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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15 views

Some properties about the Kampé de Fériet function

Prove that according to http://www.iosrjournals.org/iosr-jm/papers/Vol8-issue6/L0866770.pdf?id=7287, this special case of Kampé de Fériet function can reduce to Generalized hypergeometric function: ...
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1answer
28 views

Approximation of a generlized hypergeometric function for large parameters

I am looking for an approximation of a generalized hypergeometric function of the type $\, _0F_4$. I've stumbled upon the following approximation : assuming that none of a1,a2,…,ap is a nonpositive ...
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1answer
67 views

How to show this equality holds?

I need to show that the following equality holds for any integer i,j,m,n and p where p is probability (0<=p<=1) Could you please help me?I think I should use hyper-geometric function but I could ...
2
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1answer
32 views

Asymptotic series of Confluent Hypergeometric function $U(a,1,z) $ as $z \to 0$

Consider the Confluent hypergeometric function $U(a,b,z)$, which is a solution of the Kummer's Equation : $$zw''+(b-z)w'-aw=0$$ it has the following integral representation when $- \pi/2 < \arg ...
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1answer
66 views

how to compute the integral $\int_0^1 (1-x^p)^n dx$?

For constants $n$ and $p$, how to compute the integral $\int_0^1 (1-x^p)^n dx$ ? I saw a solution using hypergeometric function and another using incomplete beta function here: ...
8
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1answer
139 views

Expressing ${}_2F_1(a, b; c; z)^2$ as a single series

Is there a way to express $${}_2F_1\bigg(\frac{1}{12}, \frac{5}{12}; \frac{1}{2}; z\bigg)^2$$ as a single series a la Clausen? Note that Clausen's identity is not applicable here.
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2answers
31 views

What is the probability of getting 2 new balls out of 3 at the third time of picking 3 out of 12?

Here is the question: (don't bother with the title if you don't get the question) You are going to pick 3 balls out of 12 and put it back every time. The ball that gets picked is considered an old ...
2
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1answer
20 views

Simplifying hyperbolic complex function

I am trying to simplify the following fraction: $$\frac{2\tanh(z)}{1-\tanh(z))}$$ I know it equals: $e^{2z}-1$ from wolframalpha, but I have no idea why. I imagine I should be using some trig ...
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0answers
12 views

Wilf-Zeilberger context with an extra parameter

Define two sequences $(A_p(X)),(B_p(X))$ of polynomials by $A_p(X)=(-2p-8)X^2+(3p^2+22p+40)X-(p^3+11p^2+40p+48)$ and $B_p(X)=(4p+12)X^2-(3p^2+21p+34)$. Let $(g_p)_{p\geq 1}$ be the sequence of ...
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1answer
46 views

Convergence of a series $\sum_{n=0}^{+\infty} {2n\choose n} {\sum_{k=0}^n}{n\choose k} {n\choose {n-k}}({1\over3})^{2k}({1\over6})^{2n-2k}$

I have trouble showing the series $$\sum_{n=0}^{+\infty} {2n\choose n} {\sum_{k=0}^n}{n\choose k} {n\choose {n-k}}\bigg({1\over3}\bigg)^{2k}\bigg({1\over6}\bigg)^{2n-2k}$$ converges or not. I tried to ...
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1answer
35 views

Series representation of hypergeometric function reciprocal?

Basically, can you represent $\dfrac{1}{_2F_1(a,b;c;z)}$ as some kind of power series? EDIT: This question came from something I was doing with generating functions were ...
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1answer
50 views

Evaluating $\int^{x_2} _{x_1} \sqrt{a - b x^m} ~dx $

Is there any way to evaluate $$\int^{x_2} _{x_1} \sqrt{(a - b x^m)}~ dx $$ where $x_{12} = \pm (a/b)^{1/m}$ without elliptic functions or hypergeometry? Or just any way to solve it. My attempt is to ...
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1answer
45 views

Integration of this using a multi-dimensional hypergeometric function

I want to try and potentially use a Dirichlet - Hypergeometric Function in order to compute the following integral. I would appreciate some help as I'm stuck on how to go about this is a ...
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0answers
29 views

Definite integral with Bessel functions

Show that for $Re(\lambda)>0,Re(\mu)>0$ it holds the following identity $\int_0^a x J_\lambda(2a)I_\lambda(2x) J_\mu(2 \sqrt{a^2-x^2}) I_\mu(2 \sqrt{a^2-x^2}) dx = \frac{a^{2 \lambda + 2 \mu + ...
3
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0answers
44 views

Closed form expression for a sum

I want to calculate a sum of the form $$\sum_{k=0}^m \frac{\Gamma[m+1+\alpha-k]^2}{\Gamma[m+1-k]^2}\frac{\Gamma[x+k]}{\Gamma[x]k!}$$ where $m>0$ and belongs to integers and $\alpha$ takes half ...
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0answers
14 views

Is there a way to express the reciprocal of the hypergeometric function 2F1(a,b;c;z) in terms of a b and c?

