Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 021, 37 pages      arXiv:1502.07256
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications

Classes of Bivariate Orthogonal Polynomials

Mourad E.H. Ismail ab and Ruiming Zhang c
a) Department of Mathematics, King Saud University, Riyadh, Saudi Arabia
b) Department of Mathematics, University of Central Florida, Orlando, Florida 32816, USA
c) College of Science, Northwest A&F University, Yangling, Shaanxi 712100, P.R. China

Received August 04, 2015, in final form February 15, 2016; Published online February 24, 2016

We introduce a class of orthogonal polynomials in two variables which generalizes the disc polynomials and the 2-$D$ Hermite polynomials. We identify certain interesting members of this class including a one variable generalization of the 2-$D$ Hermite polynomials and a two variable extension of the Zernike or disc polynomials. We also give $q$-analogues of all these extensions. In each case in addition to generating functions and three term recursions we provide raising and lowering operators and show that the polynomials are eigenfunctions of second-order partial differential or $q$-difference operators.

Key words: disc polynomials; Zernike polynomials; 2$D$-Laguerre polynomials; $q$-2$D$-Laguerre polynomials; generating functions; ladder operators; $q$-Sturm-Liouville equations; $q$-integrals; $q$-Zernike polynomials; 2$D$-Jacobi polynomials; $q$-2$D$-Jacobi polynomials; connection relations; biorthogonal functions; generating functions; Rodrigues formulas; zeros.

