper), exceptional parameter values that would render either side of a result undefined or invalid are tacitly excluded throughout the present investigation. Thus, for example, the order α of the generalized Apostol-Bernoulli polynomials defined by the generating function (1.18) (when λ = 1) with a = 1 and b = c = e and the order α of the generalized Apostol-Genocchi polynomials defined by the generating function (1.20) (when λ = −1) with a = 1 and b = c = e should be restricted correctly to only nonnegative integer values in which cases each of these polynomial families has already been widely used in the literature when λ = 1 and λ = −1, respectively. Furthermore, in our paper, the cases λ = 1 of such results as (for example) the assertions (2.28), (2.29) and (2.30) of Theorem 3 (see also Remark 3), and the assertions (2.31), (2.32) and (2.33) of Corollary 5, ought to be viewed only as consequences of formal power series manipulations using the series in the generating function (1.20) or (wherever applicable) the series in the generating function (1.18).It should be reiterated in passing that the investigations of the corresponding generalizations of the Apostol-Euler polynomials, which are associated with any admissible (real or complex) order α, are not at all affected by the observations made here.
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