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Gegenbauer Differential Equation

The Gegenbauer differential equation is the second-order ordinary differential equation

(1-x^2)y^('')-2(mu+1)xy^'+(nu-mu)(nu+mu+1)y=0
(1)

sometimes called the hyperspherical differential equation (Iyanaga and Kawada 1980, p. 1480; Zwillinger 1997, p. 123). The solution to this equation is

y=(x^2-1)^(-mu/2)[C_1P_nu^mu(x)+C_2Q_nu^mu(x)],
(2)

where P_nu^mu(x) is an associated Legendre function of the first kind and Q_nu^mu(x) is an associated Legendre function of the second kind.

A number of other forms of this equation are sometimes also known as the ultraspherical or Gegenbauer differential equation, including

(1-x^2)y^('')-(2mu+1)xy^'+nu(nu+2mu)y=0.
(3)

The general solutions to this equation are

y=(x^2-1)^((1-2mu)/4)[C_1P_(-1/2+mu+nu)^(1/2-mu)(x)+C_2Q_(-1/2+mu+nu)^(1/2-mu)(x)].
(4)

When nu=n is a nonnegative integer, the Gegenbauer polynomial C_n^((mu))(x) gives a solution provided it is not identically zero. These polynomials are also known as ultraspherical polynomials.

The form

(1-x^2)y^('')-(2m+3)xy^'+lambday=0
(5)

is also given by Infeld and Hull (1951, pp. 21-68) and Zwillinger (1997, p. 122). It has the solution

y=(x^2-1)^(-(2m+1)/4)[C_1P_(-1/2+sqrt((1+m)^2+lambda))^(1/2+m)(x)+C_2Q_(-1/2+sqrt((1+m)^2+lambda))^(1/2+m)(x)].
(6)

See also

Gegenbauer Polynomial

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References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, 1972.Infeld, L. and Hull, T. E. "The Factorization Method." Rev. Mod. Phys. 23, 21-68, 1951.Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, 1980.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 547-549, 1953.Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 127, 1997.

Referenced on Wolfram|Alpha

Gegenbauer Differential Equation

Cite this as:

Weisstein, Eric W. "Gegenbauer Differential Equation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GegenbauerDifferentialEquation.html

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