Explicit Formulas for the Associated Jacobi Polynomials and Some Applications

Wimp, Jet

Canadian journal of mathematics, Tome 39 (1987) no. 4, pp. 983-1000

Résumé

In this paper we determine closed-form expressions for the associated Jacobi polynomials, i.e., the polynomials satisfying the recurrence relation for Jacobi polynomials with n replaced by n + c, for arbitrary real c ≧ 0. One expression allows us to give in closed form the [n — 1/n] Padé approximant for what is essentially Gauss' continued fraction, thus completing the theory of explicit representations of main diagonal and off-diagonal Padé approximants to the ratio of two Gaussian hypergeometric functions and their confluent forms, an effort begun in [2] and [19]. (We actually give only the [n — 1/n] Padé element, although other cases are easily constructed, see [19] for details.)We also determine the weight function for the polynomials in certain cases where there are no discrete point masses. Concerning a weight function for these polynomials, so many writers have obtained so many partial results that our formula should be considered an epitome rather than a real discovery, see the discussion in Section 3.

Détail

DOI : 10.4153/CJM-1987-050-4

Bibliographie

[1] 1. Askey, R. and Ismail, M., Recurrence relations, continued fractions and orthogonal polynomials, Memoirs Amer. Math. Soc. 300 (Providence, RI, 1984). Google Scholar

[2] 2. Askey, R. and Wimp, J., Associated Laguerre polynomials, Proc. Roy. Soc. Edinburgh 96 (1984), 15–37. Google Scholar

[3] 3. Bailey, W. N., Generalized hyper geometric series (Cambridge University Press, Cambridge, 1935). Google Scholar

[4] 4. Barrucand, P. and Dickinson, D., On the associated Legendre polynomials in Orthogonal expansions and their continuous analogs (Southern Illinois University Press, Carbondale, IL, 1967). Google Scholar

[5] 5. Fo, J. Bellandi and de Oliveira, E. C., On the product of two Jacobi functions of different kinds with different arguments, J. Phys. 15 (1982). Google Scholar

[6] 6. Bustoz, J. and Ismail, M., The associated ultraspherical polynomials and their q-analogues, Can. J. Math. 34 (1982), 718–736. Google Scholar

[7] 7. Cohen, M. E., On J acobi functions and multiplication theorems for intergrals of Bessel functions, J. Math. Anal. Appl. 57 (1977), 469–475. Google Scholar

[8] 8. Erdélyi, A. et al, Higher transcendental functions, 3v. (McGraw-Hill, NY, 1953). Google Scholar

[9] 9. Flensted-Jensen, M. and Koornwinder, T., The convolution structure for Jacobi function expansions, Ark. Mat. 11 (1975), 245–262. Google Scholar

[10] 10. Luke, Y. L., Mathematical functions and their approximations (Acad. Press, NY, 1975). Google Scholar

[11] 11. Luke, Y. L., The special functions and their approximation, 2v. (Acad. Press, NY. 1969). Google Scholar

[12] 12. Nevai, P., A new class of orthogonal polynomials, Proc. Amer. Math. Soc. 91 (1984), 409–415. Google Scholar

[13] 13. Pollaczek, F., Sur une famille de polynômes orthogonaux à quatre paramètres, C.R. Acad. Sci., Paris 230 (1950), 2254–2256. Google Scholar

[14] 14. Rainville, E. D., Special functions (MacMillan, NY, 1960). Google Scholar

[15] 15. Sherman, J., On the numerators of the convergents of the Stieltjes continued fraction, Trans. Amer. Math. Soc. 35 (1933), 64–87. Google Scholar

[16] 16. Wall, H. S., Analytic theory of continued fractions (Chelsea, NY, 1948). Google Scholar

[17] 17. Watson, G. N., A treatise on the theory of Bessel functions (Cambridge University Press, Cambridge, 1962). Google Scholar

[18] 18. Wimp, J., Computation with recurrence relations (Pitman Press, London, 1984). Google Scholar

[19] 19. Wimp, J., Some explicit Padé approximants for the function Φ′/Φ and a related quadrature formula involving Bessel functions, SIAM J. Math. Anal. 76 (1985), 887–895. Google Scholar