The second-order ordinary
differential equation

(1)

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

(2)

where
is an associated Legendre function
of the first kind and 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

(3)

The general solutions to this equation are

(4)

If
is an integer, then one of the solutions is known as a Gegenbauer
polynomials , also known as ultraspherical polynomials.

The form

(5)

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

(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 Web Resource. https://mathworld.wolfram.com/GegenbauerDifferentialEquation.html

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