I'm trying to use generating functions to get the value of some coefficients, namely $\displaystyle\sum_{m\geq 0} f_{2m}x^m = 1 - \Big(\displaystyle\sum_{m\geq 0} u_{2m}x^m\Big)^{-1}\\$ and $u_{2m} ...
2
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1answer
33 views

Proving that $\lim \limits_{b \rightarrow \infty} F(a,b,\frac{1}{2};\frac{z^2}{4ab})=\cosh z$

I am trying to prove that $\lim \limits_{a,b \rightarrow \infty} F(a,b,\frac{1}{2};\frac{z^2}{4ab})=\cosh z$ . Here $F$ is the hypergeometric function. Here because of two limits I am unable to ...
2
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1answer
20 views

Proving that $\lim \limits_{b \rightarrow \infty} F(1,b,1;\frac{z}{b})=e^z$

I am trying to prove that $\lim \limits_{b \rightarrow \infty} F(1,b,1;\frac{z}{b})=e^z$ without using dominated convergence theorem. Here $F$ is the hypergeometric function. I have been able to ...
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0answers
30 views

Integral over a product

How the following integral can be computed: $I = \int_0^1 (x-a_1)^{b_1}(x-a_2)^{b_2}...(x-a_n)^{b_n} dx$? Here, $a_i,b_i$ are real numbers and $n$ is a natural number. Are there any techniques for ...
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0answers
34 views

Closed form expression for 3F2 with positive unit argument

Is there any closed form expression for the Hypergeometric function ${}_3F_2(-n,-n,c;-d/2-n,-d/2-n;1)$ for $n>0$ and $d>0$. The parameter $c$ can be both positive and negative.
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1answer
23 views

${}_2F_1$ as a FINITE series: How is this result obtained?

I am using the following result: which I have found in this link: http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric2F1/03/06/07/10/0001 I am trying to find out how this result ...
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0answers
27 views

About the extensions from Confluent Hypergeometric Function of the Second Kind

I know that $\int_0^\infty t^{a-1}(1+t)^{c-a-1}e^{-yt}~dt=\Gamma(a)U(a,c,y)$ , where $\text{Re}(a),\text{Re}(y)>0$ . How about $\int_0^\infty t^{a-1}(1+t)^{c-a-1}(1+xt)^{-b}e^{-yt}~dt$ and ...
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0answers
53 views

How many solutions does Riemann's P-symbol describe?

The Papperitz-Riemann P-symbol $$ \tag 1 y(z) = P \left\{ \begin{matrix} z_1 & z_2 & z_3 & \; \\ \alpha_1 & \alpha_2 & \alpha_3 & z \\ \beta_1 & \beta_2 & ...
2
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0answers
26 views

Limit involving hypergeometric function.

I'm trying to prove a limit involving a Hypergeometric function, which seems to hold numerically, but I'm stuck. I have the following function: $ f(A,B)=\frac{2A\pi ^{5/2} (-1)^B}{{\left(A!\right)^2 ...
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3answers
75 views

Is there a simple expression for ${}_2F_1(a,a+\tfrac{1}{2};a+1;z)$?

