## On some $2D$ orthogonal $q$-polynomials

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- by Mourad E. H. Ismail and Ruiming Zhang PDF
- Trans. Amer. Math. Soc.
**369**(2017), 6779-6821 Request permission

## Abstract:

We introduce two $q$-analogues of the $2D$-Hermite polynomials which are functions of two complex variables. We derive explicit formulas, orthogonality relations, raising and lowering operator relations, generating functions, and Rodrigues formulas for both families. We also introduce a $q$-$2D$ analogue of the disk polynomials (Zernike polynomials) and derive similar formulas for them as well, including evaluating certain connection coefficients. Some of the generating functions may be related to Rogers–Ramanujan type identities.## References

- S. Twareque Ali, F. Bagarello, and G. Honnouvo,
*Modular structures on trace class operators and applications to Landau levels*, J. Phys. A**43**(2010), no. 10, 105202, 17. MR**2593994**, DOI 10.1088/1751-8113/43/10/105202 - George E. Andrews,
*The theory of partitions*, Encyclopedia of Mathematics and its Applications, Vol. 2, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. MR**0557013** - G. E. Andrews,
*$q$-hypergeometric and related functions*, NIST handbook of mathematical functions, U.S. Dept. Commerce, Washington, DC, 2010, pp. 419–433. MR**2655357** - George E. Andrews and F. G. Garvan,
*Dyson’s crank of a partition*, Bull. Amer. Math. Soc. (N.S.)**18**(1988), no. 2, 167–171. MR**929094**, DOI 10.1090/S0273-0979-1988-15637-6 - George E. Andrews, Richard Askey, and Ranjan Roy,
*Special functions*, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR**1688958**, DOI 10.1017/CBO9781107325937 - T. S. Chihara,
*An introduction to orthogonal polynomials*, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York-London-Paris, 1978. MR**0481884** - Nicolae Cotfas, Jean Pierre Gazeau, and Katarzyna Górska,
*Complex and real Hermite polynomials and related quantizations*, J. Phys. A**43**(2010), no. 30, 305304, 14. MR**2659624**, DOI 10.1088/1751-8113/43/30/305304 - Charles F. Dunkl and Yuan Xu,
*Orthogonal polynomials of several variables*, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 155, Cambridge University Press, Cambridge, 2014. MR**3289583**, DOI 10.1017/CBO9781107786134 - Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi,
*Higher transcendental functions. Vols. I, II*, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953. Based, in part, on notes left by Harry Bateman. MR**0058756** - Paul G. A. Floris,
*Addition formula for $q$-disk polynomials*, Compositio Math.**108**(1997), no. 2, 123–149. MR**1468832**, DOI 10.1023/A:1000179602766 - Paul G. A. Floris,
*A noncommutative discrete hypergroup associated with $q$-disk polynomials*, J. Comput. Appl. Math.**68**(1996), no. 1-2, 69–78. MR**1418751**, DOI 10.1016/0377-0427(95)00256-1 - P. G. A. Floris and H. T. Koelink,
*A commuting $q$-analogue of the addition formula for disk polynomials*, Constr. Approx.**13**(1997), no. 4, 511–535. MR**1466064**, DOI 10.1007/s003659900057 - Kristina Garrett, Mourad E. H. Ismail, and Dennis Stanton,
*Variants of the Rogers-Ramanujan identities*, Adv. in Appl. Math.**23**(1999), no. 3, 274–299. MR**1722235**, DOI 10.1006/aama.1999.0658 - George Gasper and Mizan Rahman,
*Basic hypergeometric series*, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 96, Cambridge University Press, Cambridge, 2004. With a foreword by Richard Askey. MR**2128719**, DOI 10.1017/CBO9780511526251 - Allal Ghanmi,
*A class of generalized complex Hermite polynomials*, J. Math. Anal. Appl.**340**(2008), no. 2, 1395–1406. MR**2390939**, DOI 10.1016/j.jmaa.2007.10.001 - Allal Ghanmi,
*Operational formulae for the complex Hermite polynomials $H_{p,q}(z,\overline {z})$*, Integral Transforms Spec. Funct.**24**(2013), no. 11, 884–895. MR**3172001**, DOI 10.1080/10652469.2013.772172 - Abdelkader Intissar and Ahmed Intissar,
*Spectral properties of the Cauchy transform on $L_2(\Bbb C,e^{-|z|^2}\lambda (z))$*, J. Math. Anal. Appl.**313**(2006), no. 2, 400–418. MR**2182508**, DOI 10.1016/j.jmaa.2005.09.056 - Mourad E. H. Ismail,
*Asymptotics of $q$-orthogonal polynomials and a $q$-Airy function*, Int. Math. Res. Not.**18**(2005), 1063–1088. MR**2149641**, DOI 10.1155/IMRN.2005.1063 - Mourad E. H. Ismail,
*Classical and quantum orthogonal polynomials in one variable*, Encyclopedia of Mathematics and its Applications, vol. 98, Cambridge University Press, Cambridge, 2009. With two chapters by Walter Van Assche; With a foreword by Richard A. Askey; Reprint of the 2005 original. MR**2542683** - Mourad E. H. Ismail,
*Analytic properties of complex Hermite polynomials*, Trans. Amer. Math. Soc.**368**(2016), no. 2, 1189–1210. MR**3430361**, DOI 10.1090/tran/6358 - Mourad E. H. Ismail and Plamen Simeonov,
*Complex Hermite polynomials: their combinatorics and integral operators*, Proc. Amer. Math. Soc.**143**(2015), no. 4, 1397–1410. MR**3314055**, DOI 10.1090/S0002-9939-2014-12362-8 - M. E. H. Ismail and J. Zeng, Combinatorial interpretations of the 2D-Hermite and 2D-Laguerre polynomials with applications, to appear.
- M. E. H. Ismail and R. Zhang, The Kibble-Slepian formula, to appear.
- Kiyosi Itô,
*Complex multiple Wiener integral*, Jpn. J. Math.**22**(1952), 63–86 (1953). MR**63609**, DOI 10.4099/jjm1924.22.0_{6}3 - R. Koekoek and R. Swarttouw,
*The Askey-scheme of hypergeometric orthogonal polynomials and its $q$-analogues*, Reports of the Faculty of Technical Mathematics and Informatics, no. 98-17, Delft University of Technology, Delft, 1998. - Srinivasa Ramanujan,
*The lost notebook and other unpublished papers*, Springer-Verlag, Berlin; Narosa Publishing House, New Delhi, 1988. With an introduction by George E. Andrews. MR**947735** - Ichir\B{o} Shigekawa,
*Eigenvalue problems for the Schrödinger operator with the magnetic field on a compact Riemannian manifold*, J. Funct. Anal.**75**(1987), no. 1, 92–127. MR**911202**, DOI 10.1016/0022-1236(87)90108-X - Gábor Szegő,
*Orthogonal polynomials*, 4th ed., American Mathematical Society Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, R.I., 1975. MR**0372517** - K. Thirulogasanthar, G. Honnouvo, and A. Krzyżak,
*Coherent states and Hermite polynomials on quaternionic Hilbert spaces*, J. Phys. A**43**(2010), no. 38, 385205, 13. MR**2718325**, DOI 10.1088/1751-8113/43/38/385205 - Alfred Wünsche,
*Laguerre $2$D-functions and their application in quantum optics*, J. Phys. A**31**(1998), no. 40, 8267–8287. MR**1651499**, DOI 10.1088/0305-4470/31/40/017 - Alfred Wünsche,
*Transformations of Laguerre 2D polynomials with applications to quasiprobabilities*, J. Phys. A**32**(1999), no. 17, 3179–3199. MR**1690370**, DOI 10.1088/0305-4470/32/17/309 - S. J. L. van Eijndhoven and J. L. H. Meyers,
*New orthogonality relations for the Hermite polynomials and related Hilbert spaces*, J. Math. Anal. Appl.**146**(1990), no. 1, 89–98. MR**1041203**, DOI 10.1016/0022-247X(90)90334-C - E. T. Whittaker and G. N. Watson,
*A course of modern analysis. An introduction to the general theory of infinite processes and of analytic functions: with an account of the principal transcendental functions*, Cambridge University Press, New York, 1962. Fourth edition. Reprinted. MR**0178117** - F. Zernike,
*Beugungstheorie des Schneidenverfahrens und Seiner Verbesserten Form, der Phasenkontrastmethode*, Physica**1**(1934), 689–704.

