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.
In this paper, firstly we introduce not only partially degenerate Hermite-Genocchi polynomials, but also a new generalization of degenerate Hermite-Genocchi polynomials. Secondly, we investigate some behaviors of these polynomials. Furthermore, we establish some implicit summation formulae and symmetry identities by making use of the generating function of partially degenerate Hermite-Genocchi polynomials. Finally, some results obtained here extend well-known summations and identities which we stated in the paper.
The paper is an accomplishment of a new 3-variable 4-parameter generating function for Humbert matrix polynomials with an approach of unifying several classes of matrix valued polynomials using standard techniques of series manipulation. The results are contained in the form of explicit expression, hypergeometric matrix representation, generating functions and three additional expansions in nexus with Legendre, Hermite and Gegenbauer polynomials within discrete sections. A range of special cases is evenly traced that accounts due to the genuine wholesome generalization of such matrix polynomials.
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