On one sum of Hankel–Clifford integral transforms of Whittaker functions

J. Choi ; A. I. Nizhnikov ; I. Shilin

Čebyševskij sbornik, Tome 20 (2019) no. 3, pp. 349-360

Résumé

In [11], the authors considered the realization $T$ of $SO(2,2)$-representation in a space of homogeneous functions on $2\times4$-matrices. In this sequel, we aim to compute matrix elements of the identical operator $T(e)$ and representation operator $T(g)$ for an appropriate $g$ with respect to the mixed basis related to two different bases in the $SO(2,2)$-carrier space and evaluate some improper integrals involving a product of Bessel-Clifford and Whittaker functions. The obtained result can be rewritten in terms of Hankel-Clifford integral transforms and their analogue. The first and the second Hankel-Clifford transforms introduced by Hayek and Pérez–Robayna, respectively, play an important role in the theory of fractional order differential operators (see, e.g., [6, 8]). The similar result have been derived recently by the authors for the regular Coulomb function in [12].

Détail

Keywords: matrix elements of representation, Hankel-Clifford integral transform, Macdonald-Clifford integral transform, Whittaker functions, Bessel-Clifford functions.
Mots-clés : group $SO(2,2)$

Bibliographie

[1] Abramowitz M., “Coulomb wave functions expressed in terms of Bessel-Clifford and Bessel functions”, Stud. Appl. Math., 29:1–4 (1950), 303–308 | MR

[2] Clifford W. K., “On Bessel's functions”, Mathematical Papers, Oxford University Press, London, 1882, 346–349

[3] Gilmore R., “Group theory”, Mathematical Tools for Physicists, Wiley-VCH, Weinheim, 2015, 159–210

[4] Erdélyi A., Magnus W., Oberhettinger F., Tricomi F. G., Tables of Intefral Transforms, v. I, McGraw-Hill Book Company, New York–Toronto–London, 1954 | MR

[5] Hayek N., “Sobre la transformación de Hankel”, Actas de la VIII Reunión Anual de Matemáticos Epañoles, 1967, 47–60 | MR

[6] Kiryakova V., Hernanden Suarez V., “Bessel–Clifford third order differential operator and corresponding Laplace type integral transform”, Dissertationes Mathematicae, 340, 1995, 143–161 | MR | Zbl

[7] Méndez Pérez J. M. R., Socas Robayna M. M., “A pair of generalized Hankel-Clifford transformations and their applications”, J. Math. Anal. Appl., 154:2 (1991), 543–557 | MR | Zbl

[8] Paneva-Konovska J., “Bessel type functions: Relations with integrals and derivatives of arbitrary orders”, AIP Conference Proceedings, 2048 (2018), 050015 | DOI

[9] Prudnikov A. P., Brychkov Yu. A., Marichev O. I., Integrals and Series, v. 1, Elementary Functions, OPA (Overseas Publishers Association), Amsterdam, 1986 | MR

[10] Shilin I. A., Choi J., “Integral and series representations of special functions related to the group $SO(2,2)$”, Ramanujan J., 44:1 (2017), 133–153 | MR | Zbl

[11] Shilin I. A., Choi J., “On matrix elements of the $SO(2,2)$-representation in a space of functions on $2\times4$-matrices”, Integral Transforms Spec. Funct., 29:10 (2018), 761–770 | MR | Zbl

[12] Shilin I. A., Choi J., Some integrals involving Coulomb functions related to three-dimensional proper Lorentz group, Submitted

[13] Temme N. M., Special Functions: An Introduction to the Classical Functions of Mathematical Physics, John Wiley and Sons, New York, 1996 | MR

[14] Vilenkin N. Ya., Pavlyuk A. P., “Representations of some semisimple Lie groups and special functions of the matrix argument”, Group Theoretical Methods in Physics, v. 1, Harwood Academic Publishers, Chur–London–Paris–New York, 1985 | MR

[15] Wang Z. X., Guo D. R., Special functions, World Scientific, Singapore–New Jersey–London–Hong Kong, 1989 | Zbl