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Beta function
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== Other identities and formulas == The integral defining the beta function may be rewritten in a variety of ways, including the following: :<math> \begin{align} \Beta(z_1,z_2) &= 2\int_0^{\pi / 2}(\sin\theta)^{2z_1-1}(\cos\theta)^{2z_2-1}\,d\theta, \\[6pt] &= \int_0^\infty\frac{t^{z_1-1}}{(1+t)^{z_1+z_2}}\,dt, \\[6pt] &= n\int_0^1t^{nz_1-1}(1-t^n)^{z_2-1}\,dt, \\ &= (1-a)^{z_2} \int_0^1 \frac{(1-t)^{z_1-1}t^{z_2-1}}{(1-at)^{z_1+z_2}}dt \qquad \text{for any } a\in\mathbb{R}_{\leq 1}, \end{align}</math> where in the second-to-last identity {{mvar|n}} is any positive real number. One may move from the first integral to the second one by substituting <math>t = \tan^2(\theta)</math>. For values <math>z=z_1=z_2\neq1</math> we have: :<math> \Beta(z,z) = \frac{1}{z}\int_0^{\pi / 2}\frac{1}{(\sqrt[z]{\sin\theta} + \sqrt[z]{\cos\theta})^{2z}}\,d\theta </math> The beta function can be written as an infinite sum<ref>{{citation|url=https://functions.wolfram.com/GammaBetaErf/Beta/06/03/0001/|title = Beta function : Series representations (Formula 06.18.06.0007)}}</ref> : <math>\Beta(x,y) = \sum_{n=0}^\infty \frac{(1-x)_n}{(y+n)\,n!}</math> If <math>x</math> and <math>y</math> are equal to a number <math>z</math> we get: :<math> \Beta(z,z) = 2\sum_{n=0}^\infty \frac{(2z+n-1)_n (-1)^n}{(z+n)n!} = \lim_{x \to 1^-}2\sum_{n=0}^\infty \frac{(-2z)_n x^n}{(z+n)n!} </math> : (where <math>(x)_n</math> is the [[falling and rising factorials|rising factorial]]) and as an infinite product : <math>\Beta(x,y) = \frac{x+y}{x y} \prod_{n=1}^\infty \left( 1+ \dfrac{x y}{n (x+y+n)}\right)^{-1}.</math> The beta function satisfies several identities analogous to corresponding identities for binomial coefficients, including a version of [[Pascal's identity]] :<math> \Beta(x,y) = \Beta(x, y+1) + \Beta(x+1, y)</math> and a simple recurrence on one coordinate: :<math>\Beta(x+1,y) = \Beta(x, y) \cdot \dfrac{x}{x+y}, \quad \Beta(x,y+1) = \Beta(x, y) \cdot \dfrac{y}{x+y}.</math><ref>{{citation|last=Mäklin|first=Tommi|year=2022|title=Probabilistic Methods for High-Resolution Metagenomics|publisher=Unigrafia|location=Helsinki|pages=27|series=Series of publications A / Department of Computer Science, University of Helsinki|issn=2814-4031|isbn=978-951-51-8695-9|url=https://helda.helsinki.fi/bitstream/handle/10138/349862/M%C3%A4klin_Tommi_dissertation_2022.pdf}}</ref> The positive integer values of the beta function are also the partial derivatives of a 2D function: for all nonnegative integers <math>m</math> and <math>n</math>, :<math>\Beta(m+1, n+1) = \frac{\partial^{m+n}h}{\partial a^m \, \partial b^n}(0, 0),</math> where :<math>h(a, b) = \frac{e^a-e^b}{a-b}.</math> The Pascal-like identity above implies that this function is a solution to the [[first-order partial differential equation]] :<math>h = h_a+h_b.</math> For <math>x, y \geq 1</math>, the beta function may be written in terms of a [[convolution]] involving the [[truncated power function]] <math>t \mapsto t_+^x</math>: :<math> \Beta(x,y) \cdot\left(t \mapsto t_+^{x+y-1}\right) = \Big(t \mapsto t_+^{x-1}\Big) * \Big(t \mapsto t_+^{y-1}\Big)</math> Evaluations at particular points may simplify significantly; for example, :<math> \Beta(1,x) = \dfrac{1}{x} </math> and :<math> \Beta(x,1-x) = \dfrac{\pi}{\sin(\pi x)}, \qquad x \not \in \mathbb{Z} </math><ref>{{citation|title=Euler's Reflection Formula - ProofWiki|url=https://proofwiki.org/wiki/Euler%27s_Reflection_Formula|access-date=2020-09-02|website=proofwiki.org}}</ref> By taking <math> x = \frac{1}{2}</math> in this last formula, it follows that <math>\Gamma(1/2) = \sqrt{\pi}</math>. Generalizing this into a bivariate identity for a product of beta functions leads to: :<math> \Beta(x,y) \cdot \Beta(x+y,1-y) = \frac{\pi}{x \sin(\pi y)} .</math> Euler's integral for the beta function may be converted into an integral over the [[Pochhammer contour]] {{mvar|C}} as :<math>\left(1-e^{2\pi i\alpha}\right)\left(1-e^{2\pi i\beta}\right)\Beta(\alpha,\beta) =\int_C t^{\alpha-1}(1-t)^{\beta-1} \, dt.</math> This Pochhammer contour integral converges for all values of {{mvar|α}} and {{mvar|β}} and so gives the [[analytic continuation]] of the beta function. Just as the gamma function for integers describes [[factorial]]s, the beta function can define a [[binomial coefficient]] after adjusting indices: :<math>\binom{n}{k} = \frac{1}{(n+1)\,\Beta(n-k+1, k+1)}.</math> Moreover, for integer {{mvar|n}}, {{math|Β}} can be factored to give a closed form interpolation function for continuous values of {{mvar|k}}: :<math>\binom{n}{k} = (-1)^n\, n! \cdot\frac{\sin (\pi k)}{\pi \displaystyle\prod_{i=0}^n (k-i)}.</math>
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