Abstract
We consider the post-quantum analog of the q-2D Itô–Hermite polynomials recently introduced by Ismail and Zhang. Their basic properties are described, such as the zeros set, the Rodrigues formula, the corresponding raising and lowering operators, operational formulas, and the (p, q)-eigenvalue problem they obey. Moreover, we establish some partial generating functions, partial Mehler’s formulas, and integral representations.
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References
Arjika, S., El Moize, O., Mouayn, Z.: Une \(q\)-déformation de la transformation de Bargmann vraie-polyanalytique. C. R. Math. Acad. Sci. Paris 356(8), 903–910 (2018)
Cao, J., Cai, T., Cai, L.P.: A Note on q-partial differential equations for generalized q-2D Hermite polynomials. In: Baigent, S., Bohner, M., Elaydi, S. (eds.) Progress on Difference Equations and Discrete Dynamical Systems. International Conference on Difference Equations. Springer (2019)
Chakrabarti, R., Jagannathan, R.: A \((p, q)\)-oscillator realization of two-parameter quantum algebras. J. Phys. A Math. Gen. 24(13), L711 (1991)
Dallinger, R., Ruotsalainen, H., Wichman, R., Rupp, M.: Adaptive pre-distortion techniques based on orthogonal polynomials. In: Conference Record of the 44th Asilomar Conference on Signals, Systems and Computers, pp. 1945–1950. IEEE (2010)
El Fardi, A., Ghanmi, A., Imlal, L., Souid-El-Ainin, M.: Analytic and arithmetic properties of the \((\Gamma,\chi )\)-automorphic reproducing kernel function and associated Hermite-Gauss series. Ramanujan J. 48(1), 47–62 (2019)
Ernst, T.: A method for \(q\)-calculus. J. Nonlinear Math. Phys. 10(4), 487–525 (2003)
Ghanmi, A.: Operational formulae for the complex Hermite polynomials \(H_{p, q}(z, {\bar{z}})\). Integral Transforms Spec. Funct. 24(11), 884–895 (2013)
Ghanmi, A.: Mehler’s formulas for the univariate complex Hermite polynomials and applications. Math. Methods Appl. Sci. 40(18), 7540–7545 (2017)
Gupta, V.: Bernstein Durrmeyer operators based on two parameters. Facta Univ. Ser. Math. Inform. 31(1), 79–95 (2016)
Gupta, V., Rassias, T.M., Agrawal, P.N., Acu, A.M.: Basics of Post-quantum Calculus. Recent Advances in Constructive Approximation Theory. Optimization and Its Applications, vol. 138. Springer, Cham (2018)
Ilarslan, H.G.I., Acar, T.: Approximation by bivariate \((p, q)\)-Baskakov–Kantorovich operators. Georgian Math. J. 25(3), 397–407 (2018)
Intissar, A., Intissar, A.: Spectral properties of the Cauchy transform on \(L^{2}(\mathbb{C}, e^{-|z|^{2}}d\lambda (z)\). J. Math. Anal. Appl. 313(2), 400–418 (2006)
Ismail, M.E.H., Simeonov, P.: Complex Hermite polynomials: their combinatorics and integral operators. Proc. Am. Math. Soc. 143, 1397–1410 (2015)
Ismail, M., Zhang, R.: On some \(2\)D orthogonal \(q\)-polynomials. Trans. Am. Math. Soc. 369(10), 6779–6821 (2017)
Itô, K.: Complex multiple Wiener integral. Jpn. J. Math. 22, 63–86 (1952)
Jia, Z.: Homogeneous q-difference equations and generating functions for the generalized 2D-Hermite polynomials. Taiwan. J. Math. 25(1), 45–63 (2021)
Liu, Z.-G.: On the complex Hermite polynomials. Filomat 34(2), 409–420 (2020)
Mursaleen, M., Ansari, K.J., Khan, A.: Some approximation results by \((p, q)\)-analogue of Bernstein–Stancu operators. Appl. Math. Comput. 264, 392–402 (2015)
Raich, R., Zhou, G.: Orthogonal polynomials for complex Gaussian processes. IEEE Trans. Signal Process. 52(10), 2788–2797 (2004)
Shigekawa, I.: Eigenvalue problems of Schrödinger operator with magnetic field on compact Riemannian manifold. J. Funct. Anal. 75, 92–127 (1987)
Acknowledgements
The authors are grateful to the anonymous reviewers for their insightful, deep and extensive comments, greatly contributed to improve this paper. The assistance of the members of “Ahmed Intissar” and “Analysis, P.D.E. & Spectral Geometry” seminars is gratefully acknowledged.
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Benahmadi, A., Ghanmi, A. Post-quantum complex Itô–Hermite polynomials. Bol. Soc. Mat. Mex. 30, 12 (2024). https://doi.org/10.1007/s40590-023-00586-0
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DOI: https://doi.org/10.1007/s40590-023-00586-0
Keywords
- q-2D-Hermite polynomials
- \((p, q)\)-Itô–Hermite polynomials
- Zeros
- Partial generating functions
- Partial Mehler’s formulas
- \((p, q)\)-Laplacian
- Integral representations