@article{Yağcı2026, 
author = {Oğuz Yağcı and Waseem Ahmad Khan and Khidir Shaib Mohamed and Azhar Iqbal and Wei Sin Koh},
title = {Degenerate two-variable    q-Legendre polynomials via    q-operational calculus and degenerate Laplace/Sumudu transforms},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {3},
pages = {7207-7234},
keywords = {q-calculus, two-variable q-Legendre polynomials, quasi-monomiality, degenerate falling factorial, degenerate Laplace transform, differential equations, process innovation, Mathematical operators, degenerate Sumudu transform},
url = {https://www.sciopen.com/article/10.3934/math.2026297},
doi = {10.3934/math.2026297},
abstract = {We introduce a degenerate    (  q  ,  λ  )-extension of the two-variable    q-Legendre-type polynomials by deforming the    q-Bessel–Tricomi kernel through the degenerate falling-factorial weights    (  1      )          k      ,      λ      . The resulting family    {            L              n      ,      q              (      λ      )        (  x  ,  y  )      }          n      ≥      0       is defined by a single generating function and interpolates both the recently studied two-variable    q-Legendre polynomials and their classical limits as    λ  →  0 and/or    q  →  1. We derive an explicit finite-sum representation, an operational Rodrigues-type formula, and a quasi-monomial structure that yields raising and lowering operators together with a fundamental    q-difference equation in the variable    y. We further compute the degenerate Laplace and Sumudu transforms of the generating kernel and obtain corresponding transform identities for              L              n      ,      q              (      λ      )      . Several reduction formulas, even/odd subsequence decompositions, and a moment-functional interpretation are presented, along with linearization coefficients, low-degree examples, and numerical table.}
}