Generalized h(x)-Fibonacci–Lucas–Polylogarithm and Legendre–Polylogarithm Polynomials Associated with Generalized Hyperharmonic Numbers

Polylogarithm-weighted sequences and h(x)-Fibonacci/Lucas polynomials have each been studied extensively, but a common formulation that incorporates generalized hyperharmonic weights into both these kernels and related Legendre-type kernels has not been formulated in a unified way. In this paper, the classical generating functions are deformed by the factor Lip(t)/(1−t)q, and the resulting coefficients are derived by Cauchy product arguments. This construction yields the h(x)-Fibonacci–polylogarithm and h(x)-Lucas–polylogarithm polynomials, explicit coefficient formulas, convolution identities, recurrence relations, and parity properties, together with a unified two-parameter family of generalized h(x)-Fibonacci–Lucas–polylogarithm polynomials Ph,na,b,p,q(x). The same deformation principle also gives rise to Legendre–polylogarithm polynomials and to a (q,λ)-extension obtained from a weighted Legendre generating kernel. These families provide a natural generating-function setting for models in which cumulative harmonic or hyperharmonic effects are intrinsic, while also making explicit the main analytic restrictions of the deformation, including convergence constraints and the loss of classical orthogonality in the Legendre setting.