Pollaczek polynomials and hypergeometric representation

The Ramanujan Journal - Tập 30 - Trang 399-402 - 2012
Jamel Benameur1, Mongi Blel1
1Department of Mathematics, College of Science, King Saud University, Riyadh, Kingdom of Saudi Arabia

Tóm tắt

This paper gives a solution, without the use of the three-term recurrence relation, of the problem posed in Ismail (Classical and Quantum Orthogonal Polynomials in One Variable, Cambridge University Press, Cambridge, 2005) (Problem 24.8.2, p. 658): that the hypergeometric representation of the general Pollaczek polynomials is a polynomial in cos(θ) of degree n. Chu solved in (Ramanujan J. 13(1–3): 221–225, 2007) the problem in a particular case. We use elementary properties of functions of complex variables and Pfaff’s transformation on hypergeometric 2 F 1-series.

Tài liệu tham khảo

Bailey, W.N.: Generalized Hypergeometric Series. Cambridge University Press, Cambridge (1935) Chihara, T.S.: An Introduction to Orthogonal Polynomials. Gordon and Breach, New York (1978) Chu, W.: Pollaczek polynomials and hypergeometric representation. Ramanujan J. 13(1–3), 221–225 (2007) Ismail, M.E.H.: Classical and Quantum Orthogonal Polynomials in One Variable. Encycl. of Math., vol. 98. Cambridge University Press, Cambridge (2005) Pollaczek, F.: Sur une généralisation des polynômes de Legendre. C. R. Acad. Sci. Paris 228, 1363–1365 (1949) Szegö, G.: On certain special sets of orthogonal polynomials. Proc. Am. Math. Soc. 1, 731–737 (1950)