I have been searching through some books and also this but I have not succeed. I wonder if there is a simple equivalent form for ${}_2F_1(a,a+\tfrac{1}{2};a+1;z)$, in terms of elementary functions ...
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0answers
25 views

Find rule generating seven 6-tuples (k=0,…,6) composed of two sets of consecutive relatively prime fractions

I have the seven 6-tuples listed below, functions of k (0,…,6), composed of four consecutive relatively prime fractions involving fifths, and two involving sixths. I would like to find a rule that ...
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3answers
303 views

Closed form of factorial and cascading power sum

Consider the following sum: $$ \sum_{i =0}^{j} \left( \frac{(j-i)^ix^i \ln(x)^{(j-i)}\ln(x)^i}{(j-i)!i!} \right) $$ I can simplify the sum to: $$ \ln(x)^j\sum_{i =0}^{j} \left( ...
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1answer
13 views

hypergeometric distribution and random sampling

Is there any simple and fast algorithm (to be implemented in Javascript) to obtain a sample from the hypergeometric distribution? My needed sample size is very large (100,000,000).
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1answer
29 views

Expectation of the number balls are drawn

A box contains 6 white balls and 4 black balls. Balls are drawn from the box without replacement until either a white ball is drawn or 3 balls have been drawn. Find the expected number of balls that ...
3
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1answer
39 views

Evaluate $\int^\infty_0 t^{a+b-1}(t+1)^{-b-1} U(a+2,a-b+2,ct)dt$

Evaluate $$ \int^\infty_0 t^{a+b-1}\left(t+1\right)^{-b-1} U\left(a+2,a-b+2,ct\right)dt $$ under the condition $a>0$, $b>0$ and $c>0$, where $U(\cdot,\cdot,\cdot)$ denotes the ...
2
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1answer
158 views

Definite integral of polynomial times exponential times hypergeometric function of imaginary argument

How would one deal with such an integral? $$I(k)\equiv \int_0^\infty r^n e^{-r(1+\mu)} e^{-{\mathrm i} kr}\:{}_1F_1({\mathrm i}/k+1;2;2{\mathrm i} kr) \, \mathrm{d} r$$ Here $n\in\{0,1\}$, $\mu\in ...
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0answers
36 views

strange hypergeometric coefficients in Concrete Mathematics book

I don't understand how in this famous book they obtained hypergeometric coefficients for $$ \sum_{k\leq n} z^k \binom{n-k}{k}.\tag{5.74} $$ They say it is $\displaystyle F{-n,\ 1+2\lceil ...
4
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3answers
54 views

Hypergeometric 2F1 with negative c

I've got this hypergeometric series $_2F_1 \left[ \begin{array}{ll} a &-n \\ -a-n+1 & \end{array} ; 1\right]$ where $a,n>0$ and $a,n\in \mathbb{N}$ The problem is that $-a-n+1$ is ...
4
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2answers
70 views

Evaluate $ \int^\infty_0\int^\infty_0 x^a y^{1-a} (1+x)^{-b-1}(1+y)^{-b-1} \exp(-c\frac{x}{y})dxdy $

Evaluate $$ \int^\infty_0\int^\infty_0 x^a y^{1-a} (1+x)^{-b-1}(1+y)^{-b-1} \exp(-c\frac{x}{y})dxdy $$ under the condition $a>1$, $b>0$, $c>0$. Note that none of $a$, $b$ and $c$ is ...
2
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1answer
33 views

How to write the integral form of a Fox H function when n=p?

I have a questions regarding Fox H functions For a general Fox H function if $n=p$, when we write the integral form, does the term with the product from $p+1$ to $n$ disappear ?
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0answers
40 views

Proof that $\int_0^\infty t^\gamma e^{t/2} \operatorname{erfc}(\sqrt{t}\,) \,dt$ leads to hypergeometric function

I'm looking for the proof of $$\DeclareMathOperator\erfc{erfc} \int_0^\infty t^\gamma e^{t/2} \erfc(\sqrt{t}\,) \, dt = \frac{2^{-2 \gamma -1} \Gamma (2 \gamma +2) \, _2F_1\left(\gamma +1,\gamma ...
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1answer
52 views

Can this series be expressed as a Hyper Geometric function

I am trying find a Hyper Geometric function representation of the following series. $$\sum\limits_{k=0}^{\infty} \frac{a^k}{k!}\frac{\Gamma \left(\frac{b+k}{c}\right)}{\Gamma ...
5
votes
2answers
92 views

Closed form for integral of inverse hyperbolic function in terms of ${_4F_3}$

While attempting to evaluate the integral $\int_{0}^{\frac{\pi}{2}}\sinh^{-1}{\left(\sqrt{\sin{x}}\right)}\,\mathrm{d}x$, I stumbled upon the following representation for a related integral in terms ...
1
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1answer
45 views

radius of convergence of hypergeometric function

Looking up information on the Bessel function there is a formula as $|z| \to \infty$: $$ I_0(z) \approx \frac{e^z}{\sqrt{2\pi z}} {}_2F_0( \tfrac{1}{2}, \tfrac{1}{2}, \tfrac{1}{2z}) = ...
2
votes
1answer
53 views

