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Abstract
In 2008, Liu and Wang established various symmetric identities for Bernoulli, Euler and Genocchi polynomials. In this paper, we extend these identities in a unified and generalized form to families of Hermite–Bernoulli, Euler and Genocchi polynomials. The procedure followed is that of generating functions. Some relevant connections of the general theory developed here with the results obtained earlier by Pathan and Khan are also pointed out.
Highlights
Let Hn(x, y) be denoted by the 2-variable Kampé de Fériet generalization of the Hermite polynomials Bell (1934) and Dattoli et al (1999) defined as [ n 2 ] yr xn−2rHn(x, y) = n! r!(n − 2r)! (1) r=0These polynomials are usually defined by the generating function as ext+yt2 = ∞ tn Hn(x, y) n! (2)
The definition and generating function of the generalized Hermite–Bernoulli, Euler and Hermite–Genocchi polynomials plays a major role in obtaining new expansions, identities and representations
Competing interests The authors declare that they have no competing interests
Summary
Let Hn(x, y) be denoted by the 2-variable Kampé de Fériet generalization of the Hermite polynomials Bell (1934) and Dattoli et al (1999) defined as [. N=0 and reduce to the ordinary Hermite polynomials Hn(x) (see Andrews 1985) when y = −1 and x is replaced by 2x. The generalized Bernoulli polynomials Bn(α)(x) of order α, α ∈ C, the generalized Euler polynomials En(α)(x) of order α, α ∈ C and the generalized Genocchi polynomials Gn(α)(x) of order α, α ∈ C, each of degree n as well as in α are defined respectively by the following generating functions (see Dere and Simsek 2015, [Erdelyi et al (1953), vol 3, p. Khan and Haroon SpringerPlus (2016)5:1920 et seq.], [Luke (1969), Section 2.8] and Luo et al 2003; Pathan 2012; Pathan and Khan 2014, 2015, 2016; Simsek 2010; Srivastava et al 2012):
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