Gamma functions of argument can be expressed in terms of gamma
functions of smaller arguments. From the definition of the beta
function,
|
(1)
|
Now, let ,
then
|
(2)
|
and ,
so
and
Now, use the beta function identity
|
(7)
|
to write the above as
|
(8)
|
Solving for and using then gives
See also
Gamma Function,
Gauss
Multiplication Formula
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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, p. 256, 1972.Arfken, G. Mathematical
Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 561-562,
1985.Erdélyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi,
F. G. Higher
Transcendental Functions, Vol. 1. New York: Krieger, p. 5, 1981.Morse,
P. M. and Feshbach, H. Methods
of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 424-425,
1953.Referenced on Wolfram|Alpha
Legendre Duplication Formula
Cite this as:
Weisstein, Eric W. "Legendre Duplication Formula."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LegendreDuplicationFormula.html
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