Ultraspherical Polynomial

The ultraspherical polynomials are solutions $P_n^{(\lambda)}(x)$ to the Ultraspherical Differential Equation for Integer $n$ and $\alpha < 1/2$. They are generalizations of Legendre Polynomials to $(n+2)$-D space and are proportional to (or, depending on the normalization, equal to) the Gegenbauer Polynomials $C_n^{(\lambda)}(x)$, denoted in Mathematica ${}^{\scriptstyle\circledRsymbol}$ (Wolfram Research, Champaign, IL) GegenbauerC[n,lambda,x]. The ultraspherical polynomials are also Jacobi Polynomials with $\alpha=\beta$. They are given by the Generating Function

$${1\over(1-2xt+t^2)^\lambda} = \sum_{n=0}^\infty P_n^{(\lambda)}(x)t^n,$$ (1)

and can be given explicitly by

$P_n^{(\lambda)}(x) = {\Gamma(\lambda+{\textstyle{1\over 2}})\over\Gamma(2\lambd... ...ver\Gamma(n+\lambda+{\textstyle{1\over 2}})}P_n^{(\lambda-1/2,\lambda-1/2)}(x),$
	(2)

where $P_n^{(\lambda-1/2,\lambda-1/2)}$ is a Jacobi Polynomial (Szegö 1975, p. 80). The first few ultraspherical polynomials are

$$P_0^{(\lambda)}(x) = 1$$ (3)

$$P_1^{(\lambda)}(x) = 2\lambda x$$ (4)

$$P_2^{(\lambda)}(x) = -\lambda+2\lambda(1+\lambda)x^2$$ (5)

$$P_3^{(\lambda)}(x) = -2\lambda(1+\lambda)x+{\textstyle{4\over 3}}\lambda(1+\lambda)(2+\lambda)x^3.$$ (6)

In terms of the Hypergeometric Functions,

$$P_n^{(\lambda)}(x) = {n+2\lambda-1\choose n}{}_2F_1(-n,n+2\lambda;\lambda+{\textstyle{1\over 2}}; {\textstyle{1\over 2}}(1-x))$$ (7)

$$= 2^n{n+\lambda-1\choose n}(x-1)^n {}_2F_1\left({-n, -n-\lambda+{\textstyle{1\over 2}}; -2n-2\lambda+1; {2\over 1-x}}\right)$$ (8)

$$= {n+2\lambda+1\choose n}\left({x+1\over 2}\right)^n {}_2F_1\left({... ...{\textstyle{1\over 2}}; \lambda+{\textstyle{1\over 2}}; {x-1\over x+1}}\right).$$ (9)

They are normalized by

$$\int_{-1}^1 (1-x^2)^{\lambda-1/2}[P_n^{(\lambda)}]^2\,dx = 2... ...mma(n+2\lambda)\over (n+\lambda)\Gamma^2(\lambda)\Gamma(n+1)}.$$ (10)

Derivative identities include

${d\over dx} P_n^{(\lambda)}(x)=2\lambda P_{n-1}^{(\lambda+1)}(x)$	(11)
$(1-x^2){d\over dx}[P_n^{(\lambda)}]=[2(n+\lambda)]^{-1}[(n+2\lambda-1)$
$\times(n+2\lambda)P_{n-1}^{(\lambda)}(x) -n(n+1)P_{n+1}^{(\lambda)}(x)]\quad$	(12)
$\quad = -nxP_n^{(\lambda)}(x)+(n+2\lambda-1)P_{n-1}^{(\lambda)}(x)$	(13)
$\quad = (n+2\lambda)xP_n^{(\lambda)}(x)-(n+1)P_{n+1}^{(\lambda)}(x)$	(14)
$nP_n^{(\lambda)}(x)=x{d\over dx}[P_n^{(\lambda)}(x)]-{d\over dx}[P_{n-1}^{(\lambda)}(x)]$	(15)
$(n+2\lambda)P_n^{(\lambda)}(x)={d\over dx}[P_{n+1}^{(\lambda)}(x)]-x{d\over dx}[P_n^{(\lambda)}(x)]$	(16)
${d\over dx}[P_{n+1}^{(\lambda)}(x)-P_{n-1}^{(\lambda)}(x)]=2(n+\lambda)P_n^{(\lambda)}P_n^{(\lambda)}(x)$	(17)
$=2\lambda[P_n^{(\lambda+1)}(x)-P_{n-2}^{(\lambda+1)}(x)]$	(18)

(Szegö 1975, pp. 80-83).

A Recurrence Relation is

$$nP_n^{(\lambda)}(x)=2(n+\lambda-1)xP_{n-1}^{(\lambda)}(x)-(n+2\lambda-2)P_{n-2}^{(\lambda)}(x)$$ (19)

for $n=2$, 3, ....

Special double-$\nu$ Formulas also exist

$$P_{2\nu}^{(\lambda)}(x) = {2\nu+2\lambda-1\choose 2\nu}{}_2F_1(-\nu,\nu+\lambda;\lambda+{\textstyle{1\over 2}};1-x^2)$$ (20)

$$= (-1)^\nu{\nu+\lambda-1\choose\nu}{}_2F_1(-\nu,\nu+\lambda;{\textstyle{1\over 2}};x^2)$$ (21)

$$P_{2\nu+1}^{(\lambda)}(x) = {2\nu+2\lambda\choose 2\nu+1}x{}_2F_1(-\nu,\nu+\lambda+1;\lambda+{\textstyle{1\over 2}};1-x^2)$$ (22)

$$= (-1)^\nu2\lambda{\nu+\lambda\choose\nu}x{}_2F_1(-\nu,\nu+\lambda+1;{\textstyle{3\over 2}}; x^2).$$ (23)

Special values are given in the following table.

$\lambda$	Special Polynomial
${\textstyle{1\over 2}}$	Legendre Polynomial
1	Chebyshev Polynomial of the Second Kind

Koschmieder (1920) gives representations in terms of Elliptic Functions for $\alpha=-3/4$ and $\alpha=-2/3$.

See also Birthday Problem, Chebyshev Polynomial of the Second Kind, Elliptic Function, Hypergeometric Function, Jacobi Polynomial

References

Abramowitz, M. and Stegun, C. 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.

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.

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.

Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., 1975.