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Questions tagged [hypergeometric-function]

In mathematics, the Gaussian or ordinary hypergeometric function ${}_2F_1(a,b;c;z)$ is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is a solution of a second-order linear ordinary differential equation (ODE). Every second-order linear ODE with three regular singular points can be transformed into this equation.

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Pfaff formula in the degenerate case

The Pfaff transformation for hypergeometric functions is true under the assumption that the parameters are not negative integers. But, as far as I understand, it also holds sometimes in the degenerate ...
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Unexpected (incorrect) solution to Lagrange Inversion solution to $x^4 - x^3 - x^2 - x - 1 = 0$ about the solution near $x = 2$

I am developing generalized hypergeometric solutions for a set of such polynomials. With this example we can write $x^4 - x^3 - x^2 - x - 1 = \frac{x^5 - 2 x^4 + 1}{x - 1}$. Lagrange Inversion ...
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Behavior of the Gaussian Hypergeometric function when one of its arguments approaches $0$ or $1$

For two positive integers $a,b$, denote by $_2F_1(a,1-b;a+1;z)$ the Gaussian Hypergeometric function whose first three parameters are fixed at $a,1-b$ and $a+1$, respectively. such function is linked ...
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Jacobi polynomials and Gram determinants

On page 294, Andrews, Askey and Roy - Special functions. For sequences of (independent) functions $\lbrace \phi(x) \rbrace_{n=0}^{\infty}$ and $\lbrace \psi(x) \rbrace_{n=0}^{\infty}$, a sequence $\...
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Simplify Sum of Hypergeometric Functions

Write the following "sum" in terms of $\,_2F_1(a,b;c;z)$: $$ _2 F_1(a,b+1;c+1;z) +\, _2 F_1(a,b-1;c-1;z) -\, _2 F_1(a,b;c;z).$$ Attempt: I played around with identities in here. In particular, to ...
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Reduction of hypergeometric function for integer parameters

Consider integers $a,b,c,d>0$ and the hypergeometric function $${}_3F_2(1-d,b,a+b;a+b+c,b+1;1)$$ I don't know much about hypergeometric functions but I understand that when the parameters are ...
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Hypergeometric representation of Fresnel $S(x)$

I am trying to find a representation for the Fresnel integral $$S(x)=\int_0^x\sin\frac{\pi t^2}{2}\,\mathrm dt$$ Then with $$\sin x=\sum_{n\geq0}\frac{(-1)^n}{(2n+1)!}x^{2n+1}$$ We have $$S(x)=\sum_{...
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Integral involving incomplete beta function

I have the following integral, $$\int_{0}^1x^{a-1}(1-x)^{b-1}B_x(c,d)dx$$ where $B_x(c,d) = \int_{0}^xt^{c-1}(1-t)^{d-1}dt$ is the incomplete beta function, and $a,b,c,d>0$. Question: Does this ...
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Expressing $G_{m,m+1}^{m+1,0}\left(x\middle| \begin{array}{c}1,\cdots,1 \\0,0,\cdots,0\\\end{array}\right)$ as a power series.

I have this family of MeijerG functions: $$ G_{m,m+1}^{m+1,0}\left(x\left| \begin{array}{c} 1,\cdots,1 \\ 0,0,\cdots,0 \\ \end{array} \right.\right) $$ which I'd like to express in terms of a power ...
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About the product of two Elliptic integrals

Let $z,x\in\left(0,1\right)$. It is possible to prove that $$\int_{0}^{1}\int_{0}^{1}\frac{1}{\sqrt{hy\left(1-h\right)\left(1-y\right)}}\frac{dydh}{\sqrt{\left(1+zhy\right)^{2}-4xzhy}}=\frac{4}{\pi^{2}...
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Special polynomials and an identity of hypergeometric series

Motivation: I have a few polynomials and am trying to find a representation for them in terms of special functions. I'm more interested in the techniques here, so I won't give any too particular ...
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Transform differential equation into hypergeometric differential equation

I would like to know if this differential equation can be transformed into the hypergeometric differential equation $ 4 (u-1) u \left((u-1) u \text{$\varphi $1}''(u)+(u-2) \text{$\varphi $1}'(u)\...
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Trying to simply an equation to find the limit of the hypergeometric distribution

