Gegenbauer Polynomial
المؤلف:
Abramowitz, M. and Stegun, I. A.
المصدر:
"Orthogonal Polynomials." Ch. 22 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover
الجزء والصفحة:
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4-8-2019
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Gegenbauer Polynomial
The Gegenbauer polynomials 
are solutions to the Gegenbauer differential equation for integer 
. They are generalizations of the associated Legendre polynomials to 
-D space, and are proportional to (or, depending on the normalization, equal to) the ultraspherical polynomials 
.
Following Szegö, in this work, Gegenbauer polynomials are given in terms of the Jacobi polynomials 
with 
by
 |
(1)
|
(Szegö 1975, p. 80), thus making them equivalent to the Gegenbauer polynomials implemented in the Wolfram Language as GegenbauerC[n, lambda, x]. These polynomials are also given by the generating function
 |
(2)
|
The first few Gegenbauer polynomials are
In terms of the hypergeometric functions,
They are normalized by
![int_(-1)^1(1-x^2)^(lambda-1/2)[C_n^((lambda))]^2dx=2^(1-2lambda)pi(Gamma(n+2lambda))/((n+lambda)Gamma^2(lambda)Gamma(n+1))](http://mathworld.wolfram.com/images/equations/GegenbauerPolynomial/NumberedEquation3.gif) |
(10)
|
for 
.
Derivative identities include
(Szegö 1975, pp. 80-83).
A recurrence relation is
 |
(19)
|
for 
, 3, ....
Special double-
formulas also exist
Koschmieder (1920) gives representations in terms of elliptic functions for 
and 
.
REFERENCES:
Abramowitz, M. and Stegun, I. A. (Eds.). "Orthogonal Polynomials." Ch. 22 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 771-802, 1972.
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 643, 1985.
Erdélyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 2. New York: Krieger, p. 175, 1981.
Infeld, L. and Hull, T. E. "The Factorization Method." Rev. Mod. Phys. 23, 21-68, 1951.
Iyanaga, S. and Kawada, Y. (Eds.). "Gegenbauer Polynomials (Gegenbauer Functions)." Appendix A, Table 20.I in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 1477-1478, 1980.
Koekoek, R. and Swarttouw, R. F. "Gegenbauer / Ultraspherical." §1.8.1 in The Askey-Scheme of Hypergeometric Orthogonal Polynomials and its 
-Analogue. Delft, Netherlands: Technische Universiteit Delft, Faculty of Technical Mathematics and Informatics Report 98-17, pp. 40-41, 1998.
Koschmieder, L. "Über besondere Jacobische Polynome." Math. Zeitschrift 8, 123-137, 1920.
Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 547-549 and 600-604, 1953.
Roman, S. "A Particular Delta Series and the Gegenbauer Polynomials." §6.3 in The Umbral Calculus. New York: Academic Press, pp. 166-174, 1984.
Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., 1975.
Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, pp. 122-123, 1997.
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