The ultraspherical polynomials are solutions
to the Ultraspherical Differential Equation for
Integer and . They are generalizations of Legendre Polynomials to
D space and are proportional to (or, depending on the normalization, equal to) the Gegenbauer
Polynomials
, denoted in Mathematica
(Wolfram Research, Champaign,
IL) GegenbauerC[n,lambda,x]. The ultraspherical polynomials are also Jacobi Polynomials with
. They are given by the Generating Function
(1) 

(2) 
(3)  
(4)  
(5)  
(6) 
In terms of the Hypergeometric Functions,
(7)  
(8)  
(9) 
(10) 
Derivative identities include
(11)  
(12)  
(13)  
(14)  
(15)  
(16)  
(17)  
(18) 
A Recurrence Relation is
(19) 
Special double Formulas also exist
(20)  
(21)  
(22)  
(23) 
Special values are given in the following table.
Special Polynomial  
Legendre Polynomial  
1  Chebyshev Polynomial of the Second Kind 
Koschmieder (1920) gives representations in terms of Elliptic Functions for and .
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. 771802, 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. 14771478, 1980.
Koschmieder, L. ``Über besondere Jacobische Polynome.'' Math. Zeitschrift 8, 123137, 1920.
Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGrawHill, pp. 547549 and 600604, 1953.
Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., 1975.
© 19969 Eric W. Weisstein