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  1. Al-Salam W.A., Chihara T.S., Convolutions of orthonormal polynomials, SIAM J. Math. Anal. 7 (1976), 16-28.
  2. Ali S.T., Bagarello F., Honnouvo G., Modular structures on trace class operators and applications to Landau levels, J. Phys. A: Math. Theor. 43 (2010), 105202, 17 pages, arXiv:0906.3980.
  3. Appell P., Sur des polynômes de deux variables analogues aux polynômes de Jacobi, Archiv Math. Phys. 66 (1881), 238-245.
  4. Appell P., Kampé de Fériet J., Fonctions hypergéométriques et hypersphériques. Polynomes d'Hermite, Gauthier-Villars, Paris, 1926.
  5. Chihara T.S., An introduction to orthogonal polynomials, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York - London - Paris, 1978.
  6. Cotfas N., Gazeau J.P., Górska K., Complex and real Hermite polynomials and related quantizations, J. Phys. A: Math. Theor. 43 (2010), 305304, 14 pages, arXiv:1001.3248.
  7. Dunkl C.F., Xu Y., Orthogonal polynomials of several variables, 2nd ed., Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2014.
  8. Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher transcendental functions. Vol. II, McGraw-Hill Book Company, New York, 1953.
  9. Finkel F., Kamran N., The Lie algebraic structure of differential operators admitting invariant spaces of polynomials, Adv. in Appl. Math. 20 (1998), 300-322, q-alg/9612027.
  10. Floris P.G.A., A noncommutative discrete hypergroup associated with $q$-disk polynomials, J. Comput. Appl. Math. 68 (1996), 69-78.
  11. Floris P.G.A., Koelink H.T., A commuting $q$-analogue of the addition formula for disk polynomials, Constr. Approx. 13 (1997), 511-535, math.QA/9509222.
  12. Ghanmi A., A class of generalized complex Hermite polynomials, J. Math. Anal. Appl. 340 (2008), 1395-1406, arXiv:0704.3576.
  13. Ghanmi A., Operational formulae for the complex Hermite polynomials $H_{p,q}(z,\overline{z})$, Integral Transforms Spec. Funct. 24 (2013), 884-895, arXiv:1211.5746.
  14. Hermite C., Sur un nouveau développement en series des fonctions, Comptes Rendus Acad. Sci. Paris 58 (1864), 93-100, 266-273.
  15. Hermite C., Sur quelques développement en series des fonctions,, Comptes Rendus Acad. Sci. Paris 60 (1865), 370-377, 432-440, 461-466, 512-518.
  16. Intissar A., Intissar A., Spectral properties of the Cauchy transform on $L_2({\mathbb C},e^{-\vert z\vert^2}\lambda(z))$, J. Math. Anal. Appl. 313 (2006), 400-418.
  17. Ismail M.E.H., Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge, 2009.
  18. Ismail M.E.H., Analytic properties of complex Hermite polynomials, Trans. Amer. Math. Soc. 368 (2016), 1189-1210.
  19. Ismail M.E.H., Zeng J., A combinatorial approach to the 2D-Hermite and 2D-Laguerre polynomials, Adv. in Appl. Math. 64 (2015), 70-88.
  20. Ismail M.E.H., Zeng J., Two variable extensions of the Laguerre and disc polynomials, J. Math. Anal. Appl. 424 (2015), 289-303.
  21. Ismail M.E.H., Zeng J., On some $2D$ orthogonal $q$-polynomials, Trans. Amer. Math. Soc., to appear, arXiv:1411.5223.
  22. Itô K., Complex multiple Wiener integral, Japan J. Math. 22 (1952), 63-86.
  23. Koekoek R., Swarttouw R.F., The Askey-scheme of hypergeometric orthogonal polynomials and its $q$-analogue, Report 98-17, Faculty of Technical Mathematics and Informatics, Delft University of Technology, 1998, available at
  24. Koornwinder T.H., The addition formula for Jacobi polynomials III. Completion of the proof, Report TW 135/72, Mathematisch Centrum, Amsterdam, 1972, available at
  25. Koornwinder T.H., Two-variable analogues of the classical orthogonal polynomials, in Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), Academic Press, New York, 1975, 435-495.
  26. Maldonado C.D., Note on orthogonal polynomials which are ''invariant in form'' to rotations of axes, J. Math. Phys. 6 (1965), 1935-1938.
  27. Myrick D.R., A generalization of the radial polynomials of F. Zernike, SIAM J. Appl. Math. 14 (1966), 476-489.
  28. Rainville E.D., Special functions, The Macmillan Co., New York, 1960.
  29. Šapiro R.L., Special functions related to representations of the group ${\rm SU}(n)$, of class I with respect to ${\rm SU}(n-1)$ $(n\ge 3)$, Izv. Vyssh. Uchebn. Zaved. Matematika (1968), no. 4, 97-107.
  30. Shigekawa I., Eigenvalue problems for the Schrödinger operator with the magnetic field on a compact Riemannian manifold, J. Funct. Anal. 75 (1987), 92-127.
  31. Szegő G., Orthogonal polynomials, Colloquium Publications, Vol. 23, 4th ed., Amer. Math. Soc., Providence, R.I., 1975.
  32. Thirulogasanthar K., Honnouvo G., Krzyżak A., Coherent states and Hermite polynomials on quaternionic Hilbert spaces, J. Phys. A: Math. Theor. 43 (2010), 385205, 13 pages.
  33. Waldron S., Orthogonal polynomials on the disc, J. Approx. Theory 150 (2008), 117-131.
  34. Wünsche A., Laguerre $2$D-functions and their application in quantum optics, J. Phys. A: Math. Gen. 31 (1998), 8267-8287.
  35. Wünsche A., Transformations of Laguerre $2D$-polynomials and their applications to quasiprobabilities, J. Phys. A: Math. Gen. 32 (1999), 3179-3199.
  36. Wünsche A., Generalized Zernike or disc polynomials, J. Comput. Appl. Math. 174 (2005), 135-163.
  37. Xu Y., Complex versus real orthogonal polynomials of two variables, Integral Transforms Spec. Funct. 26 (2015), 134-151, arXiv:1307.7819.
  38. Zernike F., Beugungstheorie des Schneidenverfahrens und seiner verbesserten Form, der Phasenkontrastmethode, Physica 1 (1934), 689-704.
  39. Zernike F., Brinkman H.C., Hypersphärische Funktionen und die in sphärischen Bereichen orthogonalen Polynome, Proc. Akad. Wet. Amsterdam 38 (1935), 161-170.

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