Complex Hermite polynomials: Their combinatorics and integral operators
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- by Mourad E. H. Ismail and Plamen Simeonov PDF
- Proc. Amer. Math. Soc. 143 (2015), 1397-1410 Request permission
Abstract:
We consider two types of Hermite polynomials of a complex variable. For each type we obtain combinatorial interpretations for the linearization coefficients of products of these polynomials. We use the combinatorial interpretations to give new proofs of several orthogonality relations satisfied by these polynomials with respect to positive exponential weights in the complex plane. We also construct four integral operators of which the first type of complex Hermite polynomials are eigenfunctions and we identify the corresponding eigenvalues. We prove that the products of these complex Hermite polynomials are complete in certain $L_2$-spaces.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
- Richard Askey, Orthogonal polynomials and special functions, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1975. MR 0481145
- Ruth Azor, J. Gillis, and J. D. Victor, Combinatorial applications of Hermite polynomials, SIAM J. Math. Anal. 13 (1982), no. 5, 879–890. MR 668329, DOI 10.1137/0513062
- 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, Encyclopedia of Mathematics and its Applications, vol. 81, Cambridge University Press, Cambridge, 2001. MR 1827871, DOI 10.1017/CBO9780511565717
- Jean Pierre Gazeau and Franciszek Hugon Szafraniec, Holomorphic Hermite polynomials and a non-commutative plane, J. Phys. A 44 (2011), no. 49, 495201, 13. MR 2860876, DOI 10.1088/1751-8113/44/49/495201
- 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
- A. Ghanmi, Operational formulae for the complex Hermite polynomials $H_{p,q}(z,\bar z)$, Integral Transforms and Special Functions 340 (2013).
- C. D. Godsil, Hermite polynomials and a duality relation for matching polynomials, Combinatorica 1 (1981), no. 3, 257–262. MR 637830, DOI 10.1007/BF02579331
- K. Gorska, private communication.
- Mourad E. H. Ismail, Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications, vol. 98, Cambridge University Press, Cambridge, 2005. With two chapters by Walter Van Assche; With a foreword by Richard A. Askey. MR 2191786, DOI 10.1017/CBO9781107325982
- Mourad E. H. Ismail, Analytic properties of complex Hermite polynomials, preprint, 2013.
- Mourad E. H. Ismail and Plamen Simeonov, Asymptotics of generalized derangements, Adv. Comput. Math. 39 (2013), no. 1, 101–127. MR 3068596, DOI 10.1007/s10444-011-9271-7
- Kiyosi Itô, Complex multiple Wiener integral, Jpn. J. Math. 22 (1952), 63–86 (1953). MR 63609, DOI 10.4099/jjm1924.22.0_{6}3
- Dmitri Karp, Holomorphic spaces related to orthogonal polynomials and analytic continuation of functions, Analytic extension formulas and their applications (Fukuoka, 1999/Kyoto, 2000) Int. Soc. Anal. Appl. Comput., vol. 9, Kluwer Acad. Publ., Dordrecht, 2001, pp. 169–187. MR 1830382, DOI 10.1007/978-1-4757-3298-6_{1}0
- Dongsu Kim and Jiang Zeng, A combinatorial formula for the linearization coefficients of general Sheffer polynomials, European J. Combin. 22 (2001), no. 3, 313–332. MR 1822720, DOI 10.1006/eujc.2000.0459
- Richard P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. MR 1676282, DOI 10.1017/CBO9780511609589
- 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
- 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
- X. G. Viennot, Une théorie combinatoire des polynômes orthogonaux généraux, Lecture Notes, Université du Québec à Montréal, Montreal, 1983.
- Jiang Zeng, Weighted derangements and the linearization coefficients of orthogonal Sheffer polynomials, Proc. London Math. Soc. (3) 65 (1992), no. 1, 1–22. MR 1162485, DOI 10.1112/plms/s3-65.1.1
Additional Information
- Mourad E. H. Ismail
- Affiliation: Department of Mathematics, University of Central Florida, Orlando, Florida 32816 – and – King Saud University, Riyadh, Saudi Arabia
- MR Author ID: 91855
- Email: mourad.eh.ismail@gmail.com
- Plamen Simeonov
- Affiliation: Department of Mathematics and Statistics, University of Houston-Downtown, Houston, Texas 77002
- Email: simeonovp@uhd.edu
- Received by editor(s): January 25, 2013
- Received by editor(s) in revised form: May 31, 2013, and July 18, 2013
- Published electronically: December 9, 2014
- Additional Notes: The first author’s research was supported by the NPST Program of King Saud University; project number 10-MAT1293-02 and King Saud University in Riyadh and by the Research Grants Council of Hong Kong grant # CityU 1014111.
- Communicated by: Jim Haglund
- © Copyright 2014 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 1397-1410
- MSC (2010): Primary 05A15, 05A18, 33C45, 45P05; Secondary 42A65
- DOI: https://doi.org/10.1090/S0002-9939-2014-12362-8
- MathSciNet review: 3314055