Gegenbauer Polynomial

The Gegenbauer polynomials C_n^((lambda))(x) are solutions to the Gegenbauer differential equation for integer n. They are generalizations of the associated Legendre polynomials to (2lambda+2)-D space, and are proportional to (or, depending on the normalization, equal to) the ultraspherical polynomials P_n^((lambda))(x).

Following Szegö, in this work, Gegenbauer polynomials are given in terms of the Jacobi polynomials P_n^((alpha,beta))(x) with alpha=beta=lambda-1/2 by


(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


The first few Gegenbauer polynomials are


In terms of the hypergeometric functions,

C_n^((lambda))(x)=(n+2lambda-1; n)_2F_1(-n,n+2lambda;lambda+1/2;1/2(1-x))
=2^n(n+lambda-1; n)(x-1)^n_2F_1(-n,-n-lambda+1/2;-2n-2lambda+1;2/(1-x))
=(n+2lambda+1; n)((x+1)/2)^n_2F_1(-n,-n-lambda+1/2;lambda+1/2;(x-1)/(x+1)).

They are normalized by


for lambda>-1/2.

Derivative identities include


(Szegö 1975, pp. 80-83).

A recurrence relation is


for n=2, 3, ....

Special double-nu formulas also exist

C_(2nu)^((lambda))(x)=(2nu+2lambda-1; 2nu)_2F_1(-nu,nu+lambda;lambda+1/2;1-x^2)
=(-1)^nu(nu+lambda-1; nu)_2F_1(-nu,nu+lambda;1/2;x^2)
C_(2nu+1)^((lambda))(x)=(2nu+2lambda; 2nu+1)x_2F_1(-nu,nu+lambda+1;lambda+1/2;1-x^2)
=(-1)^nu2lambda(nu+lambda; nu)x_2F_1(-nu,nu+lambda+1;3/2;x^2).

Koschmieder (1920) gives representations in terms of elliptic functions for lambda=-3/4 and lambda=-2/3.

See also

Birthday Problem, Chebyshev Polynomial of the First Kind, Chebyshev Polynomial of the Second Kind, Elliptic Function, Gegenbauer Differential Equation, Hypergeometric Function, Jacobi Polynomial, Legendre Polynomial

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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 q-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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Gegenbauer Polynomial

Cite this as:

Weisstein, Eric W. "Gegenbauer Polynomial." From MathWorld--A Wolfram Web Resource.

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