Gegenbauer Polynomial

The Gegenbauer polynomials are solutions to the Gegenbauer differential equation for nonnegative 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). For , this normalization agrees with the Gegenbauer polynomials implemented in the Wolfram Language as GegenbauerC[n, lambda, x]. These polynomials are also given by the generating function

(2)

For , substituting into the generating function gives . The nontrivial renormalized limit is

(3)

where is a Chebyshev polynomial of the first kind. Some sources use for this renormalized polynomial (Trott 2006). It is implemented in the Wolfram Language as GegenbauerC[n, x]. The generating function for is

(4)

The case is excluded from this renormalization because the generating function convention has .

The special cases with nonzero parameter include

			(5)
			(6)

where is a Legendre polynomial and is a Chebyshev polynomial of the second kind. For fixed nonnegative integer , a scaled limit gives the Hermite polynomial,

(7)

The first few Gegenbauer polynomials are

			(8)
			(9)
			(10)
			(11)

In terms of the hypergeometric functions,

			(12)
			(13)
			(14)

They are normalized by

(15)

for .

For and with , has real simple roots in . The roots of and interlace (NIST DLMF, §18.2(vi) and 18.16(iii)).

Derivative identities include

			(16)
			(17)
			(18)
			(19)
			(20)
			(21)
			(22)
			(23)

(Szegö 1975, pp. 80-83).

A recurrence relation is

(24)

for , 3, ....

Special double- formulas also exist

			(25)
			(26)
			(27)
			(28)

Koschmieder (1920) gives representations in terms of elliptic functions for and .