Articles of hypergeometric function

How to evaluate following integral?

Suppose the integral $$ \tag 1 I = \int \limits_{-\pi}^{\pi}dx \int \limits_{-\pi}^{\pi}\frac{dy}{\tau – \cos (2x) -2\cos(x)\cos(y)}, \quad t > 3 $$ How to evaluate it in terms of elliptic integral? My attemption. I made the substitution $$ \cos(x) = t, \quad \cos(y) = k $$ My integral then took the form $$ I = 4\int […]

Showing that the pmf of a complicated expression sums to 1 (i.e. it converges)

Background For the past week, I have struggled with a complicated probability distribution in Mathematica. I want to analytically show that it is normalized, or at least, converges. This is the last part of a project in which I have finished all numerical verifications. I have rewritten the probability distribution as nicely as I could. […]

Bounds on $ \sum\limits_{n=0}^{\infty }{\frac{a..\left( a+n-1 \right)}{\left( a+b \right)…\left( a+b+n-1 \right)}\frac{{{z}^{n}}}{n!}}$

I have a confluent hypergeometric function as $ _{1}{{F}_{1}}\left( a,a+b,z \right)$ where $z<0$ and $a,b>0$ and integer. I am interested to find the bounds on the value it can take or an approximation for it. Since $$0<\frac{a..\left( a+n-1 \right)}{\left( a+b \right)…\left( a+b+n-1 \right)}<1, $$ I was thinking that ${{e}^{z}}$ would be an upper bound. Is […]

closed form expression of a hypergeometric sum

After playing around with transforms of a certain parametric integral, I am inclined to think that the linear combination $$f(n):=\dfrac1{n-2}\left({\,}_2F_1(\dfrac{n-2}{4n},\dfrac12;\dfrac{5n-2}{4n};-1)\right)+\dfrac1{n+2}\left({\,}_2F_1(\dfrac{n+2}{4n},\dfrac12;\dfrac{5n+2}{4n};-1)\right)$$ has a closed form for integer $n$. I know for example that $f(3)=\dfrac{1}{12^{3/4}}\dfrac{\Gamma(\frac14)^2}{\sqrt{\pi}}$. Any ideas? Edit: putting $a:=\frac14-\frac1{2n}$, we can define $g(a):= \frac{8}{1-4a}f(\frac{8}{1-4a})$ to get arguments closer to the “standard” notation used in formula collections. […]

Zeros of Fox-Write Function

This question is stimulate by the previous two question here and here. We are interested in studying the following special case of Fox-Write function \begin{align} \Psi_{1,1} \left[ \begin{array}{l} (1/k,2/k) \\ (1/2,1)\end{array} ; -x^2\right], x\in \mathbb{R}, k\in (1,\infty. \end{align} Where Fox-Write function $\Psi_{1,1} \left[ \begin{array}{l} (a,A) \\ (b,B)\end{array} ; z\right]$ is defined as \begin{align} \Psi_{1,1} \left[ […]

Can you prove the following formula for hypergeometric functions?

I wanna prove the following identity for big values of $N\gg 1$ $$ {}_3F_1\left(-N+1,1,1;2;-\frac{1}{N}\right)\to\frac{1}{2}\bigg({}_2F_1\left(1,1;2;1-\frac{1}{N}\right)+\log 2+\gamma\bigg) $$ where $\gamma$ is the Euler-Mascheroni constant. By trying big values for $N$ in the above formula above it looks like the conjecture holds, but I am unable to prove it. Anybody can do better? Solution attempt following Jack D’Aurizio’s […]

On a “coincidence” of two sequences involving $a_n = {_2F_1}\left(\tfrac{1}{2},-n;\tfrac{3}{2};\tfrac{1}{2}\right)$

This was inspired by this post. Define, $$a_n = {_2F_1}\left(\tfrac{1}{2},-n;\tfrac{3}{2};\tfrac{1}{2}\right)$$ $$b_n = \sum_{k=0}^n \binom{-\tfrac{1}{2}}{k}\big(-\tfrac{1}{2}\big)^k$$ where $_2F_1$ is the hypergeometric function and binomial $\binom n k$. The first few numerators $N$ are, $$N_1(n) = \color{brown}{1, \,5, \,43, \,177, \,2867, \,11531, 92479}, \,74069, 2371495,\dots$$ $$N_2(n) = \color{brown}{1, \,5, \,43, \,177, \,2867, \,11531, 92479}, \,370345, 11857475,\dots$$ respectively, and […]

Using multivariate hypergeometric distribution to compute probability of multiple events

I am new here. I have researched this for over a month and spoken to 3 University professors and have not quite gotten the accuracy I am looking for. I am writing C# to compute probability based on user input and information in a database. I can calculate the probability of drawing at least x […]

Closed form for $\int^{\pi}_0 \frac{\sin^2 (y)}{a+\cos(y)} \cos(ny) dy$ for integer $n$

I encountered this integral when trying to obtain a Fourier series for the function inside (in connection to this question). Mathematica gives the following general solution (only valid for $|a|>1$), containing regularized generalized hypergeometric function: $$A_n=\frac{2}{\pi} \int^{\pi}_0 \frac{\sin^2 (y)}{a+\cos(y)} \cos(ny) dy=$$ $$=\frac{4}{a-1} \left(\, _3\tilde{F}_2\left(1,\frac{3}{2},2;2-n, n+2;-\frac{2}{a-1}\right)-3 \, _3\tilde{F}_2\left(1,\frac{5}{2},3;3-n, n+3;-\frac{2}{a-1}\right)\right)$$ Some information about this particular function is […]

Prove the AGM identity using only Hypergeometric series

Arithmetic Geometric Mean can be represented by a Hypergeometric function: $$\text{agm}(1,p)=\frac{1}{{_2F_1} \left(\frac{1}{2},\frac{1}{2};1;1-p^2 \right)}$$ $$0<p \leq 1$$ One of the main properties of the AGM is the following identity: $$\text{agm}(1,p)=\frac{1+p}{2}\text{agm} \left(1,\frac{2\sqrt{p}}{1+p} \right)$$ This allows the infinite product representation of the AGM. I wanted to know if it’s possible to prove this identity by directly using the […]