A New Class of Laguerre-based Generalized Apostol Polynomials

Abstract

Abstract. In this paper, we introduce a unified family of Laguerre-based Apostol Bernoulli, Euler and Genocchi polynomials and derive some implicit summation formulae and general symmetry identities arising from different analytical means and applying generating functions. The result extend some known summations and identities of generalized Bernoulli, Euler and Genocchi numbers and polynomials.

A new class of generalized polynomials associated with Laguerre and Bernoulli polynomials

“…n (x, y) (see [29]). In this regard, the results presented here can be specialized to yield or be closely connected with some known identities and formulas (see, e.g., [5,13,[18][19][20][21][22][23][24][27][28][29]40,41], and the references cited therein). Therefore, the results presented in this article seem to be potentially useful in arising problems of the aforementioned fields.…”

“…Khan et al [18][19][20][21][22][23][24], Yang [40], and Zhang and Yang [41] have established some interesting symmetry identities for various polynomials. Here we present certain symmetry identities for the generalized Laguerre-Bernoulli polynomials (20) in the following theorem.…”

Unified laguerre-based poly-Apostol-type polynomials

“…−based Apostol-Bernoulli, Laguerrebased Apostol-Euler and Laguerre-based ApostolGenocchi polynomials are studied and investigated by Khan and Usman (2016). Luo (2009), Luo and Srivastava (2005) and Srivastava (2011) introduced the Apostol-Bernoulli, Apostol-Euler and ApostolGenocchi polynomials and proved some theorems and relations for these polynomials.…”

“…Pathan and Khan (2016, 2014 introduced the Hermite-based Bernoulli polynomials, Euler polynomials and gave some relation for these polynomials. Khan and Usman (2016) introduced a new class of Laguerre-based generalized Apostol polynomials. He also gave some symmetric relations for these polynomials.…”

“…The Laguerre-based generalized Apostol-Bernoulli polynomials are defined in Khan and Usman (2016) as following generating functions…”