Articles of hypergeometric function

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 […]

Simple explanation for Hypergeometric distribution probability

I am following through the Hypergeometric distribution: The probability that we select a sample of size $n$ containing $r$ defective items from a population of $N$ items known to contain $M$ defective items is $P(X = r) = C(M,r) * C(N-M,n-r) / C(N,n)$ where C(P,Q) is the combination of P items taken Q at a […]

How to prove ${}_{2}F_{1}\left(\frac{1}{3},\frac{2}{3};\frac{3}{2}; \frac{27}{4}z^2(1-z)\right) = \frac{1}{z}$

With reference to the following post On $\int_0^1\arctan\,_6F_5\left(\frac17,\frac27,\frac37,\frac47,\frac57,\frac67;\,\frac26,\frac36,\frac46,\frac56,\frac76;\frac{n}{6^6}\,x\right)\,dx$ I used $${}_{k}F_{k-1}\left(\frac{1}{k+1} ,\cdots ,\frac{k}{k+1};\frac{2}{k} \cdots ,\frac{k-1}{k},\frac{k+1}{k};\left( \frac{z(1-z^k)}{f_k}\right)^k \right) = \frac{1}{1-z^k}$$ Where $$f_k \equiv \frac{k}{(1+k)^{1+1/k}}$$ This formula seems to be too complicated for the general case. Let us look at the easiest case $k=2$ then we have $${}_{2}F_{1}\left(\frac{1}{3},\frac{2}{3};\frac{3}{2}; \frac{27}{4}z^2(1-z^2)^2\right) = \frac{1}{1-z^2} $$ This is equivalent to proving […]

Prove ${_2F_1}\left({{\tfrac16,\tfrac23}\atop{\tfrac56}}\middle|\,\frac{80}{81}\right)=\frac 35 \cdot 5^{1/6} \cdot 3^{2/3}$

I’ve found the following hypergeometric function value by numerical observation. The identity matches at least for $100$ digits. $${_2F_1}\left(\begin{array}c\tfrac16,\tfrac23\\\tfrac56\end{array}\middle|\,\frac{80}{81}\right) \stackrel{?}{=} \frac 35 \cdot 5^{1/6} \cdot 3^{2/3}$$ Or using a Pfaff transformation in an equivalent form $$81^{1/6} \cdot {_2F_1}\left(\begin{array}c\tfrac16,\tfrac16\\\tfrac56\end{array}\middle|\,-80\right) \stackrel{?}{=} \frac 35 \cdot 5^{1/6} \cdot 3^{2/3}$$ How could we prove it? Other related problem: How could […]