Gauss Hypergeometric Representations of the Ferrers Function of the Second Kind

Howard S. Cohl   ; Justin Park   ; Hans Volkmer

Symmetry, integrability and geometry: methods and applications, Tome 17 (2021)

Résumé

We derive all eighteen Gauss hypergeometric representations for the Ferrers function of the second kind, each with a different argument. They are obtained from the eighteen hypergeometric representations of the associated Legendre function of the second kind by using a limit representation. For the 18 hypergeometric arguments which correspond to these representations, we give geometrical descriptions of the corresponding convergence regions in the complex plane. In addition, we consider a corresponding single sum Fourier expansion for the Ferrers function of the second kind. In four of the eighteen cases, the determination of the Ferrers function of the second kind requires the evaluation of the hypergeometric function separately above and below the branch cut at $[1,\infty)$. In order to complete these derivations, we use well-known results to derive expressions for the hypergeometric function above and below its branch cut. Finally we give a detailed review of the 1888 paper by Richard Olbricht who was the first to study hypergeometric representations of Legendre functions.

Détail

Keywords: Ferrers functions, associated Legendre functions, Gauss hypergeometric function.

Bibliographie

[1] Andrews G. E., Askey R., Roy R., Special functions, Encyclopedia of Mathematics and its Applications, 71, Cambridge University Press, Cambridge, 1999 | DOI | MR | Zbl

[2] Erdélyi A., Magnus W., Oberhettinger F., Tricomi F. G., Higher transcendental functions, v. I, Robert E. Krieger Publishing Co., Inc., Melbourne, Fla., 1981 | MR

[3] Ferrers N. M., An elementary treatise on spherical harmonics and subjects connected with them, Macmillan, London, 1877 | Zbl

[4] Heine E., Handbuch der Kugelfunctionen, Theorie und Anwendungen, v. 1, Druck und Verlag von G. Reimer, Berlin, 1878

[5] Heine E., Handbuch der Kugelfunctionen, Theorie und Anwendungen, v. 2, Druck und Verlag von G. Reimer, Berlin, 1881

[6] Hobson E. W., The theory of spherical and ellipsoidal harmonics, Chelsea Publishing Company, New York, 1955 | MR

[7] Karp D. B., Prilepkina E. G., “Applications of the Stieltjes and Laplace transform representations of the hypergeometric functions”, Integral Transforms Spec. Funct., 28 (2017), 710–731, arXiv: 1705.02493 | DOI | MR | Zbl

[8] Lebedev N. N., Special functions and their applications, Dover Publications, Inc., New York, 1972 | MR | Zbl

[9] Magnus W., Oberhettinger F., Soni R. P., Formulas and theorems for the special functions of mathematical physics, Die Grundlehren der mathematischen Wissenschaften, 52, Springer-Verlag, New York, 1966 | DOI | MR | Zbl

[10] Olbricht R., “Studien über die Kugel- und Cylinderfunctionen”, Nova Acta d. Kais. Leop.-Carol. Deutschen Akad. d. Naturf., 52 (1888), 1–48

[11] Olver F. W.J., Olde Daalhuis A. B., Lozier D. W., Schneider B. I., Boisvert R. F., Clark C. W., Miller B. R., Saunders B. V., Cohl H. S., McClain M. A. (eds.), NIST Digital Library of Mathematical Functions, Release 1.1.1 of 2021-03-15, , 2021 http://dlmf.nist.gov/

[12] Schäfke F. W., Einführung in die Theorie der speziellen Funktionen der mathematischen Physik, Die Grundlehren der mathematischen Wissenschaften, 118, Springer-Verlag, Berlin – Göttingen – Heidelberg, 1963 | DOI | MR | Zbl

[13] Zhurina M. I., Karmazina L., Tables and formulae for the spherical functions $P^{m}_{-1/2+i\tau} (z)$, Pergamon Press, Oxford – New York – Paris, 1966 | MR