Problems with a Definite Integral Resulting in the Hypergeometric Function

I am attempting to find $$\int_{a}^{b}(\cos^n{x} )dx$$ by using $$\int(\cos^n{x}) dx=-\frac{\cos^{n+1}x}{n+1}{_2}F_1(\frac{n+1}{2},\frac{1}{2};\frac{n+3}{2};\cos^{2}x)+c.$$ However, when $a$ and $b$ ...
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1answer
37 views

asymptotic expansion for Bessel function $I_0(z)$ in terms of Gauss hypergeometric functions ${}_2F_1$

On the Wikipedia page one can asympotoic formula of the Bessel function $$ I_0(z) \propto \frac{e^z}{\sqrt{2\pi z}} $$ On the Wolfram page there is a more detailed asymptotic formula for the Bessel ...
3
votes
1answer
101 views

Any way to simplify integral of Confluent Hypergeometric Function of the First Kind?

The integral is this: $$\int_{-\log n}^{0}e^{t(1-s)} \cdot z \cdot {}_1F_1(1-z, 2, t) dt $$ Is there a way to write this in terms of special functions that eliminates the integral and doesn't use ...
0
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1answer
28 views

Is $\frac{(\alpha)^n (\beta)^n} {(\delta)^n} > \frac{(\alpha+1)^n (\beta+1)^n} {(\delta+1)^n}$ for any $n$ ?(in this specific case)

Let $\alpha=(K-1)a$, $\beta=K$ and $\delta=Ka$, where $K>a\ge 1$ ($\delta>\alpha>\beta$). Can we claim that $\frac{(\alpha)^n (\beta)^n} {(\delta)^n} > \frac{(\alpha+1)^n (\beta+1)^n} ...
0
votes
0answers
28 views

Analyze the variation of $f(x)=(1+\frac{1}{x})^{-K}$ $_2F_1((K-1)a,K,Ka,\frac{1}{1+x})$ w.r.t. $x$

Is there any way to analyze the variation(w.r.t. $x$) of the following function: $f(x)=(1+\frac{1}{x})^{-K}$$ _2F_1((K-1)a,K,Ka,\frac{1}{1+x})$, where $ _2F_1$ is the Gauss' Hypergeometric Function, ...
3
votes
1answer
55 views

Identities for hypergeometric functions ${}_2F_1$ with z=1/2

Is there a closed form (or approximation) for a hypergeometric function of form: $_2F_1(1,b+c;c;\frac{1}{2}) \quad \text{where} \; b,c \in \mathbb{N}$ ? I researched all identities in ...
0
votes
0answers
31 views

an integral involves mutiply of two Kummer Confluent Hypergeometric function

i met an integral involving the mutiply of two Kummer Confluent Hypergeometric functions as follows: $$\int_{ - \infty }^\infty {\frac{1}{{{x^\alpha }{{\left( {x + k} \right)}^\beta ...
1
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1answer
18 views

Identity relating hypergeometric function and Legendre polynomial

In my notes I have written down the following relation: $_2F_1(a,a+\frac{1}{2};c;z)=2^{c-1}z^{(1-c)/2}(1-z)^{-a+(c-1)/2}L_{2a-c}^{1-c}\big(\frac{1}{\sqrt{1-z}}\big)\ ,$ where $_2F_1(a,b;c;z)$ is the ...
1
vote
1answer
106 views

Evaluate $\int^\pi_{-\pi}(1-a\cos\theta)^{-b-2}\log(1-a\cos\theta)d\theta$

$$ \int^\pi_{-\pi} \left(1-a\cos\theta\right)^{-b-2} \log\left(1-a\cos\theta\right)d\theta$$ Under the condition $0<a<1$ and $b>0$. Mathematica found the following form. $$ ...
0
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0answers
51 views

Show that this Hypergeometric Function equal to this gamma function

I have a question related to hypergeometric functions: Show that ...
0
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0answers
48 views

Asymptotics of Hypergeometric series

Suppose we have a $_pF_q$ hypergeometric series that terminates for all $r>t$ for some positive integer $t$, and consider the expression \begin{multline*}\lim_{n\to\infty} ...