My textbook states that, $\frac{((1-p)N)^{(n-x)}}{N^{(x)}(N-x)^{(n-x)}} = (1-p)^{(n-x)}$ where $a^{(b)} = aP_b$ I tried expanding the numerator and denominator, and then factoring out the $(1-p)*$ ...
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Simplifying this hypergeometric function with variable b

The student-t CDf has a hypergeometric function as a component $$_2F_1\left(\frac{1}{2}, \frac{\nu + 1}{2}; \frac{3}{2}; -\frac{x^2}{\nu}\right)$$ where $\nu$ is the distributions degree of freedom. ...
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Evaluating $\int_0^1\frac{x^{2/3}(1-x)^{-1/3}}{1-x+x^2}dx$

How can we prove $$\int_0^1\frac{x^{2/3}(1-x)^{-1/3}}{1-x+x^2}\mathrm{d} x=\frac{2\pi}{3\sqrt 3}?$$ Thought 1 It cannot be solved by using contour integration directly. If we replace $-1/3$ with $-2/...
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Integral involving the logarithm of a confluent hypergeometric function

I am trying to find the solution of the integral \begin{align} I =\int_{0}^{\infty}e^{-t}t^{\alpha+1}\left[ _{1}F_{1}(-m; \alpha +1;t)\right]^{2}\log\left[ _{1}F_{1}(-m; \alpha +1;t)\right]^{2}dt \...
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Indefinite integral involving the product of two generalized Laguerre polynomials

I am trying to find the indefinite integral \begin{align} \int{x^{\alpha +1}e^{-x}\left(L_{m}^{\alpha}(x)\right)^{2}dx} \end{align} where $L_{m}^{\alpha}(x)$ is the generalized Laguerre Polynomial, ...
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Sum involving hypergeometric 2F2 function

I'm trying to simplify the following sum: $$ \sum_{i=0}^n\frac{z^i}{(n-i)!}\,\frac{1}{(1+a)_i\,(1-a)_i}\sum_{j=0}^i(-1+a)_j\,(-1-a)_j\frac{(-z)^j}{j!}, $$ where $n=1,2,\ldots$, $z>0$, $0<a<1$,...
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hypergeometric function and its asymptotic expansion near z=1

my dear fellows, I have a question to make. Given the hypergeometric function $_{2}F_{1}[a,b,c,z]$ in the interval $z \in (1, \infty)$. What is the proper asymptotic expansion of the aforesaid ...
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Is it true that $\int_0^1 \big(K(k^{1/2})\big)^2\,dk = \frac{7}2\zeta(3)$?

Define the complete elliptic integral of the first kind as, $$K(k) = \tfrac{\pi}{2}\,_2F_1\left(\tfrac12,\tfrac12,1,\,k^2\right)$$ Part I. From the link above, we find some of the evaluations below, ...
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Symmetry in function given by double sum

I had to deal with this function: $$ f_n(x_1,x_2)=(x_2-x_1)^{n-1}\sum_{m=0}^{n-1}\sum_{j=0}^{n-m-1}C(n,m,j)\left(\frac{x_2}{x_2-x_1}\right)^m\left(\frac{x_2(1-x_1)}{x_2-x_1}\right)^j $$ where $$C(n,...
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Integration of Hypergeometric 2F1 on the real axis with arbitrary extrema of integrations

I have to compute the integral $$ \int_x^1 dz \, z^\alpha (1-z)^\beta \; _2F_1(a_1,a_2;b_1;z) $$ where $x$ is some positive real number $0 \le x<1$. If $x=0$ then the answer is a $_3F_2$ (see ...
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Infinite binomial sum

Let $\displaystyle\pi_{lr}\left(p\right) := {l \choose r}p^{r}\left(1 - p\right)^{l - r}\quad$ ( i.e., the binomial probability with parameters $\displaystyle l$ and $\displaystyle r$ ). I'...
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What is $\int_0^1 \left(\tfrac{\pi}2\,_2F_1\big(\tfrac13,\tfrac23,1,\,k^2\big)\right)^3 dk$?

