Some families of Genocchi type polynomials and their interpolation functions

Abstract

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.

Unified (p,q)-analog of Aspostol type polynomials of order α

“…Apostol type polynomials and numbers firstly introduced by Apostol [1] and also Srivastava [21]. Some relationships between Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials were introduced and studied extensively by Luo and Srivastava [10][11][12][13], Lu and Srivastava [4] and Srivastava [19][20][21][22][23][24][25]. Motivated by their works, many mathematicians have studied and investigated Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials and numbers, cf.…”

Explicit relations for the modified degenerate Apostol-type polynomials

“…Keeping in view many of the above-mentioned and other related investigation by Carlitz (see [1,2]), Dolgy et al [3], Kim et al [5], He et al [6], Kim [7] , Kwon et al [8], Kurt [9,10], Liu and Wang [11], Luo [12,13], Srivastava [16], Kurt [18], Yang [21], Young [22]. We systematically study the above defined the generalized higher order modified degenerate Apostol-Bernoulli polynomials and the generalized higher order modified degenerate Apostol-Euler polynomials.…”

On the <i>p</i>-adic Gamma Function and Changhee Polynomials

“…The q-calculus (or quantum calculus) appeared in the 18th century and it continues to develop rapidly and has been studied by many scientists (cf. [3][4][5][6][7][8]). Many generalizations of special functions with a q-parameter recently were obtained using p-adic q-integral on p  (cf.…”