# Gegenbauer polynomials

In mathematics, **Gegenbauer polynomials** or **ultraspherical polynomials** *C*^{(α)}_{n}(*x*) are orthogonal polynomials on the interval [−1,1] with respect to the weight function (1 − *x*^{2})^{α–1/2}. They generalize Legendre polynomials and Chebyshev polynomials, and are special cases of Jacobi polynomials. They are named after Leopold Gegenbauer.

## Characterizations

A variety of characterizations of the Gegenbauer polynomials are available.

- The polynomials can be defined in terms of their generating function (Stein & Weiss 1971, §IV.2):

- The polynomials satisfy the recurrence relation (Suetin 2001):

- Gegenbauer polynomials are particular solutions of the Gegenbauer differential equation (Suetin 2001):

- When
*α*= 1/2, the equation reduces to the Legendre equation, and the Gegenbauer polynomials reduce to the Legendre polynomials. - When
*α*= 1, the equation reduces to the Chebyshev differential equation, and the Gegenbauer polynomials reduce to the Chebyshev polynomials of the second kind.[1]

- They are given as Gaussian hypergeometric series in certain cases where the series is in fact finite:

- (Abramowitz & Stegun p. 561). Here (2α)
_{n}is the rising factorial. Explicitly,

- They are special cases of the Jacobi polynomials (Suetin 2001):

- in which represents the rising factorial of .
- One therefore also has the Rodrigues formula

## Orthogonality and normalization

For a fixed *α*, the polynomials are orthogonal on [−1, 1] with respect to the weighting function (Abramowitz & Stegun p. 774)

To wit, for *n* ≠ *m*,

They are normalized by

## Applications

The Gegenbauer polynomials appear naturally as extensions of Legendre polynomials in the context of potential theory and harmonic analysis. The Newtonian potential in **R**^{n} has the expansion, valid with α = (*n* − 2)/2,

When *n* = 3, this gives the Legendre polynomial expansion of the gravitational potential. Similar expressions are available for the expansion of the Poisson kernel in a ball (Stein & Weiss 1971).

It follows that the quantities are spherical harmonics, when regarded as a function of **x** only. They are, in fact, exactly the zonal spherical harmonics, up to a normalizing constant.

Gegenbauer polynomials also appear in the theory of Positive-definite functions.

The Askey–Gasper inequality reads

## See also

- Rogers polynomials, the
*q*-analogue of Gegenbauer polynomials - Chebyshev polynomials
- Romanovski polynomials

## References

- Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 22".
*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables*. Applied Mathematics Series.**55**(Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 773. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253. - Bayin, S.S. (2006),
*Mathematical Methods in Science and Engineering*, Wiley, Chapter 5. - Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Orthogonal Polynomials", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.),
*NIST Handbook of Mathematical Functions*, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248 - Stein, Elias; Weiss, Guido (1971),
*Introduction to Fourier Analysis on Euclidean Spaces*, Princeton, N.J.: Princeton University Press, ISBN 978-0-691-08078-9. - Suetin, P.K. (2001) [1994], "Ultraspherical polynomials", in Hazewinkel, Michiel (ed.),
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4.

- Specific

- Arfken, Weber, and Harris (2013) "Mathematical Methods for Physicists", 7th edition; ch. 18.4