A new generalization of Apostol type Hermite–Genocchi polynomials and its applications

By using the modified Milne-Thomson’s polynomial given in Araci et al. (Appl Math Inf Sci 8(6):2803–2808, 2014), we introduce a new concept of the Apostol Hermite–Genocchi polynomials. We also perform a further investigation for aforementioned polynomial and derive some implicit summation formulae and general symmetric identities arising from different analytical means and generating functions method. The results obtained here are an extension of Hermite–Bernoulli polynomials (Pathan and Khan in Mediterr J Math 12:679–695, 2015a) and Hermite–Euler polynomials (Pathan and Khan in Mediterr J Math 2015b, doi:10.1007/s00009-015-0551-1) to Apostol type Hermite–Genocchi polynomials defined in this paper.

Definition 1 Let c be positive integer. The generalized 2-variable 1-parameter Hermite Kamp'e de Feriet polynomials H n (x, y, c) for nonnegative integer n are stated by which is an extention of 2-variable Hermite Kamp'e de Feriet polynomials H n (x, y) defined by ∞ n=0 H n (x, y) t n n! = e xt+yt 2 (see Bell 1934;Pathan and Khan 2015a, b).
It immediately follows from Definition 1 that and by (11), we have Motivated by their importance and potential for applications in certain problems in number theory, combinatorics, classical and numerical analysis and other fields of applied mathematics, several kinds of some special numbers and polynomials were recently studied by many authors (see Milovanović and Rassias 2014;Borwein and Erdelyi 1995;Agarwal 2014;Choi and Agarwal 2014;Srivastava et al. 2014;Agarwal 2012;Luo et al. 2014;Agarwal and Koul 2003;Apostol 1951;Araci 2014;Araci et al. 2014a, b;Bell 1934;Dattoli et al. 1999;Dere and Simsek 2015;Dere et al. 2013;Guo and Qi 2002;Gaboury and Kurt 2012;He et al. 2015;Jolany et al. 2013;Khan et al. 2008;Khan 2015Khan , 2016aKim and Hu 2012;Kim and Adiga 2004;Kim 2007Kim , 1999Kurt and Kurt 2011;Luo et al. 2003a, b;Luo 2006Luo , 2009Luo , 2011Luo and Srivastava 2005, 2006Milne Thomsons 1933;Khan 2014a, b, 2015a, b, c, d;Srivastava and Manocha 1984;Srivastava 2000Srivastava , 2011Yang 2008;Zhang and Yang 2008). In Kurt and Kurt (2011), Kurt and Kurt first introduced the definition of Hermite-Apostol-Genocchi polynomials and derived some explicit formulas. Gaboury and Kurt (2012) also gave the generating function of Hermite-Apostol-Genocchi polynomials with three parameters. Their definitions are motivated us to write this paper. In summary, we introduce a new family of the generalized Apostol type Genocchi polynomials G (α) n (x, y; a, b, c; ) as Definition 2 in the next section, which generalizes the concepts stated above and then research their basic properties and relationships with Genocchi numbers G n , Genocchi polynomials G n (x) and the generalized Apostol Genocchi numbers G n (a, b; ), generalized Apsotol Genocchi polynomials G n (x; a, b, c; ) of Jolany et al. (2013), Hermite-Genocchi polynomial H G n (x, y) of Dattoli et al. (1999) and generalized Apostol Hermite-Genocchi polynomials H G (α) n (x, y; ). The remainder of this paper is organized as follows: We modify generating functions for the Milne-Thomson's polynomials as defined in Luo and Srivastava (2006) and derive some identities related to Hermite polynomials and Genocchi polynomials. Some implicit summation formulae and general symmetric identities are derived arising from different analytical means and applying generating functions. These results extend some known summations and identities of Hermite-Bernoulli, Euler and Hermite-Genocchi polynomials studied earlier by Dattoli et al. (1999), Jolany et al. (2013), Khan (2015Khan ( , 2016a, Luo (2009Luo ( , 2011, Khan (2014a, 2015a), Yang (2008), Zhang and Yang (2008).

On the generalized Apostol type Hermite-Genocchi polynomials
In this section, by (4) and f (t, α; ) = 2t b t +a t α , we derive a new class of Apostol Hermite-Genocchi polynomials and investigate its properties. Now we start at the following definition.
j=0 n j (log c) n−j x n−2j y j (see Pathan and Khan 2015a, b).
Definition 2 Let a, b and c be positive integers with the condition a � = b. A new generalization of Apostol-Genocchi polynomials G (α) n (x, ν; a, b, c; ) for nonnegative integer n is defined by Setting h(t, y) = yt 2 in (14), we get the following corollary.

Corollary 1 Let a, b and c be positive integers with the condition
n (x, y; a, b, c; ) for nonnegative integer n are defined by Gaboury and Kurt (2012) For α = 1 in (15), we have In the case x = 0 in (15), we see that Also in the case x = y = 0 and c = 1 in Definition 1, it leads to the extension of the generalized Apostol-Genocchi numbers denoted by G (α) n (a, b; ) for nonnegative integer n defined earlier in Jolany et al. (2013) and holds.
Corollary 2 Taking c = e in Eq. (15), we have Gaboury and Kurt (2012) By using Corollary 1, we state the following theorem.
Theorem 4 Let a, b and c be positive integers, by a � = b. Then, for x, y ∈ R and n ≥ 0, we have Proof By (11) and (15) where [.] is Gauss' notation, and represents the maximum integer which does not exceed a number in the square brackets.
Replacing n by n − 2j in the right hand side, we have Hence, our assertion follows from (37). □
Theorem 6 Let a, b and c be positive integers, by a � = b. Then, for x, y ∈ R and n ≥ 0, we have where [.] is Gauss' notation, and represents the maximum integer which does not exceed a number in the square brackets.
Proof It follows from (15) that

General symmetry identities
In this section, we investigate and derive symmetric identities for the generalized Apostol type Hermite-Genocchi polynomials H G (α) n (x, y; a, b, c; ) and Apostol Genocchi numbers G (α) n (a, b; ). It turns out that some well known identities of Khan et al. (2008), Khan (2015, a), Milne Thomsons (1933), Khan (2014a, b, 2015a, b, c), Srivastava (2011) andYang (2008). As it has been mentioned in previous sections, α will be considered as an arbitrary real or a complex parameter.
Theorem 10 Let a, b and c be positive integers, by a � = b. Then, for x, y ∈ R and n ≥ 0 , we have Proof Let us consider Then we see that g(t) is symmetric in a and b, and therefore we consider g(t) in two ways: Firstly Secondly By comparing the coefficients of t n on the right hand sides of two ways, we arrive at the desired result.