On Buschman–Erdelyi and Mehler–Fock transforms related to the group 𝑆𝑂0(3, 1)
https://doi.org/10.22405/2226-8383-2023-24-1-228-236
Abstract
By using a functional defined on a pair of the assorted represention spaces of the connected subgroup of the proper Lorentz group, a formula for the Buschman–Erdelyi transform of the Legendre function (up to a factor) is derived. Also a formula for the Mehler–Fock transform of the Legendre function of an inverse argument is obtained. Moreover, a generalization of one known formula for the Mehler–Fock transform is derived.
About the Author
Ilya Anatol’evich Shilin
National Research University “MPEI”;
Moscow Pedagogical State University
Russian Federation
Russian Federation
References
1. Gradshteyn, I. S. & Ryzhik, I. M., 2007, Tables of integrals, series, and products, Academic
2. Press, Amsterdam.
3. Kratzer, A. & Franz W., 1960, Transzendente Functionen, Academische Verlafsgesellschapt,
4. Leipzig.
5. Pridnikov, A. P., Brychkov, Yu. A.& Marichev, O. I., 1986, Integrals and Series. Vol. 3: More
6. Special Functions, Gordon & Breach Science Publishers, Amsterdam.
7. Pridnikov, A. P., Brychkov, Yu. A.& Marichev, O. I., 1986, Integrals and Series. Vol. 2: Special
8. Functions, Gordon & Breach Science Publishers, Amsterdam.
9. Sitnik, S. M. 2015, “Operators of Buschman–Erdelyi transforms, their classification, basic
10. properties, and applications”, Sci. Notes Belgorod State Univ., Series “Mathematics. Physics”,
11. vol. 39, no. 11, pp. 60–76. (In Russian.)
12. Fock, V. A., 1943, “On the representation of an arbitrary function by an integral involving
13. Legendre’s function with a complex index”, Docklady Math., vol. 39, no. 7, pp. 279–283.
14. Shilin, I. A. & Choi J., 2022, “Method of continual addition theorems and integral relations
15. between the Coulomb functions and the Appell function 𝐹1”, Comp. Math. & Mathem. Phys.,
16. vol. 62, no. 9, pp. 1486–1495.
17. Buschman, R. G., 1962, “An inversion integral for a Legendre transformation”, American Math.
18. Monthly, vol. 69, no. 4, pp. 288–289.
19. Copson, E. T., 1957, “On a singular boundary value problem for an equation of hyperbolic
20. type”, Rational Mech. Analysis, vol. 1, pp. 349–356.
21. Erdelyi, A. 1964, “An integral equation involving Legendre functions”, J. SIAM, vol. 12, no. 1,
22. pp.. 15–30.
23. Mehler, F. G., 1881, “Ueber eine mit den Kugel- und cylinderfunctionen verwandte function
24. und ihre anwendung in der theorie elektricitatatsvertheilung”, Math. Annalen, vol 18, no. 2,
25. pp. 161–194.
26. Rosenthal, P., 1974, “On an inversion theorem for the general Mehler-Fock transform pair”,
27. Pacific J. Math., vol. 52, no. 2, pp. 539–545.
28. Shilin, I. A. & Choi J., 2023, “On some relations between hyper Bessel–Clifford, Macdonald and
29. Meijer functions and hyper Hankel–Clifford integrsl transforms”. Int. Transforms. Spec. Func.
30. doi.org/10.1080/10652469.2023.2191320.
31. Shilin, I. A. & Choi J., 2023, “Maximal subalgebras in so(2, 1), addition theorems and Bessel–
32. Clifford functions”, J. Analysis„ Vol. 31. № 2. pp. 719-732.
33. Sneddon, I. N., 1972, The Use of Integral Transforms, McGraw-Hill, New York.
34. Yakubovich, S. B., 1996, Index transforms, World Scientific, Singapore.
Review
For citations:
Shilin I.A. On Buschman–Erdelyi and Mehler–Fock transforms related to the group 𝑆𝑂0(3, 1). Chebyshevskii Sbornik. 2023;24(1):228-236. (In Russ.) https://doi.org/10.22405/2226-8383-2023-24-1-228-236
Views:
500
Email this article 