Results in Mathematics

, 74:186 | Cite as

On a Novel Class of Polyanalytic Hermite Polynomials

  • Abdelhadi Benahmadi
  • Allal GhanmiEmail author


We discuss some algebraic and analytic properties of a general class of orthogonal polyanalytic polynomials, including their operational formulas, recurrence relations, generating functions, integral representations and different orthogonality identities. We establish their connection and rule in describing the \(L^2\)-spectral theory of some special second order differential operators of Laplacian type acting on the \(L^2\)-Gaussian Hilbert space on the whole complex plane. We will also show their importance in the theory of the so-called rank-one automorphic functions on the complex plane. In fact, a variant subclass leads to an orthogonal basis of the corresponding \(L^2\)-Gaussian Hilbert space on the strip \({\mathbb {C}}/{\mathbb {Z}}\).


Holomorphic Hermite polynomial polyanalytic complex Hermite polynomial generating function orthogonality relation integral representation polyanalytic functions rank-one autmorphic theta functions 

Mathematics Subject Classification

Primary 33E05 



  1. 1.
    Abreu, L.D., Feichtinger, H.G.: Function spaces of polyanalytic functions. In: Vasil’ev, A. (ed) Harmonic and Complex Analysis and Its Applications. Trends in Mathematics, pp. 1–38. Birkhäuser/Springer, Cham (2014) Google Scholar
  2. 2.
    Agorram, F., Benkhadra, A., El Hamyani, A., Ghanmi, A.: Complex Hermite functions as Fourier–Wigner transform. Integral Transforms Spec. Funct. 27(2), 94–100 (2016)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Balk, M.B.: Polyanalytic Functions, Mathematical Research, vol. 63. Akademie-Verlag, Berlin (1991)zbMATHGoogle Scholar
  4. 4.
    Benahmadi, A., Ghanmi, A.: Non-trivial 1d and 2d integral transforms of Segal–Bargmann type. Integral Transforms Spec. Funct. 30(7), 547–563 (2019)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Burgatti, P.: Sulla funzioni analitiche d’ordini \(n\). Boll. Unione Mat. Ital. 1(1), 8–12 (1922)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Chihara, H.: Holomorphic Hermite functions and ellipses. Integral Transforms Spec. Funct. 28(8), 605–615 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Cotfas, N., Gazeau, J.-P., Górska, K.: Complex and real Hermite polynomials and related quantizations. J. Phys. A 43(30), 305304 (2010)MathSciNetCrossRefGoogle Scholar
  8. 8.
    de Gosson, M.: The Wigner Transform. Advanced Textbooks in Mathematics. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2017)CrossRefGoogle Scholar
  9. 9.
    El Gourari, A., Ghanmi, A.: Spectral analysis on planar mixed automorphic forms. J. Math. Anal. Appl. 383(2), 474–481 (2011)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Ferapontov, E.V., Veselov, A.P.: Integrable Schrödinger operators with magnetic fields: factorization method on curved surfaces. J. Math. Phys. (2) 42, 590–607 (2001)CrossRefGoogle Scholar
  11. 11.
    Folland, G.B.: Harmonic Analysis in Phase Space. Princeton University Press, Princeton, NJ (1989)CrossRefGoogle Scholar
  12. 12.
    Gazeau, J.P., Szafraniec, F.H.: Holomorphic Hermite polynomials and a non-commutative plane. J. Phys. A 44(49), 495201 (2011)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Ghanmi, A.: A class of generalized complex Hermite polynomials. J. Math. Anal. Appl. 340, 1395–1406 (2008)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Ghanmi, A.: Operational formulae for the complex Hermite polynomials \(H_{p, q}(z,{\bar{z}})\). Integral Transforms Spec. Funct. 24(11), 884–895 (2013)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Ghanmi, A.: Mehler formulas for the univariate complex Hermite polynomials and applications. Math. Methods Appl. Sci. 40(18), 7540–7545 (2017)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Ghanmi, A., Intissar, A.: Construction of concrete orthonormal basis for \(L^2-(\Gamma,\chi )\)-theta functions associated to discrete subgroups of rank-one in \((+)\). J. Math. Phys. 54(6), 063514 (2013)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Hermite, C., Sur un nouveau développement en série des fonctions, Compt. Rend. Acad. Sci. Paris 58, t. LVII I, : 94–100 et 266–273 ou Oeuvres complètes, tome 2. Paris 1908, 293–308 (1864)Google Scholar
  18. 18.
    Intissar, A., Intissar, A.: Spectral properties of the Cauchy transform on \(L^2({\mathbb{C}},e^{-|z|^{2}}d\lambda )\). J. Math. Anal. Appl. 313(2), 400–418 (2006)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Ismail, M.E.H., Simeonov, P.: Complex Hermite polynomials: their combinatorics and integral operators. Proc. Am. Math. Soc. 143(4), 1397–1410 (2015)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Ismail, M.E.H.: Analytic properties of complex Hermite polynomials. Trans. Am. Math. Soc. 368(2), 1189–1210 (2016)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Itô, K.: Complex multiple Wiener integral. Jpn. J. Math. 22, 63–86 (1952)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Karp, D., Holomorphic spaces related to orthogonal polynomials and analytic continuation of functions, Analytic extension formulas and their applications (Fukuoka, 1999, Kyoto, : Int. Soc. Anal. Appl. Comput., vol. 9, Kluwer Acad. Publ. Dordrecht 2001, pp. 169–187 (2000)Google Scholar
  23. 23.
    Matsumoto, H.: Quadratic Hamiltonians and associated orthogonal polynomials. J. Funct. Anal. 136(1), 214–225 (1996)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Mehler, F.G.: Uber die Entwicklung einer Function von beliebig vielen Variabeln nach Laplaceschen Functionen höherer Ordnung. J. Reine Angew. Math. 66, 161–176 (1866)MathSciNetGoogle Scholar
  25. 25.
    Rainville, E.D.: Special Functions. Chelsea Publishing Co., Bronx, NY (1971)zbMATHGoogle Scholar
  26. 26.
    Shigekawa, I.: Eigenvalue problems of Schrödinger operator with magnetic field on compact Riemannian manifold. J. Funct. Anal. 75, 92–127 (1987)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Souid, El Ainin M.: Concrete description of the \((\Gamma,\chi )\)-theta Fock–Bargmann space for rank-one in high dimension. Complex Var. Elliptic Equ. 60(12), 1739–1751 (2015)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Szegö, G.: Orthogonal Polynomials, 4th edn. American Mathematical Society, Providence, RI (1975)zbMATHGoogle Scholar
  29. 29.
    Thangavelu, S.: Lectures on Hermite and Laguerre Expansions. Princeton University Press, Princeton (1993)zbMATHGoogle Scholar
  30. 30.
    Thangavelu, S.: Harmonic Analysis on the Heisenberg Group. Progress in Mathematics, 159. Birkhäuser Boston Inc, Boston, MA (1998)CrossRefGoogle Scholar
  31. 31.
    van Eijndhoven, S.J.L., Meyers, J.L.H.: New orthogonality relations for the Hermite polynomials and related Hilbert spaces. J. Math. Anal. Appl. 146(1), 89–98 (1990)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Wong, M.W.: Weyl Transforms. Springer, Berlin (1998)zbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Analysis, P.D.E. and Spectral Geometry, Department of Mathematics, Faculty of SciencesMohammed V UniversityRabatMorocco

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