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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Integral involving power of trigonometric functions

I'm having a technical problem evaluating the following integral: $$\int_{r=0}^1\int_{\theta=0}^{\pi \over2} \cos^{2\epsilon -1}\theta \sin^{\epsilon-1}\theta e^{-ikr\sin^\epsilon\theta}d\theta dr$$ ...
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12 views

Limits of Kummer Confluent Hypergeometric function for fixed z

I have the following: \begin{equation} \sum_{j=0}^{\alpha_{2}}C\, e^{az}\, M(-\alpha_{2}+j,-\alpha_{3}+j,-\lambda z)\Bigg|_{z=-\infty}^{0} \end{equation} $C$ is a constant, $a,\alpha_{2},\alpha_{3}, ...
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61 views

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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16 views

Solving summation with incomplete gamma function

I solved an indefinite integral that gave me the following result: \begin{equation} 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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23 views

Integral of combination of power, exponential, and kummer hypergeometric function

I am trying to solve a couple integrals of the form: \begin{equation} \int_{0}^{\infty} x \, e^{-a(x-b)^{2}}\, M(-\alpha,-\beta,\lambda x) \end{equation} $\alpha > 0$ and $\beta > 0$ are ...
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1answer
38 views

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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Limit of this monster function [closed]

Here's a function, and I am interested in finding its limit when $z\rightarrow \infty$. Any idea would be really appreciated. First thing I need to know is if this limit really exists. ...
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2answers
61 views

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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27 views

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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Derivation of the hypergeometric function $\frac{\partial {}_{3}F_{2}(a_{1}, a_{2}, a_{3}; b_{1}, b_{2}; \frac{1}{z})}{\partial z}$

We know that the first order derivative of the generalized hypergeometric function ${}_{3}F_{2}(a_{1}, a_{2}, a_{3}; b_{1}, b_{2}; z)$ is expressed as follows: \begin{equation} \frac{\partial ...
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19 views

Relations between hypergeometric functions

I am trying to find a relation between hypergeometrics $${}_2F_1(a,b,c;z)\,\,\text{and}\,\,{}_2F_1(a+1,b+1,c+1;z)$$ I can see that $$\frac{\partial}{\partial z}{}_2F_1(a,b,c;z) = ...
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27 views

Solution to Sextic Polynomial with Two Real Roots

I have the polynomial $$f(x;a)=3ax^6+6x^5-9ax^4-4x^3+9ax^2+6x-3a$$ where the variable $a$ is a random variable from the uniform distribution in the range $[0,1)$. When I analyze this function using ...
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32 views

Analytical Abel transform of these basis functions

I am trying to perform Abel transforms on basis functions $f_k(r)P_l(cos(\theta))$ for Abel inversion. The typical radial basis functions used are Gaussians $e^{-(\frac{r-r_k}{\sqrt{2}\sigma})^2}$. I ...
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1answer
42 views

Integrate $\int(\log(\sin x \cos x))^n dx$ with hypergeometric function form

Evaluate $$\int({\log(\sin x\cos x)})^{n} \, \mathrm{d}x$$ with result in hypergeometric function form Could anyone help me with that?
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33 views

Does this reduce down to the PolyGamma function?

Does this reduce down to the PolyGamma function? $H_n=$ $$\lim_{s\to 0} \, \left(-\frac{\left(\frac{1}{s}+1\right)^n (s+1)^{-n} \left(\sum _{k=0}^{\infty } \frac{\left(-\frac{1}{s}\right)^k ...
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34 views

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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28 views

The inverse Laplace transform of $\Gamma\left(\zeta\right) \, W_{\zeta,\mu}(z) $

Someone has a reference that addresses an integral of the followns type $$I = \frac{1}{2i\pi} \int_{\sigma-i\infty}^{\sigma+i\infty} e^{t\zeta} \, \Gamma\left(\zeta\right) \, W_{\zeta,\mu}(z) \, ...
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28 views

What is the closed form of this series: $\sum_{n\geq 1}\frac{n^k{(-1)}^{n+1}}{n!}$ for $k<-10$ and for $k>1$?

