Gegenbauer Differential Equation

The second-order ordinary differential equation


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


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


The general solutions to this equation are


If -1/2+mu+nu is an integer, then one of the solutions is known as a Gegenbauer polynomials C_n^((lambda))(x), also known as ultraspherical polynomials.

The form


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


See also

Gegenbauer Polynomial

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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 Web Resource.

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