Geodesic
A Uniform Asymptotic Expansion of the Jacobi Polynomials with Error Bounds
Canadian journal of mathematics, Tome 37 (1985) no. 5, pp. 979-1007

Voir la notice de l'article provenant de la source Cambridge University Press

In a recent investigation of the asymptotic behavior of the Lebesgue constants for Jacobi polynomials, we encountered the problem of obtaining an asymptotic expansion for the Jacobi polynomials , as n → ∞, which is uniformly valid for θ in . The leading term of such an expansion is provided by the following well-known formula of “Hilb's type” [13, p. 197]: (1.1) where α > – 1, β real and ; c and are fixed positive numbers. Note that the remainder in (1.1) is always θ 1/2 O(n –3/2). 
Frenzen, C. L.; Wong, R. A Uniform Asymptotic Expansion of the Jacobi Polynomials with Error Bounds. Canadian journal of mathematics, Tome 37 (1985) no. 5, pp. 979-1007. doi: 10.4153/CJM-1985-053-5
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[1] 1. Abramowitz, M. and Stegun, I., Handbook of mathematical functions, N.B.S. Applied Math. Ser. 55 (Washington, D.C., 1964). Google Scholar

[2] 2. Askey, R., Orthogonal polynomials and special functions, CBMS 21 (SIAM, Philadelphia, 1975). Google Scholar | DOI

[3] 3. Askey, R., The collected papers of Gabor Szegö, Vol. 2 (Birkhauser, Cambridge, MA, 1982). Google Scholar | DOI

[4] 4. Elliott, D., Uniform asymptotic expansions of the Jacobi polynomials and an associated function, Math. Comp. 25 (1971), 309–314. Google Scholar

[5] 5. Gasper, G., Formulas of the Dirichlet-Mehler type, fractional calculus and its applications 457 (Springer-Verlag Lecture Notes in Mathematics), 207–215. Google Scholar | DOI

[6] 6. Gatteschi, L., Limitazione degli errori nelle formule asintotiche per le funzioni speciali, Rend. Semin. Mat. Univ. e Pol., Torino 16 (1956-1957), 83–94. Google Scholar

[7] 7. Gatteschi, L., Una nuova rappresentazione asintotica dei polinomi di Jacobi, Rend. Semin. Mat. Univ. e Pol., Torino 27 (1967-68), 165–184. Google Scholar

[8] 8. Gradshteyn, I. S. and Ryzhik, I. M., Tables of integrals, series and products (Academic Press, New York, 1965). Google Scholar

[9] 9. Hahn, E., Asymptotik bei Jacobi-Polynomen und Jacobi-Funktionen, Math. Zeit. 171 (1980), 201–226. Google Scholar

[10] 10. Olver, F. W. J., Asymptotics and special functions (Academic Press, New York, 1974). Google Scholar

[11] 11. Szegö, G., Über einige asymptotische Entwicklungen der Legendreschen Funktionen, Proc. Lond. Math. Soc. (2) 36 (1932), 427–450. Google Scholar

[12] 12. Szegö, G., Asymptotische Entwicklungen der Jacobischen Polynome, Schr. der König. Gelehr. Gesell. Naturwiss. Kl. 70 (1933), 33–112. Google Scholar

[13] 13. Szegö, G., Orthogonal polynomials, Colloquium Publications (American Mathematical Society, Providence, R.I., 1967). Google Scholar

[14] 14. Watson, G. N., A treatise on the theory of Bessel functions (Cambridge University Press, Cambridge, 1944). Google Scholar

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