Structural Properties and Zero Distributions of q-Laguerre–Hahn–Tricomi–Appell Polynomials in the Framework of Quantum q-Calculus

Khan, Waseem Ahmad; Yağci, Oğuz; Mohamed, Khidir Shaib; Suhail, Muntasir; Himadan, Ahmed; Ibrahim, Habeeb; Mohammed, Naglaa

doi:10.3390/sym18081268

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Structural Properties and Zero Distributions of q-Laguerre–Hahn–Tricomi–Appell Polynomials in the Framework of Quantum q-Calculus

by

Waseem Ahmad Khan

¹

,

Oğuz Yağci

²

,

Khidir Shaib Mohamed

^3,*

,

Muntasir Suhail

^3,*,

Ahmed Himadan

³

,

Habeeb Ibrahim

³ and

Naglaa Mohammed

³

¹

Department of Electrical Engineering, Prince Mohammad Bin Fahd University, P.O. Box 1664, Al Khobar 31952, Saudi Arabia

²

Department of Mathematics, Kırıkkale University, Kırıkkale 71450, Türkiye

³

Department of Mathematics, College of Science, Qassim University, Buraidah 51452, Saudi Arabia

^*

Authors to whom correspondence should be addressed.

Symmetry 2026, 18(8), 1268; https://doi.org/10.3390/sym18081268

Submission received: 30 June 2026 / Revised: 21 July 2026 / Accepted: 24 July 2026 / Published: 26 July 2026

(This article belongs to the Special Issue Symmetry in Various Polynomial Applications)

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Abstract

We introduce a bivariate Hahn-factorial q-Laguerre–Tricomi–Appell polynomial class obtained by multiplying a nonsingular Appell factor by a Hahn-factorial deformation of the two-variable q-Laguerre–Tricomi generating kernel. The construction is interpreted formally, coefficientwise, and is not presented as an orthogonality or Laguerre–Hahn functional characterization. Its defining product yields a finite q-binomial convolution, which is the principal mechanism for transferring algebraic and operational properties from the base family to the Appell deformation. We establish direct and inverse connection formulas, a Hessenberg determinant representation, recurrence and higher Hahn-difference formulas, Hahn-shift identities, operational transfer formulas, and quasi-monomiality relations through a basis-defined raising operator. We also derive reductions to the underlying Hahn-factorial q-Laguerre–Tricomi family, one-variable Hahn-Appell and q-Appell families, translated q-Laguerre specializations, admissible Bernoulli–Euler-type subclasses, a singular Genocchi-type convolution, and the joint classical limit

q \to 1^{-}

,

w \to 0

. A finite coefficient scheme is then used to study representative zero distributions and graphical behavior. The results show that the proposed class is a coherent Appell-type deformation of a Hahn-factorial q-Laguerre–Tricomi kernel and that its structural identities follow from explicitly invertible q-binomial transforms whenever the Appell factor has a nonzero constant term.

Keywords: Hahn-factorial q-Laguerre–Tricomi polynomials; q-Appell polynomials; q-binomial convolution; monomiality principle; differential equations; formal Hahn difference relations; zero distribution

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MDPI and ACS Style

Khan, W.A.; Yağci, O.; Mohamed, K.S.; Suhail, M.; Himadan, A.; Ibrahim, H.; Mohammed, N. Structural Properties and Zero Distributions of q-Laguerre–Hahn–Tricomi–Appell Polynomials in the Framework of Quantum q-Calculus. Symmetry 2026, 18, 1268. https://doi.org/10.3390/sym18081268

AMA Style

Khan WA, Yağci O, Mohamed KS, Suhail M, Himadan A, Ibrahim H, Mohammed N. Structural Properties and Zero Distributions of q-Laguerre–Hahn–Tricomi–Appell Polynomials in the Framework of Quantum q-Calculus. Symmetry. 2026; 18(8):1268. https://doi.org/10.3390/sym18081268

Chicago/Turabian Style

Khan, Waseem Ahmad, Oğuz Yağci, Khidir Shaib Mohamed, Muntasir Suhail, Ahmed Himadan, Habeeb Ibrahim, and Naglaa Mohammed. 2026. "Structural Properties and Zero Distributions of q-Laguerre–Hahn–Tricomi–Appell Polynomials in the Framework of Quantum q-Calculus" Symmetry 18, no. 8: 1268. https://doi.org/10.3390/sym18081268

APA Style

Khan, W. A., Yağci, O., Mohamed, K. S., Suhail, M., Himadan, A., Ibrahim, H., & Mohammed, N. (2026). Structural Properties and Zero Distributions of q-Laguerre–Hahn–Tricomi–Appell Polynomials in the Framework of Quantum q-Calculus. Symmetry, 18(8), 1268. https://doi.org/10.3390/sym18081268

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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Structural Properties and Zero Distributions of q-Laguerre–Hahn–Tricomi–Appell Polynomials in the Framework of Quantum q-Calculus

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