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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integration with small expansion

I am reading this set of lecture notes: https://www.southampton.ac.uk/~doug/qft/aqft5.pdf and I would like to understand how to go from the relation (page 52 no Eq number) $4^\epsilon \int (\sin \...
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Series of hypergeometric 2F1 functions / Moment generating function of random variable

Consider a random variable $X\in\mathbb N$ with probability mass function $$\Pr(X=k) = \frac{\binom{N+k+r-1}{r-1}}{\binom{N+r-1}{r-1}}p^k (1-p)^{N+r} {}_2F_1(N,k+N+r;k+N+1;p),$$ whereby ${}_2F_1$ ...
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How to find the integral value of hypergeometric functions

We know that, 1. $$ \int_0^1 t^{b-1} (1-t)^{c-b-1} (1-tz)^{-a} dt = \beta(b, c-b) ~\,_2F_1~(a, b, c, z) $$ 2. $$ \int_0^1 t^{a-1} (1-t)^{b-1} e^{ct} dt = \beta(a, b) ~\,_1F_1~(a, b, c) $$ where $\...
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Ramanujan's infinite series for $\frac{x^3(3x-2)}{(2x-1)^3}$ for all positive integers $x$

Here we go, yet another wild infinite series by Ramanujan. If $x$ is a positive integer, then$$1+3\Bigg(\cfrac{x-1}{x+1}\Bigg)^3\cfrac{3x-1}{3x-3}+5\Bigg\{\cfrac{(x-1)(x-2)}{(x+1)(x+2)}\Bigg\}^3\...
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How to prove Dixon's theorem evaluating a well-poised 3F2 generalized hypergeometric series at 1 using the Selberg integral?

In Wikipedia (https://en.wikipedia.org/wiki/Dixon%27s_identity) I encounter the following Dixon's theorem evaluating a well-poised 3F2 generalized hypergeometric series at 1 is $$_3F_2 (a,b,c;1+a-b,1+...
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derivative of hypergeometric function

I am doing an integral in Mathematica and I find the solution contains derivatives of hypergeometric functions. I would like (ideally) a simple analytic form for these. I have tried HypExp mathematica ...
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Computation of incomplete confluent hypergeometric function of the first kind

In summary I need to compute "upper incomplete confluent hypergeometric function of the first kind": $U(a, b, x, z) = \sum_{n=0}^{\infty} \frac{ x^n }{n!} \frac{\Gamma(a + n, z)}{\Gamma(b + n)}$ ====...
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How to compute $\int_{0}^\pi \sin^a(\theta) \sin^b(x\theta) \sin^c(y\theta) d\theta$?

I need to compute integrals of the form, $\int_{0}^\pi \sin^a(\theta) \sin^b(x\theta) \sin^c(y\theta) d\theta$ where $a,b,c,x,y$ are rational numbers, e.g., $ a=1,\quad b=-1/3, \quad c=1/3,\quad x=...
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Hypergeometric extension to deal with double summation

I would like to be able to find an expression for summations of the form $$ \sum_{l,m=1}^{\infty}\frac{x^{l}y^{m}}{a^{2}+l^{2}+m^{2}}, $$ at least in terms of special functions, although I can't even ...
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Testing from which of two jars my sample set has come from and with what significance? What if the significance is a function of the success events?

Assume I have two jars, each having two sets of balls, black and white. The balls have different weights, and for the white (success draw) balls, the heavier weight makes a more significant success if ...
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A certain sum of products of binomial coefficients

Let $M$ and $N$ be positive integers such that $2N-3M\geq 0$. I would like to know if the (finite) sum $$ \sum_{i=0}^{\infty}{{2N-3M}\choose{N-3i}}{M\choose i}^3 $$ has a nice closed form (in terms ...
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Another bizarre convolutional identity.

