AM  Vol.6 No.12 , November 2015
Generating Functions for Products of Special Laguerre 2D and Hermite 2D Polynomials
Author(s) Alfred Wünsche
ABSTRACT
The bilinear generating function for products of two Laguerre 2D polynomials with different arguments is calculated. It corresponds to the formula of Mehler for the generating function of products of two Hermite polynomials. Furthermore, the generating function for mixed products of Laguerre 2D and Hermite 2D polynomials and for products of two Hermite 2D polynomials is calculated. A set of infinite sums over products of two Laguerre 2D polynomials as intermediate step to the generating function for products of Laguerre 2D polynomials is evaluated but these sums possess also proper importance for calculations with Laguerre polynomials. With the technique of operator disentanglement some operator identities are derived in an appendix. They allow calculating convolutions of Gaussian functions combined with polynomials in one- and two-dimensional case and are applied to evaluate the discussed generating functions.

Cite this paper
Wünsche, A. (2015) Generating Functions for Products of Special Laguerre 2D and Hermite 2D Polynomials. Applied Mathematics, 6, 2142-2168. doi: 10.4236/am.2015.612188.
References
[1]   Erdélyi, A. (1953) Higher Transcendental Functions. Volume 2, Bateman Project, McGraw-Hill, New York.

[2]   Rainville, E.D. (1960) Special Functions. Chelsea Publishing Company, New York.

[3]   Magnus, W., Oberhettinger, F. and Soni, R.P. (1966) Formulas and Theorems for the Special Functions of Mathematical Physics. Springer-Verlag, Berlin.

[4]   Bell, W.W. (1968) Special Functions for Scientists and Engineers. D. Van Nostrand Company, London.

[5]   Suetin, P.K. (1988) Orthogonal Polynomials in Two Variables. Nauka, Moskva. (In Russian)

[6]   Dunkl, C.F. and Xu, Y. (2014) Orthogonal Polynomials of Several Variables. 2nd Edition, Cambridge University Press, Cambridge.

[7]   Wünsche, A. (1988) Some Remarks about the Glauber-Sudarshan Quasiprobability. Acta Physica Slovaca, 48, 385- 408.

[8]   Wünsche, A. (1998) Laguerre 2D—Functions and Their Application in Quantum Optics. Journal of Physics A: Mathematical and General, 31, 8267-8287.
http://dx.doi.org/10.1088/0305-4470/31/40/017

[9]   Wünsche, A. (1999) Transformation of Laguerre 2D Polynomials with Application to Quasiprobabilities. Journal of Physics A: Mathematical and General, 32, 3179-3199.
http://dx.doi.org/10.1088/0305-4470/32/17/309

[10]   Wünsche, A. (2000) General Hermite and Laguerre Two-Dimensional Polynomials. Journal of Physics A: Mathematical and General, 33, 1603-1629; Corrigendum Journal of Physics A: Mathematical and General, 33, 3531.
http://dx.doi.org/10.1088/0305-4470/33/17/501

[11]   Wünsche, A. (2001) Hermite and Laguerre 2D Polynomials. Journal of Computational and Applied Mathematics, 133, 665-678.
http://dx.doi.org/10.1016/s0377-0427(00)00681-6

[12]   Wünsche, A. (2001) Hermite and Laguerre 2D Polynomials. In: Dattoli, G., Srivastava, H.M. and Cesarano, C., Eds., Advanced Special Functions and Integration Methods, Aracne Editrice, Roma, 157-198.

[13]   Abramochkin, E. and Volostnikov, V. (1991) Beam Transformations and Nontransformed Beams. Optics Communications, 83, 123-135.
http://dx.doi.org/10.1016/0030-4018(91)90534-K

[14]   Abramochkin, E.G. and Volostnikov, V.G. (2004) Generalized Gaussian Beams. Journal of Optics A: Pure and Applied Optics, 6, 157-161.
http://dx.doi.org/10.1088/1464-4258/6/5/001

[15]   Fan, H.-Y. and Ye, X. (1993) Hermite Polynomial States in Two-Mode Fock Space. Physics Letters A, 175, 387-390.
http://dx.doi.org/10.1016/0375-9601(93)90987-B

[16]   Fan, H.-Y., Zou, H. and Fan, Y. (2001) A Complete and Orthonormal Representation in Two-Mode Fock Space Gained by Two-Variable Hermite Polynomials. International Journal of Modern Physics A, 16, 369-375.
http://dx.doi.org/10.1142/S0217751X01002294

[17]   Dattoli, G., Lorenzutta, S., Mancho, A.M. and Torre, A. (1999) Generalized Polynomials and Associated Operational Identities. Journal of Computational and Applied Mathematics, 108, 209-218.
http://dx.doi.org/10.1016/S0377-0427(99)00111-9

[18]   Bastiaans, M.J. and Alieva, T. (2005) Bi-Orthonormal Sets of Gaussian-Type Modes. Journal of Physics A: Mathematical and General, 38, 9931-9939.
http://dx.doi.org/10.1088/0305-4470/38/46/003

[19]   Bastiaans, M.J. and Alieva, T. (2005) Propagation Law for the Generating Function of Hermite-Gaussian-Type Modes in First-Order Optical Systems. Optics Express, 13, 1107-1112.
http://dx.doi.org/10.1364/OPEX.13.001107

[20]   Bastiaans, M.J. and Alieva, T. (2005) Generating Function for Hermite-Gaussian Modes Propagating through First-Order Optical Systems. Journal of Physics A: Mathematical and General, 38, L73-L78.
http://dx.doi.org/10.1088/0305-4470/38/6/L01

[21]   Shahwan, M.J.S. (2012) Incomplete 2D Hermite Polynomials and Their Generating Relations. Applied Mathematics & Information Sciences, 6, 109-112.

