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

Oğuz Yağcı; Waseem Ahmad Khan; Khidir Shaib Mohamed; Azhar Iqbal; Wei Sin Koh; Oğuz Yağcı; Waseem Ahmad Khan; Khidir Shaib Mohamed; Azhar Iqbal; Wei Sin Koh

doi:10.3934/math.2026297

AIMS Mathematics

2026, Volume 11, Issue 3: 7207-7234. doi: 10.3934/math.2026297

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

1.
Department of Mathematics, Kırıkkale University, Kırıkkale 71450, Türkiye
2.
Department of Electrical Engineering, Prince Mohammad Bin Fahd University, P. O. Box 1664, Al Khobar 31952, Saudi Arabia
3.
Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia
4.
Department of Mathematics and Natural Sciences, Prince Mohammad Bin Fahd University, P. O. Box 1664, Al Khobar 31952, Saudi Arabia
5.
Faculty of Business and Communications, INTI International University, Nilai 71800, Negeri Sembilan, Malaysia

Received: 22 January 2026 Revised: 28 February 2026 Accepted: 10 March 2026 Published: 19 March 2026
MSC : 05A30, 33C45, 33D45, 44A10

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}^{(\lambda)}_{n, q}(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}^{(\lambda)}_{n, q} $. Several reduction formulas, even/odd subsequence decompositions, and a moment-functional interpretation are presented, along with linearization coefficients, low-degree examples, and numerical table.
- $ q $-calculus,
- two-variable $ q $-Legendre polynomials,
- quasi-monomiality,
- degenerate falling factorial,
- degenerate Laplace transform,
- differential equations,
- process innovation,
- Mathematical operators,
- degenerate Sumudu transform
Citation: Oğuz Yağcı, Waseem Ahmad Khan, Khidir Shaib Mohamed, Azhar Iqbal, Wei Sin Koh. Degenerate two-variable $ q $-Legendre polynomials via $ q $-operational calculus and degenerate Laplace/Sumudu transforms[J]. AIMS Mathematics, 2026, 11(3): 7207-7234. doi: 10.3934/math.2026297

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Abstract

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}^{(\lambda)}_{n, q}(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}^{(\lambda)}_{n, q} $. Several reduction formulas, even/odd subsequence decompositions, and a moment-functional interpretation are presented, along with linearization coefficients, low-degree examples, and numerical table.

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