Ultraspherical polynomials

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Gegenbauer polynomials

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
How to Cite This Entry:
Ultraspherical polynomials. P.K. Suetin (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098