[22]   Ismail, M.E.H. and Zeng, J. (2015) Two-Vaiable Extensions of the Laguerre and Disc Polynomials. Journal of Mathematical Analysis and Applications, 424, 289-303.
http://dx.doi.org/10.1016/j.jmaa.2014.11.015

[23]   Ismail, M.E.H. and Zeng, J. (2015) A Combinatorial Approach to the 2D-Hermite and 2D-Laguerre Polynomials. Advances in Applied Mathematics, 64, 70-88.
http://dx.doi.org/10.1016/j.aam.2014.12.002

[24]   Twareque Ali, S., Bagarello, F. and Gazeau, J.P. (2015) D-Pseudo-Bosons, Complex Hermite Polynomials and Integral Quantization. Sigma, 11, 078 (23 p.).

[25]   Wünsche, A. (1999) Ordered Operator Expansions and Reconstruction from Ordered Moments. Quantum and Semiclassical Optics, 1, 264-288.
http://dx.doi.org/10.1088/1464-4266/1/2/010

[26]   Srivastava, H.M. and Manocha, H.L. (1984) A Treatise on Generating Functions. Ellis Horwood, John Wiley, New York.

[27]   Fan, H.-Y., Liu, Z.-W. and Ruan, T.-N. (1984) Does the Creation Operator a+ Possess Eigenvectors. Communications in Theoretical Physics (Beijing), 3, 175-188.
http://dx.doi.org/10.1088/0253-6102/3/2/175

[28]   Nieto, M.M. and Truax, D.R. (1995) Arbitrary-Order Hermite Generating Functions for Obtaining Arbitrary-Order Coherent and Squeezed States. Physics Letters A, 208, 8-16.
http://dx.doi.org/10.1016/0375-9601(95)00761-Q

[29]   Wünsche, A. (1996) The Coherent States as Basis States on Areas Contours and Paths in the Phase Space. Acta Physica Slovaca, 46, 505-516.

[30]   Fernández, F.M. and Castro, E.A. (1996) Algebraic Methods in Quantum Chemistry and Physics. CRC Press, Boca Raton. (Cited According to [31])

[31]   Fernández, F.M. (1998) Generating Functions for Hermite Polynomials of Arbitrary Order. Physics Letters A, 237, 189-191.
http://dx.doi.org/10.1016/S0375-9601(97)00853-0

[32]   Dattoli, G., Torre, A. and Carpanese, M. (1998) Operational Rules and Arbitrary Order Hermite Generating Functions. Journal of Mathematical Analysis and Applications, 227, 98-111.
http://dx.doi.org/10.1006/jmaa.1998.6080

[33]   Dattoli, G., Torre, A. and Lorenzutta, S. (1999) Operational Identities and Properties of Ordinary and Generalized Special Functions. Journal of Mathematical Analysis and Applications, 236, 399-414.
http://dx.doi.org/10.1006/jmaa.1999.6447

[34]   Wünsche, A. (2005) Generalized Zernike or Disc Polynomials. Journal of Computational and Applied Mathematics, 174, 135-163.
http://dx.doi.org/10.1016/j.cam.2004.04.004

[35]   Roman, S. (1984) The Umbral Calculus. Academic Press, New York.

[36]   Prudnikov, A.P., Brychkov, Y.A. and Marichev, O.I. (1991) Integrals and Series: Special Functions. Taylor and Francis, London. (Russian Original: Nauka, Moskva 1983)

[37]   Wünsche, A. (2001) Relation of Quasiprobabilities to Bargmann Representation of States. Journal of Optics B: Quantum and Semiclassical Optics, 3, 6-15.
http://dx.doi.org/10.1088/1464-4266/3/1/302

[38]   Wünsche, A. (2004) Quantization of Gauss-Hermite and Gauss-Laguerre Beams in Free Space. Journal of Optics B: Quantum and Semiclassical Optics, 6, S47-S59.

[39]   Fan, H.-Y. and Wünsche, A. (2005) Wavefunctions of Two-Mode States in Entangled-State Representation. Journal of Optics B: Quantum and Semiclassical Optics, 7, R88-R102.
http://dx.doi.org/10.1088/1464-4266/7/6/R02

[40]   Perelomov, A.M. (1977) Generalized Coherent States and Some of Their Applications. Soviet Physics Uspekhi, 20, 703-720.
http://dx.doi.org/10.1070/PU1977v020n09ABEH005459

[41]   Perelomov, A.M. (1986) Generalized Coherent States and Their Application. Springer-Verlag, Berlin.

[42]   Vourdas, A. and Wünsche, A. (1998) Resolutions of the Identity in Terms of Line Integrals of SU(1, 1) Coherent States. Journal of Physics A: Mathematical and General, 31, 9341-9352.
http://dx.doi.org/10.1088/0305-4470/31/46/024

[43]   Wünsche, A. (2003) Squeezed States. In: Dodonov, V.V. and Man’ko, V.I., Eds., Theory of Nonclassical States of Light, Taylor and Francis, London and New York, 95-152.

 
 
Top