×

Integral representations for Jacobi polynomials and some applications. (English) Zbl 0172.08803


Full Text: DOI

References:

[1] Askey, R., Orthogonal expansions with positive coefficients, (Proc. Am. Math. Soc., 16 (1965)), 1191-1194 · Zbl 0136.05103
[2] Askey, R., Jacobi polynomial expansions with positive coefficients and imbeddings of projective spaces, Bull. Am. Math. Soc., 74, 301-304 (1968) · Zbl 0167.35003
[3] Askey, R., Dual equations and classical orthogonal polynomials, J. Math. Anal. Appl., 24, 672-685 (1968) · Zbl 0185.12601
[4] Askey, R.; Fitch, J.; Gasper, G., On a positive trigonometric sum, (Proc. Am. Math. Soc., 19 (1968)), 1507 · Zbl 0174.35704
[5] Bailey, W. N., Generalized Hypergeometric Functions (1935), Cambridge Press: Cambridge Press Cambridge, England · Zbl 0011.02303
[6] Bailey, W. N., Associated hypergeometric series, Quar. J. Math., 8, 115-118 (1937) · JFM 63.0321.02
[7] Bateman, H., The solution of linear differential equations by means of definite integrals, Trans. Camb. Phil. Soc., 21, 171-196 (1909) · JFM 40.0381.04
[8] Bochner, S., Positive zonal functions on spheres, (Proc. Nat. Acad. Sci., 40 (1954)), 1141-1147 · Zbl 0058.29101
[9] Chaudhuri, J., On the operational representation of some hypergeometric polynomials, Rend. Sem. Mate, 38, 27-32 (1967), Padova · Zbl 0148.29902
[10] Erdélyi, A., (Higher Transcendental Functions, vol. I (1953), McGraw Hill: McGraw Hill New York) · Zbl 0051.30303
[11] Erdélyi, A., (Higher Transcendental Functions, vol. II (1953), McGraw Hill: McGraw Hill New York) · Zbl 0052.29502
[12] Erdélyi, A., (Tables of Integral Transforms, vol. II (1954), McGraw Hill: McGraw Hill New York) · Zbl 0055.36401
[13] Fejér, L., Über die Laplacesche Reihe, Math. Ann., 67, 76-109 (1909) · JFM 40.0499.01
[14] Fejér, L., Ultrasphärikus polynomok összegéröl, Mat. Fiz. Lapok., 38, 161-163 (1931) · JFM 57.1433.01
[15] Feldheim, E., Relations entre les polynomes de Jacobi, Laguerre et Hermite, Acta Math., 75, 117-138 (1943) · JFM 68.0152.04
[16] Feldheim, E., Contributions à la théorie des polynomes de Jacobi, Mat. Fiz. Lapok., 48, 453-504 (1941), [Hungarian, French summary] · JFM 67.0238.02
[17] Feldheim, F., Contributi alla teoria della funzioni ipergeometriche di più variabili, Annali della Scuola Norm. Super. di Pisa, Series II, 12, 17-60 (1943) · Zbl 0063.01340
[18] Feldheim, E., On the positivity of certain sums of ultraspherical polynomials, J. d’Anal. Math., 11, 275-284 (1963) · Zbl 0113.28001
[19] Gegenbauer, L., Zur Theorie der Funktionen \(C_n^{v\) · JFM 16.0452.02
[20] Gegenbauer, L., Das Additionstheorem der Functionen \(C_n^{v\) · JFM 25.0830.01
[21] Koshliakov, N. S., On Sonine’s Polynomials, Mess. Math., 55, 152-160 (1926) · JFM 52.0353.03
[22] Kummer, E. E., Über die hypergeometrische Reihe, J. Reine Angew. Math., 15, 127-172 (1836) · ERAM 015.0533cj
[23] Landau, E., Über eine trigonometrische Ungleichung, Math. Zeit., 37, 36 (1933) · JFM 59.0371.01
[24] Lyness, J. N.; Moler, C., Problem 67-6, SIAM Review, 10, 226-227 (1968)
[25] Miller, W., Lie Theory and Special Functions (1968), Academic Press: Academic Press New York · Zbl 0174.10502
[26] Orihara, A., Bessel functions and the Euclidean motion group, Tôhoku Math. J., 13, 66-74 (1961) · Zbl 0118.29401
[27] Seidel, W.; Szász, O., On positive harmonic functions and ultraspherical polynomials, J. London Math. Soc., 26, 36-41 (1951) · Zbl 0042.07602
[28] Slater, L. J., Generalized Hypergeometric Functions (1966), Cambridge Press: Cambridge Press Cambridge, England · Zbl 0135.28101
[29] Szegö, G., Ultrasphaerikus polinomok összegéröl, Mat. Fiz. Lapok., 45, 36-38 (1938) · JFM 64.0353.02
[30] Szegö, G., Orthogonal polynomials, (Am. Math. Soc. Coll. Pub., 23 (1959)) · JFM 65.0278.03
[31] Turán, P., On a trigonometrical sum, Ann. Soc. Polonaise Math., 25, 155-161 (1952) · Zbl 0048.30404
[32] Vilenkin, N. Ja, Some relations for Gegenbauer functions, Uspehi Math. Nauk, 13, No. 3 (81), 167-172 (1958), (N. S.) · Zbl 0081.29301
[33] Vilenkin, N. Ja, Group representations and special functions (1965), Moscow · Zbl 0144.38003
[34] Watson, G. N., Theory of Bessel functions (1944), Cambridge Press: Cambridge Press Cambridge, England · Zbl 0063.08184
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.