Sheffer-type lacunary bivariate $ q $-Laguerre polynomials with applications to heterogeneous graph filters

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

doi:10.3934/nhm.2026053

Networks and Heterogeneous Media

2026, Volume 21, Issue 4: 1393-1425. doi: 10.3934/nhm.2026053

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Sheffer-type lacunary bivariate $ q $-Laguerre polynomials with applications to heterogeneous graph filters

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
4.
Department of Mathematics and Natural Sciences, Prince Mohammad Bin Fahd University, P.O. Box 1664, Al Khobar 31952, Saudi Arabia
5.
Cognitive Intelligence and AI Systems, Prince Mohammad Bin Fahd University, P.O. Box 1664, Al Khobar 31952, Saudi Arabia
6.
Faculty of Business and Communications, INTI International University, Nilai 71800, Negeri Sembilan, Malaysia

Received: 02 May 2026 Revised: 23 June 2026 Accepted: 23 June 2026 Published: 03 July 2026
33D45, 39A13, 05A30, 05C50, 65M70

We introduce a lacunary bivariate $ q $-Laguerre–Sheffer family generated by the product of a Sheffer factor, a first $ q $-exponential, and a $ q $-Tricomi–Bessel kernel. The construction has three independent controls: the lacunarity index fixes which powers enter the finite sums, the kernel order changes the Bessel-type shifts, and the Sheffer factor supplies an additional deformation of the sequence. We derive explicit coefficient and convolution formulas, lowering relations, a quasi-monomial operator structure, a $ q $-difference equation, and a $ q $-translation identity. We then evaluate the polynomials at a mass-weighted graph Laplacian associated with heterogeneous network diffusion. This gives polynomial spectral filters whose vertex-domain support is limited to a prescribed graph-hop radius. Numerical examples illustrate the locality bound, spectral response tuning, comparison with a Chebyshev polynomial filter, kernel truncation accuracy, and a one-parameter calibration problem.
- $ q $-Laguerre polynomials,
- $ m $-lacunary polynomials,
- Sheffer sequences,
- quasi-monomiality,
- $ q $-Tricomi–Bessel functions,
- numerical methods,
- process innovation,
- graph Laplacians,
- heterogeneous networks,
- spectral graph filters
Citation: Waseem Ahmad Khan, Oğuz Yağcı, Khidir Shaib Mohamed, Azhar Iqbal, Wei Sin Koh, Naglaa Mohammed. Sheffer-type lacunary bivariate $ q $-Laguerre polynomials with applications to heterogeneous graph filters[J]. Networks and Heterogeneous Media, 2026, 21(4): 1393-1425. doi: 10.3934/nhm.2026053

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Abstract

We introduce a lacunary bivariate $ q $-Laguerre–Sheffer family generated by the product of a Sheffer factor, a first $ q $-exponential, and a $ q $-Tricomi–Bessel kernel. The construction has three independent controls: the lacunarity index fixes which powers enter the finite sums, the kernel order changes the Bessel-type shifts, and the Sheffer factor supplies an additional deformation of the sequence. We derive explicit coefficient and convolution formulas, lowering relations, a quasi-monomial operator structure, a $ q $-difference equation, and a $ q $-translation identity. We then evaluate the polynomials at a mass-weighted graph Laplacian associated with heterogeneous network diffusion. This gives polynomial spectral filters whose vertex-domain support is limited to a prescribed graph-hop radius. Numerical examples illustrate the locality bound, spectral response tuning, comparison with a Chebyshev polynomial filter, kernel truncation accuracy, and a one-parameter calibration problem.