I would like to check the closed form of this sum $$\sum_{n\geq 1}\frac{n^k{(-1)}^{n+1}}{n!}$$ , for an integer $k>1$ and $k<-10$. Note : I run some computation in wolfram alpha i have got ...
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96 views

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 ...
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267 views

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): ...
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30 views

Inverse function of hypergeometric function, e.g., ${}_{2}F_{1}(1,1;1.2;x)$

I want to know whether it is able to express the inverse function of hypergeometric function using some special function. For instance, the Gauss hypergeometric function ...
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33 views

Asymptotic of $ _1F_1(a;b;z)$

How it can be shown that $$ _1F_1(a;b;z) = \frac{\Gamma(b)}{\Gamma(a)}\, e^{z} \, z^{a-b}\, [1+ O(\mid z\mid^{-1})]; \quad (\Re(z)>0)$$ or $$ _1F_1(a;b;z) = \frac{\Gamma(b)}{\Gamma(b-a)}\, ...
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25 views

Proof of Hypergeometric Contiguous relation

I want to prove the following recursive relation: $$c(c+1)_2F_1(a,b;c;z)=c(c-a+1)_2F_1(a,b+1;c+2;z)+a[c-(c-b)z]_2F_1(a+1,b+1;c+2;z)$$ I tried using both the series ...
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88 views

Is this a known polynomial?

This expression has appeared in my work: $$P_n(x)=\sum _{j=0}^n\binom{n}{j}\, j^n x^j $$ It seems too simple not to be already somewhere (perhaps in a different form). Has anyone seen it before?
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60 views

explicit formula for $ _2F_2(1,1;2;2;z) $

Is it an explicit formula for $$ _2F_2(1,1;2;2;z) ,$$ where $$_2F_2(a,b;c;d;z)=\sum_{n\geq 0}\frac{(a)_n(b)_n}{(c)_n(d)_n n!}z^n .$$ thanks you in advance
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Waring's Problem and the floor function - solving a recurrence relation by hand

First-time poster here. While doing some research on Waring's problem and the term $\{(3/2)^n\},$ I determined that the following recurrence relation holds for a certain sequence (here $n$ is a fixed, ...
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36 views

Finding a relation between hypergeometric functions $_2F_1$

I would write the following Gauss hypergeometric function $$ _2F_1 \left(a,b; c-n; x\right) $$ in terms of $$ _2F_1 \left(a,b; c; x\right) $$ Where $a,b,c\in \mathbb C , x\in \mathbb R$ and $n\in ...
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31 views

Exact solution of an simple Meijer-G function

I am trying to simplify the following Meijer-G funtion \begin{equation} G^{2,2}_{2,2}\Bigl({}^{0,\, 1-m}_{0,\,0} |x \Bigr) \end{equation} But the Matlab(MuPAD) and WolframAlpha give me different ...
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27 views

Integral of the principal value of a hypergeometric function

I am looking to write the hypergeometric function $${}_2F_1\left(1,1,2+\epsilon, -\frac{\alpha}{\beta}\right) = \int_0^1\,dt\,\frac{(1-t)^{\epsilon}}{1-tz + i\delta},$$ where $z=-\alpha/\beta$ and ...
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48 views

Does a limit to this Hypergeometric Function Exist Analytically?

I am interested in evaluating limit $$\lim_{x\rightarrow\pi/2}\left[(\cos x)^n\, _2F_1\left(-\frac{n}{2},-n-m+1;\frac{1}{2}-n;-\frac{16m c}{\cos^2x}\right)\right], $$ where $n$ is a positive even ...
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Upper Bound for General Hyper Geometric Function $_2F_3$

I am not working in physics field, but I still somehow encountered a problem involving integration of Bessel functions. Currently the problem can be reduced to finding an point-wise upper and lower ...
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28 views

Non trivial sum involving Gamma functions and hypergeometric

I have the following sum that I wish to evaluate: $$ \sum_{n=0}^{\infty} \left(\int_0^1 du\,u^{1+n+\epsilon}\right) \frac{\Gamma(n+2) \Gamma(1+\epsilon)}{\Gamma(n+3+\epsilon)} {}_2F_1\left(1, n+2, ...
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19 views

Integral in terms of hypergeometric

I am considering the following integral $$\int_{-t}^{\infty} \frac{dy}{y-s} \frac{1}{y^2} \frac{1}{y^{\epsilon}} {}_2F_1(1,1,2+\epsilon, -t/y)$$ Rewriting the hypergeometric using its integral ...
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107 views

Two formulae for $\pi$, probably known?

I stumbled upon (in the literature) two identities for $\pi$, but they were not referenced as they are probably well-known. Hoping someone could point out who found them first. Basically, the ...
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39 views

Relation between hypergeometric functions $_2F_1$ of $z$ and $\frac{1}{1+z}$.

