Orthogonal polynomials on the interval with the weight function ; a particular case of the Jacobi polynomials for (); the Legendre polynomials are a particular case of the ultraspherical polynomials: .
For ultraspherical polynomials one has the standardization
and the representation
The ultraspherical polynomials are the coefficients of the power series expansion of the generating function
The ultraspherical polynomial satisfies the differential equation
More commonly used are the formulas
For references see Orthogonal polynomials.
See Spherical harmonics for a group-theoretic interpretation. Ultraspherical polynomials are also connected with Jacobi polynomials by the quadratic transformations
See [a1] for -ultraspherical polynomials.
|[a1]||R.A. Askey, M.E.H. Ismail, "A generalization of ultraspherical polynomials" P. Erdös (ed.) , Studies in Pure Mathematics to the Memory of Paul Turán , Birkhäuser (1983) pp. 55–78|
Ultraspherical polynomials. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ultraspherical_polynomials&oldid=14267