References

[1]	F. H. Jackson, $q$-difference equations, Amer. J. Math., 32 (1910), 305–314. https://doi.org/10.2307/2370183
[2]	V. Kac, P. Cheung, Quantum Calculus, Springer, New York, 2002. https://doi.org/10.1007/978-1-4613-0071-7
[3]	G. Gasper, M. Rahman, Basic hypergeometric series, in Encyclopedia of Mathematics and its Applications, 2nd ed., Cambridge University Press, Cambridge, (2004), 1–37. https://doi.org/10.1017/CBO9780511526251
[4]	R. Koekoek, P. A. Lesky, R. F. Swarttouw, Hypergeometric Orthogonal Polynomials and Their $q$-Analogues, Springer Monographs in Mathematics, Springer, Berlin, 2010. https://doi.org/10.1007/978-3-642-05014-5
[5]	D. S. Moak, The $q$-analogue of the Laguerre polynomials, J. Math. Anal. Appl., 81 (1981), 20–47. https://doi.org/10.1016/0022-247X(81)90048-2 doi: 10.1016/0022-247X(81)90048-2
[6]	R. Askey, Limits of some $q$-Laguerre polynomials, J. Approx. Theory, 46 (1986), 213–216. https://doi.org/10.1016/0021-9045(86)90062-6 doi: 10.1016/0021-9045(86)90062-6
[7]	J. Cigler, Operator methods for $q$-identities. II. $q$-Laguerre polynomials, Monatsh. Math., 91 (1981), 105–117. https://doi.org/10.1007/BF01295141 doi: 10.1007/BF01295141
[8]	J. Cao, N. Raza, M. Fadel, Two-variable $q$-Laguerre polynomials from the context of quasi-monomiality, J. Math. Anal. Appl., 535 (2024), 128126. https://doi.org/10.1016/j.jmaa.2024.128126 doi: 10.1016/j.jmaa.2024.128126
[9]	Q. Bao, D. Yang, Notes on $q$-partial differential equations for $q$-Laguerre polynomials and little $q$-Jacobi polynomials, Fundam. J. Math. Appl., 7 (2024), 59–76. https://doi.org/10.33401/fujma.1365120 doi: 10.33401/fujma.1365120
[10]	N. Ahmad, W. A. Khan, A new generalization of $q$-Laguerre-based Appell polynomials and quasi-monomiality, Symmetry, 17 (2025), 439. https://doi.org/10.3390/sym17030439 doi: 10.3390/sym17030439
[11]	H. Qawaqneh, W. A. Khan, H. Aydi, U. Duran, C. S. Ryoo, A comprehensive study of generalized bivariate $q$-Laguerre polynomials: Structural properties and applications, Eur. J. Pure Appl. Math., 18 (2025), 6668. https://doi.org/10.29020/nybg.ejpam.v18i3.6668 doi: 10.29020/nybg.ejpam.v18i3.6668
[12]	M. Zayed, W. A. Khan, C. S. Ryoo, U. Duran, An exploratory study on bivariate extended $q$-Laguerre-based Appell polynomials with some applications, AIMS Math., 10 (2025), 12841–12867. https://doi.org/10.3934/math.2025577 doi: 10.3934/math.2025577
[13]	M. Kumar, N. Raza, W. Ramírez, On the theory and applications of $q$-Mittag-Leffler-Laguerre polynomials, Dolomites Res. Notes Approx., 18 (2025), 118–134. https://doi.org/10.25430/pupj-DRNA-2025-1-10 doi: 10.25430/pupj-DRNA-2025-1-10
[14]	P. Appell, On a class of polynomials, Ann. Sci. Éc. Norm. Supér., 9 (1880), 119–144. https://doi.org/10.24033/asens.186
[15]	I. M. Sheffer, Some properties of polynomial sets of type zero, Duke Math. J., 5 (1939), 590–622. https://doi.org/10.1215/S0012-7094-39-00549-1 doi: 10.1215/S0012-7094-39-00549-1
[16]	G. C. Rota, D. Kahaner, A. Odlyzko, On the foundations of combinatorial theory. VIII. Finite operator calculus, J. Math. Anal. Appl., 42 (1973), 684–760. https://doi.org/10.1016/0022-247X(73)90172-8 doi: 10.1016/0022-247X(73)90172-8
[17]	S. Roman, The umbral calculus, in Pure and Applied Mathematics, Academic Press, New York, 111 (1984). Available from: https://search.worldcat.org/title/11866043.
[18]	S. Khan, N. Raza, Monomiality principle, operational methods and family of Laguerre–Sheffer polynomials, J. Math. Anal. Appl., 387 (2012), 90–102. https://doi.org/10.1016/j.jmaa.2011.08.064 doi: 10.1016/j.jmaa.2011.08.064
[19]	Z. Özat, B. Çekim, M. A. Özarslan, Laguerre-type general-Appell polynomials, Integr. Transforms Spec. Funct., 36 (2025), 571–589. https://doi.org/10.1080/10652469.2024.2418889 doi: 10.1080/10652469.2024.2418889
[20]	W. A. Khan, K. S. Mohamed, F. A. Costabile, S. A. Wani, A. Adam, A new generalization of $m$th-order Laguerre-based Appell polynomials associated with two-variable general polynomials, Mathematics, 13 (2025), 2179. https://doi.org/10.3390/math13132179 doi: 10.3390/math13132179
[21]	M. Fadel, W. Ramírez, C. Cesarano, S. Díaz, $q$-Legendre based Gould–Hopper polynomials and $q$-operational methods, Ann. Univ. Ferrara, 71 (2025), 32. https://doi.org/10.1007/s11565-025-00587-z doi: 10.1007/s11565-025-00587-z
[22]	G. Dattoli, M. Migliorati, H. M. Srivastava, Some families of generating functions for the Bessel and related functions, Georgian Math. J., 11 (2004), 219–228. https://doi.org/10.1515/GMJ.2004.219 doi: 10.1515/GMJ.2004.219
