Several Structural Properties and Characterisations of Affine Gould–Hopper-Based Appell Polynomials

Abstract

In this paper, we introduce and systematically study a new two-parameter family of affine (q,η)-Gould–Hopper-based Appell polynomials. These polynomials arise by fusing the affine (q,η)-exponential Gould–Hopper kernel with an invertible Appell multiplier. A central contribution is the affine quasi-monomial construction: an explicit raising operator obtained from a logarithmic-difference quotient of the combined kernel, together with the lowering operator and the resulting q-commutation structure. An explicit double-sum series representation and an affine diffusion equation of order j connecting differences in the two variables, and the full quasi-monomial framework identifying the operator pair (Pq,η+,Pq,η−) are developed. Additional results include a governing difference equation, a converse characterisation, a determinantal representation, an affine addition formula, and an order recursion. Further, the Bernoulli and Euler sub-families are obtained as invertible specialisations, while the Genocchi family is treated separately as a derived noninvertible family through its exact relation with the Euler family. The corresponding structural results are stated with these hypotheses made explicit. Surface plots, numerical value tables, and a numerical illustration of real zeros accompany the theoretical development. The diffusion relation provides a discrete affine analogue of a higher-order evolution equation; no claim of a fully developed physical model is made in the present work.