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

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

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.

Keywords

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

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

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

[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

[7] J. Cigler, Operator methods for q-identities. II. q-Laguerre polynomials, Monatsh. Math., 91(1981), 105-117. https://doi.org/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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[20] W. A. Khan, K. S. Mohamed, F. A. Costabile, S. A. Wani, A. Adam, A new generalization of mth-order Laguerre-based Appell polynomials associated with two-variable general polynomials, Mathematics, 13(2025), 2179. https://doi.org/10.3390/math13132179

[21] M. Fadel, W. Ramírez, C. Cesarano, S. Díaz, q-Legendre based Gould-Hopper polynomials andq-operational methods, Ann. Univ. Ferrara, 71(2025), 32. https://doi.org/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

[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

[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

[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

[26] U. Zainab, M. Kumar, N. Raza, Borel transform and integral identities involving λ-Bessel and λ-Tricomi matrix functions, Math. Methods Appl. Sci., 48(2025), 3253-3271. https://doi.org/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

[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

[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

[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

[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

[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

[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

[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

[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

[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