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Legendre Duplication Formula

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Gamma functions of argument 2z can be expressed in terms of gamma functions of smaller arguments. From the definition of the beta function,

B(m,n)=(Gamma(m)Gamma(n))/(Gamma(m+n))=int_0^1u^(m-1)(1-u)^(n-1)du.
(1)

Now, let m=n=z, then

(Gamma(z)Gamma(z))/(Gamma(2z))=int_0^1u^(z-1)(1-u)^(z-1)du
(2)

and u=(1+x)/2, so du=dx/2 and

(Gamma(z)Gamma(z))/(Gamma(2z))=int_(-1)^1((1+x)/2)^(z-1)(1-(1+x)/2)^(z-1)(1/2dx)
(3)
=1/2int_(-1)^1((1+x)/2)^(z-1)((1-x)/2)^(z-1)dx
(4)
=1/(2^(1+2(z-1)))int_(-1)^1(1-x^2)^(z-1)dx
(5)
=2^(1-2z)[2int_0^1(1-x^2)^(z-1)dx].
(6)

Now, use the beta function identity

B(m,n)=2int_0^1x^(2m-1)(1-x^2)^(n-1)dx
(7)

to write the above as

(Gamma(z)Gamma(z))/(Gamma(2z))=2^(1-2z)B(1/2,z)=2^(1-2z)(Gamma(1/2)Gamma(z))/(Gamma(z+1/2)).
(8)

Solving for Gamma(2z) and using Gamma(1/2)=sqrt(pi) then gives

Gamma(2z)=(2pi)^(-1/2)2^(2z-1/2)Gamma(z)Gamma(z+1/2)
(9)
=(2^(2z-1)Gamma(z)Gamma(z+1/2))/(sqrt(pi)).
(10)

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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