# Tagged Questions

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 ...

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### How to isolate and solve for k in a Sigma notation probability mass function equation?

"isolate and solve for k:" $$P(X = k) = \sum_{k=0}^n {{{K \choose k} {{N-K} \choose {n-k}}}\over {N \choose n}}$$ If the above equation is a function of P, how would the equation be stated as a ...
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### turning a certain chebychev polynomial-like expression into a hypergeometric form

Can the following expression be represented in terms of hypergeometric function $$\sqrt{3}\sin(\arcsin(7/25)/3)-\cos(\arcsin(7/25)/3)$$ It looks similar to the one presented on [this site][1] [1] ...
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### How to proof this about Hypergeometric function?

I am trying to understand why the following four properties are true $$_{2}F_{1}(a,b,c,1)=\frac{\Gamma(c)\Gamma(-a-b)}{\Gamma(c-a)\Gamma(c-b)}$$ $$_{2}F_{1}(-n,b,c,1)=\frac{(c-b)_{n}}{(c)_{n}}$$ ...
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### 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 ...
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### Solving summation with incomplete gamma function

I solved an indefinite integral that gave me the following result: f(u) = -\sum_{k=0}^{n}\binom{n}{k}\frac{\mathrm{sgn}(u)^{k+1}\, b^{n-k}}{2\, a^{\frac{k+1}{2}}}\, ...
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### Integral of combination of power, exponential, and kummer hypergeometric function

I am trying to solve a couple integrals of the form: $$\int_{0}^{\infty} x \, e^{-a(x-b)^{2}}\, M(-\alpha,-\beta,\lambda x)$$ $\alpha > 0$ and $\beta > 0$ are ...
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### Integrals involving whittaker functions.

I want to compute the following integrals: $$\int y^{a} e^{\frac{1}{2}y}M_{k,m}(y)dy$$ where a is an arbitrary constant and $M_{k,m}$ is a whittaker function of the first kind. I already know that ...
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### Expected number of successes before first failure (Hypergeometric distribution)

Think of the following scenario: We are a group of $42$ people. I tag you, and you tag another person. This other person tags another person, etc. The "chain" of tagging stops when a person has been ...
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### Prove that ${}_2F_1(0,b;c;z)=1$

I do not know how I could prove that ${}_2F_1(0,\beta;\gamma;t)=1$ because when I apply the definition I get $0$, namely.. $$\sum_{n=0}^{\infty}\frac{(0)_n(\beta)_n}{n!(\gamma)_n}t^n=0$$ someone ...
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### Evaluate the limit of the Appell F1 function.

I am interested in the following $$\lim_{x\rightarrow\infty}xF_1(1/2;n+1/2,-1/2;3/2;-x^2,-c x^2),$$ where $0<n<1/2$ and $c>0$. $F_1$ is the Appell series. Any idea on how to obtain the ...
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### Evaluating a certain integral which generalizes the ${_3F_2}$ hypergeometric function

Euler gave the following well-known integral representations for the Gauss hypergeometric function ${_2F_1}$ and the generalized hypergeometric function ${_3F_2}$: for ...
### Closed form for $\sum_{n=0}^\infty\frac{\Gamma\left(n+\tfrac14\right)}{2^n\,(4n+1)^2\,n!}$
I was experimenting with hypergeometric-like series and discovered the following conjecture (so far confirmed by more than $5000$ decimal digits): ...