## Additional Information

**Mourad E. H. Ismail**- Affiliation: College of Science, Northwest A&F University, Yangling, Shaanxi 712100, People’s Republic of China – and – Department of Mathematics, University of Central Florida, Orlando, Florida 32816
- MR Author ID: 91855
- Email: mourad.eh.ismail@gmail.com
**Ruiming Zhang**- Affiliation: College of Science, Northwest A&F University, Yangling, Shaanxi 712100, People’s Republic of China
- MR Author ID: 257230
- Email: ruimingzhang@yahoo.com
- Received by editor(s): November 5, 2014
- Received by editor(s) in revised form: June 16, 2015, and August 14, 2015
- Published electronically: July 7, 2017
- Additional Notes: The research of the first author was supported by the DSFP at King Saud University in Riyadh, by Research Grants Council of Hong Kong contract #1014111, and by the National Plan for Science, Technology and innovation (MAARIFAH), King Abdelaziz City for Science and Technology, Kingdom of Saudi Arabia, Award No. 14-MAT623-0

The research of the second author was supported by Research Grants Council of Hong Kong, Contract #1014111, and the National Science Foundation of China, grant No. 11371294. The second author is the corresponding author - © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**369**(2017), 6779-6821 - MSC (2010): Primary 33C50, 33D50; Secondary 33C45, 33D45
- DOI: https://doi.org/10.1090/tran/6824
- MathSciNet review: 3683094