Let $\lambda^M \ge0$ and $\Lambda\ge 0$ and $q \in (0,1)$. Now define another three numbers $(a,b,c)$ by solving the following set of non-linear equations below: \begin{eqnarray} c \cdot (c-2) &=&...
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Find the function $f(x)=\sum_{n=1}^{\infty}\frac{H_{n-1}(-x)^n}{n!}$

I want to find the function defines by : $$f(x)=\sum_{n=1}^{\infty}\frac{H_{n-1}(-x)^n}{n!}$$ Where $H_n=1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}$ is the Harmonic series. My work We have the ...
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Hypergeometric even part

The following series represents an Even function: $$ {}_0F_3\left(\left.\begin{array}{c} -,-,-\\ 3/2,5/2,2 \end{array}\right| x^2\right) $$ Determine the function F(x) such that f_even(x) ...
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Find a tight lower bound to $ - \frac{\partial}{\partial a} {}_{1}F_{1}(a; \frac{1}{2}; z) $, simplified to a series

Motivation: I have an expression involving a derivative of the hypergeometric function that is difficult to evaluate numerically, so I am looking for a tight lower bound that can be evaluated easily. ...
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Solution to Laguerre differential equation

I am slightly confused by the solution to the Laguerre differential equation $$xy''+(\alpha +1-x)y' + ny=0 .$$ The solution is $$y = c_1U(-n,1+\alpha, x) +c_2 L_n^\alpha(x),$$ where $U$ is a ...
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What is this mystery function that wolfram alpha says my exponential generating function is equal too?

The $E_n(x^n)$ is the mystery function $$\sum_{d=0}^{\infty}\frac{x^{dn}}{\Gamma(dn+1)}=E_n(x^n)$$ Here are the first 3 values of the function
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What is the probability of drawing at least 1 ace, at least 1 king, and at least 1 queen, in a 5 card poker hand from a standard 52 card deck?

I use the following terms to define the different events: A = draw at least 1 ace B = draw at least 1 king C = draw at least 1 queen I use the following expression to define the problem: P(A∩B∩C) =...
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Help with the Euler-type integral $\int_{0}^{m}\frac{1-e^{2\pi i x}}{x-j}\frac{x^{s-1}}{(1+x)^{z}}dx$

Consider the integral : $$I=\int_{0}^{m}\frac{1-e^{2\pi i x}}{x-j}\frac{x^{s-1}}{(1+x)^{z}}dx\;\;\;\;s,z\in\mathbb{C}\;\;\;\;j,m \in \mathbb{N}\;\;0\leq j\leq m$$ i have tried using the Mellin ...
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Value of $\sum_{n=1}^{\infty}\frac{1}{\sqrt{n!}}$?

I was reading this old question and fascinated by the second infinite sum $$\sum_{n=1}^{\infty}\dfrac{1}{\sqrt{n!}}.$$ This is clearly convergent (by comparison or ratio test) and, we can obtain some ...
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Closed form of $\sum_{k=0}^{\infty} \sum_{m=0}^{\infty} r^m \cdot t^k \binom{m+k}{k} \binom{m+k+1}{k}$ for fixed $r, t$

I would like to find the closed form for the double sum $$\sum_{k=0}^{\infty} \sum_{m=0}^{\infty} r^m \cdot t^k \binom{m+k}{k} \binom{m+k+1}{k} \tag 1$$ where $r, t$ are known values. When I plugged ...
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An ODE with Hyperbolic function as solution [duplicate]

Consider the following ODE, $$ x \ddot{x} - \dot{x}^2 - 1 = 0. $$ I can guess that the solution must be $$ x = a \cosh \left(\frac{t - b}{a} \right) $$ How to solve this ODE?
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Integral involving Hypergeometric Function 2F1 and its derivative

When working on a probability problem involving Binomial distributions I came across this integral: $P = \int_0^1 dp_1\,\left(p_1^{k_1}(1-p_1)^{n-k_1} \int_{\max(0,p1-\epsilon)}^{p_1}dp_2\,p_2^{k_2}(...
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approximate the hypergeometric function to get unbiased squared correlation

Following Olkin and Pratt (1958), the unbiased minimum variance estimator of the square correlation $r^2$ (here considering only a single predictor) is $1-\frac{n-2}{n-1}(1-r^2)_2F_1(1;1;\frac{n+1}{...
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Hypergeometric representation degenerate cases

In Concrete Mathematics, section 5.5, the book discussed degenerate cases for hypergeometric series. But slightly later on, after establishing the gamma function, $\Gamma(z+1)=z!$, $(-z)!\Gamma(z)=\...
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Reduction of ${}_{4}F_{3}(\cdots; 1)$

