A unified family of multivariable Legendre poly-Genocchi polynomials

In this paper, we introduce a new class of Legendre poly-Genocchi polynomials and give some identities of these polynomials related to the Stirling numbers of the second kind. The concept of poly-Bernoulli numbers Bₙ⁽ᵏ⁾(a,b), poly-Bernoulli polynomials Bₙ⁽ᵏ⁾(x,a,b) of Jolany et al., Hermite-Bernoulli polynomials HBₙ(x,y) of Dattoli et al., HBₙ(α)(x,y) of Pathan et al. and HGₙ⁽ᵏ⁾(x,y) of Khan are generalized to the one SGₙ⁽ᵏ⁾(x,y,z). Some implicit summation formulae and general symmetry identities are derived by using different analytical means and applying generating function. These results extended some known summation and identities of Hermite poly-Genocchi numbers and polynomials.