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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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Integral $\int_0^x \ln^{n}\left(2\sin\frac{t}2\right)\,dt$

Evaluate $$\mathcal{L}_n(x)=\int_0^x \ln^n\left(2\sin\frac{t}2\right)\,dt\qquad n\in\Bbb N_0, x\in[0,\pi/2]$$ In this answer to a question of mine, @TitoPiezasIII claims that $$\frac{(-1)^n}{n!}\...
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How do we prove that $\sum_{i=1}^{n} X_{i}$ is a sufficient statistic in the present context?

Let $n$ items be drawn in order without replacement from a shipment of $N$ items of which $N\theta$ are bad. Let $X_{i} = 1$ if the $i$-th drawn is bad and $X_{i} = 0$ otherwise. Show that $\sum X_{i}$...
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Finding a substitution to an equation $ \frac{(C x)^2}{2} y''(x) + D x (1 - E x) y'(x) - F y(x) = 0 $ to transform it to Kummer's equation

It seems that the ordinary differential equation $$ \frac{(C x)^2}{2} y''(x) + D x (1 - E x) y'(x) - F y(x) = 0 $$ can be presented (suggested by Wolfram alpha) with a certain choice of $a$ and $b$ (...
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Transformations relating 3F2 at z with 3F2's at 1/z

I am searching for some transformations for a 3F2 hypergeometric function which send the argument z to 1/z. I am aware of the one given in NIST book (p. 410, Formula 16.8.8) in the special case q=2 (...
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Please provide a function approximating the following hypergeometric series?

Stipulation: Would prefer polynomial asymptotic with shrinking error term and no (Riemann) Zeta functions. Series: $${_1F}_2(1;m+\frac{1}{2};m+1;\frac{1}{4}) =\ ?$$ Put differently, it looks like: $$...
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About a solution to ODE $ (c x)^2/2 y''(x) + a x (1 - b x) y'(x) - d y(x) = 0 $

The ordinary differential equation $$ \frac{(c x)^2}{2} y''(x) + a x (1 - b x) y'(x) - d y(x) = 0 $$ should have two independent solutions which are given by a confluent hypergeometric function $U$ ...
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A twisted hypergeometric series $\sum_{n=1}^\infty\frac{H_n}{n}\left(\frac{(2n)!}{4^n(n!)^2}\right)^2$

I was given that $$S=\sum_{n=1}^\infty\frac{H_n}{n}\left(\frac{(2n)!}{4^n(n!)^2}\right)^2=\frac{32}\pi G\ln2+\frac{64}\pi\Im\operatorname{Li}_3\left(\frac{1+i}2\right)-2\ln^22-\frac53\pi^2,$$ where $...
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Prove an transformation formula for Gauss hypergeometric function $_2F_1(a,b;c;z)$

In " Special functions: an introduction to classical functions of mathematical physics" by Nico M. Temme, at page 113 is reported this formula: $$_2F_1(a,b;c;z)=\frac{\Gamma(c)\Gamma(b-a)}{\Gamma(b)\...
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The solution of associated legendre function in hypergeometric function.

Im learning about invers scattering theory that have been told in a book entitled soliton : an introduction by P. G. Drazin and R. S. Johnson. In the 46'th page there was a paragraph said that the ...
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Can you exactly integrate $x_i^2 x_j^2 e^{\sum_{k=1}^4 \lambda_k x_k^2}$ over $S^3$?

I am concerned with the following integral $$ I = \int_{S^3} x_i^2 x_j^2 e^{\sum_{k=1}^4 \lambda_k x_k^2} d\mu(x) $$ where the $x_{i,j,k}$ are components of $x \in S^3$, each $\lambda_k < 0$, and ...
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Finding constants in Wronskian of standard solutions of confluent hypergeometric function

I've got this question on special functions take-home exam. I will add some literature in the nearest time. Let us have an equation $xy''+(c-x)y'-ay=0$ which is a confluent hypergeometric function ...
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Binomial sums of the form $\sum_{k=\ell+1}^{n}(-1)^{(k-\ell+1)}\binom{k}{\ell}$

I'm trying to compute the sum $$\sum_{k=\ell+1}^{n}(-1)^{(k-\ell+1)}\binom{k}{\ell}$$ with $\ell \in [2,n-1]$ in an efficient way. I know that this reduces to something like $$ 1 -\sum_{i=0}^{n/2 -1}...
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Hypergeometric solution in a particular case

I have a question to make in relation to the solution of the hypergeometric differential equation. Let us consider the aforesaid equation, \begin{equation} y(1-y)h'' + [c-(1+a+b)y]h' -abh=0, \end{...
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Kummer transform of the confluent hypergeometric function of second kind

I can see the kummer transformation of the confluent hypergeometric function of first kind throught the integral representation. However, I failed to see that for the second kind. More specificially, ...
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Discontinuity of Hypergeometric function along the branch cut

I am trying to evaluate an expression involving the hypergeometric function evaluated near its (principal) branch cut discontinuity, which is placed on the real line from $1$ to infinity. For $x>1$...
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Hypergeometric function: parameters as integer values

I would like to know if exists a simple form to express $ \, _2F_1(n-1,2-n;2;u) $ or $ \, _2F_1(3-n,n;2;u) $ in terms of some sort of polynomic expression. n is a integer number. $ n=0, -1, -2, ....
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Convert a polynomial to Hypergeometric equation

How can I convert this polynomial to Hypergeometric equation. $$Pn=n!\sum_{r=0}^{\frac{n}{2} } (-1)^r\frac{(p-1)!}{(p-n+r-1)!} \frac{1}{r! (n-2r)!}2x^{n-2r}$$
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Hypergeometric series for $\mathrm{Cl}_2(\pi/3)$

I am trying to find a hypergeometric series for $\mathrm{Cl}_2(\pi/3)$, where $$\mathrm{Cl}_2(x)=-\int_0^x\log\left|2\sin\frac{t}2\right|dt=\sum_{k\geq1}\frac{\sin kx}{k^2}$$ Is the Clausen ...
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Simplification of $\sin(\pi^x)$ , with $x$ being a positive irrational number

How to simplify if $a > 0$ and $\cos(a) < 0$ Was a previous post. Correction, it was suppose to be if a > 0 & cos(a) > 0. An answer was given. https://math.stackexchange.com/a/1274372/...
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Please help with a Hypergeometric probability question [closed]

Need help with this question, my textbook has the solution but I don't know how to get it. 15 coins in a bag. Three 5 rand coins Five 2 rand coins Seven 1 rand coins What is the prob that if I pull ...
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Yet another difficult logarithmic integral

This question is a follow-up to MSE#3142989. Two seemingly innocent hypergeometric series ($\phantom{}_3 F_2$) $$ \sum_{n\geq 0}\left[\frac{1}{4^n}\binom{2n}{n}\right]^2\frac{(-1)^n}{2n+1}\qquad \...
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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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Integration of $\int^{1}_{-1} \frac {1}{3} \sinh^{-1} \left( \frac {3\sqrt 3}{2} (1-t^2) \right) dt$

Recently I came across with respect to this post of mine hyperbolic solution to the cubic equation for one real root given by $$ t=-2\sqrt \frac {p}{3} \sinh \left( \frac {1}{3} \sinh^{-1} \left( \...
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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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3answers
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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 ...