It is suspected that $$ {}_{4}F_{3}\left(\frac{1}{4}, \frac{1}{4}, \frac{3}{4}, \frac{3}{4}; 1 , \frac{3}{2}, \frac{3}{2}; 1 \right) = \frac{8}{\pi} - \frac{4 \, \sqrt{2} \, \Gamma^2\left(\frac{3}{4}\...
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How to find infinite sum of trigonometric, hyperbolic function?

$$u(x,y)=\sum_{n=1}^{\infty}\dfrac{-2 \cos n \pi}{n \pi}\dfrac{\sinh n \pi(y-1)}{\sinh(-n \pi)}\sin (n \pi x)$$ I tried putting random values of $x$ and $y$ and simplified. But still stuck on it. How ...
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An integral involving hypergeometric functions and leading to a sum with Gamma functions

I am interested in computing the following integral $$ I=\int_{-1}^{1}\mathrm{d} u~_2F_1\left(-k,\alpha+k-\frac{1}{2};\alpha, 1-u^2\right)(1-u^2)^{\alpha-1}u^{2m} $$ where $_2F_1$ is the standard ...
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Is there a closed form expression of $_3F_2\left(a,b,-\frac{1}{2};a-\frac{1}{2},a+b;1\right)$ where $a,b>0$?

I want to see if I can find a closed form expression for the following $$_3F_2\left(a,b,-\frac{1}{2};a-\frac{1}{2},a+b;1\right)=\sum_{l\ge 0}\frac{(-1/2)_l(a)_l(b)_l}{(a-1/2)_l(a+b)_l}\frac{1}{l!},$$ ...
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Expressing Hermite polynomials using confluent hypergeometric functions

On Wikipedia it's said that one could express the Hermite polynomials using the $_1F_1$ function and the following formulae are provided: $$H_{2n}(x)=(-1)^n \frac{(2n)!}{n!}(_1F_1)(-n,1/2;x^2),$$ $$...
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$\int_{0}^{e}\frac{\sin(\operatorname{W(x)})-1}{\sin(\operatorname{W(x)})+1}\frac{\sin(\operatorname{2W(x)})-1}{\sin(\operatorname{2W(x)})+1}dx=?$

Solve the following integral : $$\int_{0}^{e}\frac{\sin(\operatorname{W(x)})-1}{\sin(\operatorname{W(x)})+1}\frac{\sin(\operatorname{2W(x)})-1}{\sin(\operatorname{2W(x)})+1}dx=?$$ My attempt: We ...
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How to calculate the limit of the variance of the moment estimate of the geometric distribution?

Let $\overline X$ be the sample variance, so that a low-order moment estimate of the geometric distribution can be obtained:$\;\hat{p}=1/\overline X$. I want to verify whether this estimator is an ...
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How to calculate this limit related to hypergeometric functions

How to calculate this limit $$ \lim_{n\rightarrow+\infty} np^n \sum_{k=n}^{+\infty} \frac1k \binom{k-1}{n-1} (1-p)^{k-n} ,\quad\,where\;0<p<1 $$
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Moment Estimation on Geometric Distribution

I recently had trouble calculating the moment estimates for the parameter $p$ of the geometric distribution: $$ P(X=k)=(1-p)^{k-1}p,\quad k=1,2,\cdots $$ We know that there are two kinds of moment ...
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About gaussian hypergeometric function

i am novice with this argument and i'm trying to figure out how transform this linear combination: $$ A \,z^h \, _2F_1(h, h, 2h, z) + B\,z^{(1-h)}\, _2F_1(1- h, 1 - h, 2- 2h, z) \rightarrow \dfrac{\...
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Evaluating limit involving convergent and divergent hypergeometric functions

Let $a,b,c$ be positive real numbers satisfying $c - a - b = 1 > 0$. After much manipulation, my problem reduces to evaluating the following limit: $$ \lim_{z\rightarrow 1^-} \left| \frac{1}{_2F_1(...
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Different ways to guarantee compact support

I am studying computer algebra. To be more precise, I am trying to understand the problem of hypergeometric summation. Definition1: A function $F(n,k)$ is hypergeometric in both arguments if $\frac{...
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Associated Legendre Function of Second Kind

The associated Legendre Function of Second kind is related to the Legendre Function of Second kind as such: $$ Q_{n}^m(z)= (-1)^m (1-z^2)^{m/2} \frac{d^m}{dz^m}(Q_{n}(z)) $$ The recurrence relations ...
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Independent Check of Wolfram's Evaluation of the Clausen ($_3F_2$) Hypergeometric Function

Could someone be kind enough to independently verify the calculations of the Generalised Hypergeometric Function (Hypergeometric3F2) at http://functions.wolfram.com/webMathematica/FunctionEvaluation....
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How can I compute this integral involving $\Gamma$-functions?