[23]	M. Riyasat, T. Nahid, S. Khan, $q$-Tricomi functions and quantum algebra representations, Georgian Math. J., 28 (2021), 793–803. https://doi.org/10.1515/gmj-2020-2079 doi: 10.1515/gmj-2020-2079
[24]	T. Nahid, H. P. Rani, Mittag-Leffler based Bessel and Tricomi functions via umbral approach, Rep. Math. Phys., 92 (2023), 1–17. https://doi.org/10.1016/S0034-4877(23)00051-4 doi: 10.1016/S0034-4877(23)00051-4
[25]	M. Fadel, N. Raza, W. S. Du, Characterizing $q$-Bessel functions of the first kind with their new summation and integral representations, Mathematics, 11 (2023), 3831. https://doi.org/10.3390/math11183831 doi: 10.3390/math11183831
[26]	U. Zainab, M. Kumar, N. Raza, Borel transform and integral identities involving $\lambda$-Bessel and $\lambda$-Tricomi matrix functions, Math. Methods Appl. Sci., 48 (2025), 3253–3271. https://doi.org/10.1002/mma.10483 doi: 10.1002/mma.10483
[27]	D. Babusci, G. Dattoli, K. Górska, K. A. Penson, Lacunary generating functions for the Laguerre polynomials, Séminaire Lotharingien de Combinatoire, 76 (2017), B76b. Available from: https://arXiv.org/abs/1302.4894.
[28]	F. R. K. Chung, Spectral graph theory, in CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, 92 (1997). Available from: https://bookstore.ams.org/cbms-92.
[29]	D. I. Shuman, S. K. Narang, P. Frossard, A. Ortega, P. Vandergheynst, The emerging field of signal processing on graphs: Extending high-dimensional data analysis to networks and other irregular domains, IEEE Signal Process. Mag., 30 (2013), 83–98. https://doi.org/10.1109/MSP.2012.2235192 doi: 10.1109/MSP.2012.2235192
[30]	D. Mugnolo, Semigroup Methods for Evolution Equations on Networks, Understanding Complex Systems, Springer, Cham, 2014. https://doi.org/10.1007/978-3-319-04621-1
[31]	T. Liu, R. Xue, A convergent multi-step efficient iteration method to solve nonlinear equation systems, J. Appl. Math. Comput., 71 (2025), 2571–2588. https://doi.org/10.1007/s12190-024-02324-9 doi: 10.1007/s12190-024-02324-9
[32]	T. Liu, B. Ding, A radial basis function neural network approach for solving a diffusion partial differential equation efficiently, Appl. Math. Comput., 509 (2026), 129651. https://doi.org/10.1016/j.amc.2025.129651 doi: 10.1016/j.amc.2025.129651
[33]	T. Liu, R. Xue, M. Barfeie, A unified framework for high-order compact finite differences using infinitely- and piecewise-smooth RBFs with polynomials, Calcolo, 63 (2026), 18. https://doi.org/10.1007/s10092-026-00684-1 doi: 10.1007/s10092-026-00684-1
[34]	Y. Liu, Y. Li, T. Liu, An RBF–FD method for pricing under the Bates model: Handling stochastic volatility and jump processes, Eng. Anal. Bound. Elem., 183 (2026), 106622. https://doi.org/10.1016/j.enganabound.2025.106622 doi: 10.1016/j.enganabound.2025.106622
[35]	Y. Liu, Y. Ding, F. Soleymani, Polynomial approximation and tensor-product iterative methods for Rayleigh–Tail and fractional Rayleigh–Tail integral equations, Comput. Appl. Math., 45 (2026), 173. https://doi.org/10.1007/s40314-025-03477-4 doi: 10.1007/s40314-025-03477-4
[36]	F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, et al., NIST Digital Library of Mathematical Functions, Release 1.2.7, 2026. Available from: https://dlmf.nist.gov/10.2.
[37]	D. K. Hammond, P. Vandergheynst, R. Gribonval, Wavelets on graphs via spectral graph theory, Appl. Comput. Harmon. Anal., 30 (2011), 129–150. https://doi.org/10.1016/j.acha.2010.04.005 doi: 10.1016/j.acha.2010.04.005
[38]	M. Defferrard, X. Bresson, P. Vandergheynst, Convolutional neural networks on graphs with fast localized spectral filtering, Adv. Neural Inf. Process. Syst., 29 (2016). Available from: https://arXiv.org/abs/1606.09375.
[39]	S. Banerjee, S. Ganguly, On Laplacian and distance Laplacian spectra of generalized fan graph and a new graph class, Int. J. Math. Comput. Eng., 3 (2025), 293–306. https://doi.org/10.2478/ijmce-2025-0021 doi: 10.2478/ijmce-2025-0021
[40]	S. Banerjee, Signless Laplacian spectrum of power graph of certain finite non-commutative groups, Int. J. Math. Comput. Eng., 4 (2026), 33–44. https://doi.org/10.2478/ijmce-2026-0003 doi: 10.2478/ijmce-2026-0003
[41]	A. Kiran, M. Yaseen, A. Khan, T. Abdeljawad, M. A. Alqudah, R. Thinakaran, Solving time fractional diffusion-wave equation using hyperbolic polynomial B-splines: A uniform grid approach, Ain Shams Eng. J., 17 (2026), 103868. https://doi.org/10.1016/j.asej.2025.103868 doi: 10.1016/j.asej.2025.103868

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