On a Biparametric Appell Extension: Analytical Properties and Structural Analysis

This paper introduces and investigates a novel two-parameter sequence, termed the biparametric Appell extension (B-App-Ex) and denoted by Bn(x;λ,α). Standard classical Appell sequences often lack sufficient structural parameters, which can limit their operational flexibility in certain advanced spectral schemes. To address this limitation, we construct an enhanced operational framework by integrating a binomial structural kernel (1+w)λ with a linear exponential scaling eαxw entirely within the Appell class. We provide a rigorous logical deduction of the fundamental properties of this sequence, including its explicit power series representation, a characteristic three-term recurrence relation, and a governing second-order differential equation (DEq.). A significant contribution of this work is the establishment of analytically exact connection and inverse connection formulas between the B-App-Ex basis and various classical orthogonal polynomial (COP) families. Numerical verification via a collocation-based projection framework demonstrates that these algebraic kernels achieve near-machine epsilon precision (≈10−15), remaining stable even for high-order approximations. Furthermore, by isolating the dilation factor α, we establish an O(N) computational complexity that offers a reduction in latency by approximately two orders of magnitude compared to classical matrix-based transformations. The results demonstrate that the proposed biparametric (Bip.) extension offers a versatile and highly optimized analytical template for modeling complex dynamic systems where structural shifting and spatial scaling must be tuned simultaneously.