What are the relation between hypergeometric functions $_2F_1$ of $z$ and $\frac{1}{1+z}$. Specifically, I need a transformation that transforms: $_2F_1\left(a,b;c; -\sinh^2(x)\right)$ to ...
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1answer
70 views

Relation between these expressions involving the Hypergeometric function and the Gegenbauer polynomials

I would like to find the relation between the solutions of a differential equation obtained by two different authors. The first solution is given in terms of the hypergeometric function $_2F_1$: ...
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43 views

Product of two hypergeometric functions

For $\Re a, \Re b, \Re c, \Re a', \Re b', \Re c'>0$, I would calculate the following product $$ {}_2 F_1(a, b; c; x^{-1}) \times \, {}_2 F_1(a', b'; c'; 1-\frac{x}{y}) $$ For all $y>x>1$. ...
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26 views

Hypergeometric conditional probability

Question: Bowl 1 contains 8 red and 6 black marbles. Four marbles are randomly selected from bowl 1 and placed in bowl 2. Then one marble is randomly selected from bowl 2. What is the probability ...
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35 views

Integration of hypergeometric functions?

I would calculate the following integral \begin{equation} I_x = \int_{0}^{1} y^{b+\mu-1} (1-y)^{\nu-1}\, _2F_1(a,b+\nu +\mu;c; xy) \, dy. \end{equation} Such that $\quad \Re a,\Re b,\Re \mu, \Re \nu ...
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70 views

A series related to Catalan numbers

Recall the definition of Catalan numbers: $$C_n=\frac1{n+1}\binom{2n}n=\frac{2^n(2n-1)!!}{(n+1)!}.\tag1$$ Now consider the following series with a parameter $n\in\mathbb N^+$: ...
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129 views

Generating function for the number of surjections

Let $S_k^n$ be the number of possible surjections from a set of $k$ elements to a set of $n$ elements. We have $$\begin{align} &S_0^0 = 1,\qquad\forall k>0: S_k^0 = 0,\\ &S_n^n = ...
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34 views

Using hypergeometric functions to solve this integral

After looking at calculations, I realized that the exponent needs to be 1/4 instead of -1/4 I have this equation and I am trying to solve the integral of it. $$((R^2) - (y^2))^{1/4} dy$$ I tried ...
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70 views

Chudnovsky binary splitting and factoring

In this article, a fast recursive formulation of the Chudnovsky pi formula using binary splitting is given. For $S(a,b)$: $$ m = (a + b) / 2 $$ $$P(a,b) = P(a,m) P(m,b)$$ $$Q(a,b) = Q(a,m) Q(m,b)$$ ...
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1answer
56 views

Show that $\int_{-1}^1 \frac{du}{(1-u^4)^{1/4}} = \frac{\pi}{\sqrt 2}$.

Show that $\int_{-1}^1 \frac{du}{(1-u^4)^{1/4}} = \frac{\pi}{\sqrt 2}$. Not sure if it is helpful to anyone but $\int \frac{du}{(1-u^4)^{1/4}} = f^{-1}(x)$ is a solution to the differential equation ...
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17 views

Partial derivatives of the hypergeometric ${_2F_1}$

Do formulas for the partial derivatives of the hypergeometric function ${_2F_1}$ exist? I mean I am interested in $$\frac{\partial}{\partial a}\ {_2F_1}(a,b,c,z)$$$$\frac{\partial}{\partial b}\ ...
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3answers
122 views

Evaluating an infinite sum involving possibly hypergeometric terms

I was considering the following infinite sum $$ A(n) = \sum_{k=2}^{\infty}\left[\frac{(-1)^{k+n-1}}{k^n}(0k -1)(k-1)(2k-1)...((n-1)k-1) \right] $$ Some cases: $$ A(1) = ...
5
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140 views

Relation between hypergeometric and gamma functions

Show that, for a positive integer $n$, ...
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1answer
29 views

Binomial experiment vs Hypergeometric experiment

I was doing a problem from a textbook. It said that there were 209 waste treatment facilities in the US and 8 of them treat hazardous waste on site it then said that if 10 were randomly sampled then: ...
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28 views

Confluent Hypergeometric Function behaviour when $x \rightarrow \infty$

I'm very new to confluent hypergeometric functions so please bear with me. What I'm trying to prove is that $$M \left (\frac{c+m}{2m}, \frac{1}{2}, \frac{m}{2d}x^2 \right ) \rightarrow \infty \quad ...