I would like to find a closed form for the following integral: $$I=\int_0^1 d\alpha\ \alpha^{\omega-5/2} (1-\alpha)^{-1/2} \int_0^\alpha d\beta\ \beta^{2\omega-3} (1-\alpha-\beta)^{5/2-2\omega} (\...
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Trigonometric formula coming from hypergeometric functions

While playing with hypergeometric functions, I numerically stumbled upon the identity: $$\mathrm{cos}\left(\dfrac{\pi}{6} - \dfrac{1}{6} \mathrm{arctan}\left( \dfrac{3\sqrt{15}}{11} \right) \right) = \...
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Euler/Kummer Transformations for the Clausen and Appell Hypergeometric Functions

The series expansion for the Clausen $_3F_2$ Hypergeometric function defined by the series $$ _3F_2(a,b,c;d,e;x) =\sum_{k=0}^\infty\frac{(a)_k(b)_k(c)_k}{(d)_k(e)_kk!}x^k $$ is unconditionally ...
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Eliminating Value from Bessel Function [closed]

How I can get the value for "k" as a function of "r" from equality $J1(kr)=Dr$, where D is constant. $J1(kr)$ is the Bessel function with n=1.
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General Solution of Hypergeometric equation in range $x \in (0,1) $

I'm trying to find the general solution of the hypergeometric equation \begin{equation} x(1-x)\partial_x^2y+\left[\gamma-(\alpha+\beta+1)x \right]{\partial_x} y -\alpha\beta y=0, \end{equation} in the ...
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simplify ${}_1F_1(2\alpha , 2 ,ik\pi) +{}_1F_1(2\alpha , 2 ,-ik\pi)$

1)Is it possible to simplify $$J_{k,\alpha}={}_1F_1(2\alpha , 2 ,ik\pi) +{}_1F_1(2\alpha , 2 ,-ik\pi)$$ $k\in \{1,2,\cdots\}$ and $\alpha \in (0,1)$. ${}_1F_1$ is confluent hypergeometric ...
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57 views

Expressing the Hypergeometric Function $_3F_2(a,a,b;p,p;x) $ in terms of $_2F_1()$

Is it possible to express the Clausen Hypergeometric Function $_3F_2(a,a,b;p,p;x)$ (the first two parameters and the last two are identical) in terms of the Gauss Hypergeometric Function $_2F_1()$ and ...
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25 views

generalised gauss hypergeometric series

Can ${}_4\!F_3(a,a,a+1/2,a+1/2;b,b,b;z)$ be written successively in terms of ${}_2\!F_3$ and ${}_1\!F_2$? $a$ and $b$ are integers in this case.
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Literature request for polynomials $P_n(x,y)$ generated as $(1-z)^x {}_2F_1(\alpha, \alpha+x-y;2s;t)=\sum_{n\geq 0}P_n(x,y)t^n$

Consider the following generating function $$ \phi(t) = (1-t)^x {}_2F_1(\alpha,\alpha+x-y; 2\alpha;t)=(1-t)^{y} {}_2F_1(\alpha,\alpha+y-x; 2\alpha;t) $$ where we have used the Euler transform ${}_2F_1(...
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When is Confluent Hypergeometric Function of the Second Kind square integratable?

Confluent Hypergeometric Function of the Second Kind has the Maclaurin series of $z^{1-b}$ and asymptotic series of $\frac{1}{z^a}$, so it appeared that if $b>1$ (for $z\ll1$) and $a>0$(for $z\...
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Generating function containing Meijer G function

Consider the following terminology for Meijer g function : $$G_{p,q}^{m,n}\left(z\left|\begin{smallmatrix}a_1,.......,a_p\\ b_1,......,b_q\end{smallmatrix}\right.\right)$$ Now consider the ...

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