As in this post, define the ff: $$K(k)=K_2(k)={\tfrac{\pi}{2}\,_2F_1\left(\tfrac12,\tfrac12,1,\,k^2\right)}$$ $$K_3(k)={\tfrac{\pi}{2}\,_2F_1\left(\tfrac13,\tfrac23,1,\,k^2\right)}$$ $$K_4(k)={\...
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Looking for the Closed Form of a Two-Variable Geometric Sum

Is there any closed form of the equation $$\sum_{i=0}^n a^{n-i} \cdot b^i$$ for real values $a$ and $b$ and integer $n \ge 0$?
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What is going on with these asymptotics for $\mathrm{Re}\left\{ \ _2F_1\left(\tfrac{1}{2}, \tfrac{5}{2} ; 2 ; x \right) \ \right\}$

I am interested in the large $x$ asymptotics for the function $$ \mathrm{Re}\left\{ \ _2F_1\left(\tfrac{1}{2}, \tfrac{5}{2} ; 2 ; x \right) \ \right\} $$ When I check the series expansion at $x = \...
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Hypergeometric functions and modular forms

May I please ask if it is possible to write Hypergeometric functions in terms of Jacobi theta functions? I am trying to bring the following Hypergeometric expression (pg.9, eq 4.3) into a known ...
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What is the amount of draws necessary to see all red cards from a standard deck of 52 cards if you draw 5 cards from the deck?

Problem abstraction A standard deck of $52$ cards has $26$ red cards: it has $13$ hearts, $13$ diamonds, as well as $26$ black cards ($13$ spades, as well as $13$ clubs). Let us draw $5$ cards from ...
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Formula for ${}_2F_1(h,-n, 2h; 2)$.

Does anyone know a closed form for the following evaluations of the Hypergeometric function $$ {}_2F_1(h,-n, 2h; t^{-1}) $$ with $h>0,n\geq 0$ both integers and $0\leq t\leq 1$ a real. For the most ...
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Evaluating $\lim_{\epsilon\to 0^{+}}\ \frac{ _2F_1\left( \tfrac{1}{2} - \nu, \tfrac{1}{2} + \nu; \epsilon; y \right) }{\Gamma(\epsilon)}$

For $\nu \in \mathbb{C}$ and negative $y<0$ is there a way to compute the limit $$ f(\nu,y) \equiv \lim_{\epsilon \to 0^{+}} \ \frac{ _2F_1\left( \tfrac{1}{2} - \nu, \tfrac{1}{2} + \nu; \epsilon; y ...
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An intriguing pattern in Ramanujan's theory of elliptic functions that stops?

I. Define the ff integrals, $$K(k)=K_2(k)=\int_0^{\pi/2}\frac{1}{\sqrt{1-k^2 \sin^2 x}}dx=\large{\tfrac{\pi}{2}\,_2F_1\left(\tfrac12,\tfrac12,1,\,k^2\right)}$$ $$K_3(k)=\int_0^{\pi/2}\frac{\cos\left(...
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Asymptotics of Hypergeometric $_2F_1(a;b;c;z)$ for large $|z| \to \infty$?

I found this list of asymptotics of the Gauss Hypergeometric function $_2F_1(a;b;c;z)$ here on Wolfram's site for large $|z| \to \infty$ In particular there is a general formula for $|z| \to \infty$ $...
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Probability of absence and presence of an object whilst sampling without replacement in an unknown pool

Sorry if this is a duplicate, or this is the wrong place! Statistics isn't really one of my stronger points. I have a problem I'm trying to solve, but can't get my head around it. I have a pool of ...
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How do I prove that any second-order equation with three regular singular points can be transformed into a hypergeometric equation?

Consider the hypergeometric equation $$z(1-z)u''+(c-(a+b+1)z)u'-abu=0.$$ I read that it should be possible to show that any second-order equation with three regular singular points can be transformed ...
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Asymptotics of gauss hypergeometric function

My problem is focused on obtaining $r(\rho)$ which is the inverse of the $\rho$ given below. Along the way, I will be integrating a differential equation containing $r(\rho)$ in the regime $\rho>&...
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Evaluation of a partial sum of hypergeometric functions

I would like to determine a closed form expression as a function of N to the following summation: $\sum_{k=1}^{N-1}\left[\frac{{}_2{\rm F}_1\left(1,k-1;\frac{3}{2}+N-k;-1\right)}{\Gamma\left(\frac{3}{...
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Hypergeometric function 3F2 with unit argument

Recently I obtained the following expression $${}_3F_2(-n,a - b ,1-b-n; b + 1, 1-a-n; 1), $$ with $b>a>0$ and $n\in\mathbb{N}$. My question is: If someone knows a closed form solution to the ...
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What closed forms exist for this basic hypergeometric series?

