Degenerate two-variable $q$-Legendre polynomials via $q$-operational calculus and degenerate Laplace/Sumudu transforms

We introduce a degenerate $(q,\lambda )$-extension of the two-variable $q$-Legendre-type polynomials by deforming the $q$-Bessel–Tricomi kernel through the degenerate falling-factorial weights $(1{)}_{k,\lambda }$. The resulting family $\{{\mathcal{L}}_{n,q}^{(\lambda )}(x,y){\}}_{n\ge 0}$ is defined by a single generating function and interpolates both the recently studied two-variable $q$-Legendre polynomials and their classical limits as $\lambda \to 0$ and/or $q\to 1$. 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 ${\mathcal{L}}_{n,q}^{(\lambda )}$. Several reduction formulas, even/odd subsequence decompositions, and a moment-functional interpretation are presented, along with linearization coefficients, low-degree examples, and numerical table.