A Deformed Gegenbauer-Type Polynomial System With Numerical Illustration for Spectral Diffusion Models

PurposeThis study aims to develop a mathematically cautious and reproducible q,τ-deformed Gegenbauer-type (DGT) polynomial (q,τ-DGT) for spectral approximation and illustrative diffusion modeling. The revision is motivated by the need to avoid circular definitions, unsupported orthogonality claims, and unstable diffusion sign conventions.MethodsAn explicit positive q,τ-number and a finite q,τ-Pochhammer product are introduced as the algebraic basis of the deformation. A DGT polynomial family is then constructed from a generating series, and an explicit coefficient formula is derived. Exact orthogonality is not assumed under the deformation. Instead, the approximation problem is formulated in a positive weighted Hilbert space using a Gram-matrix least-squares projection. The resulting basis is orthonormalized by the Gram–Schmidt procedure. Bessel and Parseval identities are then stated for this orthonormal system. Finally, a stable q,τ-modal diffusion model is formulated using a positive spectral operator, which ensures exponentially decaying modal amplitudes.ResultsThe proposed construction provides a well-defined deformed polynomial system, a reproducible finite-dimensional approximation procedure, and a stable spectral diffusion representation with monotone energy decay. Numerical illustrations for fx=ex demonstrate the deformation of the polynomial modes, projection error, pointwise approximation error, normalized modal energy decay, and the influence of q and τ on spectral localization and smoothing.ConclusionThe revised framework offers a safer and more transparent formulation of q,τ-DGT spectral approximation. It supports reproducible numerical computation and provides an illustrative diffusion model without overclaiming a complete deformed Sturm–Liouville theory. The results suggest that the deformation parameters can be used to tune spectral concentration and diffusion smoothing, while a full operator theory remains an important direction for future work.