I've run into: $$\sum_{x=1}^n {x^a\over 1-q^{x}}, \ s.t.\ q\in \mathbb N>1 \ or \ q\in (0, 1),\ a \in \mathbb N$$ I am interested mostly in the cases where $a = 1$ or $ a = 2$ Things I've done ...
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An identity on $\small{}_pF_q\left(\left.\begin{array}{c} a_1+1,a_2+1,\dots ,a_p+1\\ b_1+1,b_2+1,\dots ,b_q+1\end{array}\right| z\right)$

I stumbled upon this relation while trying to answer this post. I was trying to find a relation between the two generalized hypergeometric functions, $$A=\,_3F_2\left(\color{blue}{\tfrac12,\tfrac12},\...
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On $\sum_{n=0}^{\infty}{2n+3\choose n+1} \left(\frac{1}{2^n}\cdot\frac{3}{2n+1}\right)^4$ and Gieseking's constant

I. Intro While trying to solve this post about the function, $$F(k)=\sum_{n=0}^{\infty}{2n+3\choose n+1} \left(\frac{1}{2^n}\cdot\frac{3}{2n+1}\right)^k$$ for $k=3$, I found out Mathematica can ...
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Rewriting Appell's Hypergeometric Function $F_1$ in terms of Gauss' Hypergeometric Function $_2F_1$

While going through David H. answer on What is $\int_0^1 \frac{\log \left(1-x^2\right) \sin ^{-1}(x)^2}{x^2} \, dx$? I have encountered a step in between I do not really understand. Within the second ...
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Correction terms in the asymptotics of hypergeometric function

I am interested in obtaining the asymptotic expansion of $r(\rho)$ $($which is the inverse of $\rho$ below$)$, $$\rho=\frac{2b}{1-q}\left(1-\left(\frac br\right)^{1-q}\right)^{1/2}\left(_2F_1\left(\...
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What hypergeometric transformation rules might I apply to try to simplify a certain expression?

I have (https://mathematica.stackexchange.com/questions/189538/sum-a-certain-hypergeometric-function-based-expression-pertaining-to-an-integrat) a Mathematica expression involving the following six (...
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An identity for generalized hypergeometric function

I think the following identity is true, $$ \frac{4 (4 s+9)}{3 \Gamma \left(s+\frac{5}{2}\right) \Gamma \left(s+\frac{7}{2}\right)}-\frac{16 (s+2)}{3 \Gamma (s+3)^2}=\frac{\, _3F_2\left(2,s+\frac{5}{2}...
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Hypergeometric identity

I was trying to solve this integral problem and I noticed something that may be true $$ \int((1-x^r)^{1/r}-x)^{2 n} \mathrm dx = \frac{1}{2 n+1}\sum _{j=1}^{2 n+1} (-1)^{j+1} x^j \binom{2 n+1}{j} \, ...
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Integral involving a Gaussian hypergeometric function and a rational function

Let $x_0,x \in (0,1/10)$ and define: \begin{equation} g(x):= F_{2,1}\left[\frac{1}{13},\frac{1}{17},\frac{1}{5}; 100 x^2 \right] \end{equation} Then the following identity holds true: \begin{eqnarray} ...
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Integral involving hypergeometric function

I've worked out the projection of a spherically symmetric power law volume density profile $\rho(r)=br^a$, i.e. its surface density $\sigma(R)$, and am now trying to integrate this in a series of ...
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Identities involving the Gaussian hypergeometric function

By applying the algorithm from Solving linear ordinary, 2nd order differential equations via global integral bases. to the five parameter family of ODEs defined in my first answer to Gauge ...
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How to find integration of function, in form of hypergeometric function, given below?

I would like to prove the left side to right hand side which is in form of hypergeometeric function. Looking for your hints, suggestions and solultions. $$ \alpha_{1} \int_{0}^{1} (1-z)^{\alpha_{1}+\...
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which one is worse in terms of probability

There are 2N white balls and N red balls (all balls are same except for the color), to put into K different boxes, such that every box contains 3N/k balls. We say event A happens, if any box has more ...