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Perhaps the best-known value of the gamma function at a non-integer argument is =, which can be found by setting = in the reflection or duplication formulas, by using the relation to the beta function given below with = =, or simply by making the substitution = in the integral definition of the gamma function, resulting in a Gaussian integral.
The duplication formula and the multiplication theorem for the gamma function are the prototypical examples. The duplication formula for the gamma function is (+) = ().It is also called the Legendre duplication formula [1] or Legendre relation, in honor of Adrien-Marie Legendre.
Duplication, or doubling, multiplication by 2; Duplication matrix, a linear transformation dealing with half-vectorization; Doubling the cube, a problem in geometry also known as duplication of the cube; A type of multiplication theorem called the Legendre duplication formula or simply "duplication formula"
The difference equation for the G-function, in conjunction with the functional equation for the gamma function, can be used to obtain the following reflection formula for the Barnes G-function (originally proved by Hermann Kinkelin):
The duplication matrix is the unique (+) matrix which, for any ... The explicit formula for calculating the duplication matrix for a ...
An exercise of elementary analytic geometry shows that in all three cases, both the x - and y-coordinates of the newly defined point satisfy a polynomial of degree no higher than a quadratic, with coefficients that are additions, subtractions, multiplications, and divisions involving the coordinates of the previously defined points (and ...
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Legendre's formula can be used to prove Kummer's theorem. As one special case, it can be used to prove that if n is a positive integer then 4 divides ( 2 n n ) {\displaystyle {\binom {2n}{n}}} if and only if n is not a power of 2.