Questions tagged [hypergeometric-function]
Hypergeometric functions often refer to a family of functions ${}_p F_q$ represented by a corresponding series, where $p,q$ are non-negative integers. The case ${}_2F_1(a,b;c;z)$ is a special case of particular importance; it is known as the Gaussian, or ordinary, hypergeometric function.
1,101
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Closed form for this power series looking like an hypergeometric?
I would like to "resum" the following expression:
$$\sum_{k=0}^a \frac{(-a)_k (-b)_k}{(c)_k} x^k\,, \tag{1}$$
with $a, b, c$ positive even numbers and $x > 0$ real. Is there a known ...
2
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0answers
37 views
A hypergeometric beast: Simple closed-form for combination of $_4F_3(1)$?
Conjecture:
$$
\frac{1}{4}
-\frac{1}{4}{_4F}_3\left({-\frac{1}{5},\frac{1}{5},\frac{2}{5},\frac{3}{5}\atop\frac{1}{4},\frac{1}{2},\frac{3}{4}};1\right)
-\frac{1}{\sqrt[4]
{5}}{_4F}_3\left({\frac{1}{...
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1answer
33 views
Understanding the behavior of HypergeometricPFQ
I am using HypergeometricPFQ functions (more exactly $_3F_2$) as approximants for other more complicated functions. Here are three of them (corresponding to N=3, N=4 and N=5, respectively, in the ...
4
votes
2answers
104 views
Find a closed-form solution to the following summation
I am solving a summation that appears in a paper, it claims that
$$\sum_{n=1}^{\infty} \binom{2n}{n+k}z^n=\bigg(\frac{4z}{(1+\sqrt{1-4z})^2}\bigg)^k$$
I found this identity here in equation (66)
$$\...
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2answers
35 views
Inequality for Binomial distribution function
Suppose $F(y;n,p)$ is the binomial distribution function, i.e. the probability that there are $y$ or fewer successes out of $n$ independent Bernoulli trials each with success probability $p$.
Is it ...
2
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2answers
75 views
Estimating alternating sum of product of binomial coefficients $\sum_{i=0}^{k-1} \binom{n}{m+i} \binom{m+i}{k} \binom{k-1}{i} (-1)^i$
I am interested in getting a lower bound on the expression
$$\sum_{i=0}^{k-1} \binom{n}{m+i} \binom{m+i}{k} \binom{k-1}{i} (-1)^i .$$
for $1 \le k,m \le n$. In particular, $m = n/2 + C\sqrt{n}$ for ...
4
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0answers
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Weighted Twist of Vandermonde's Identity
I've been stuck on the following expression, trying to determine a "simple" closed form for it:
$$\sum_{i=0}^{m-1}\frac{1}{m-i}{x-a \choose i-a}{2m-x-b \choose m-1-i-b}.$$
I feel like there ...
1
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1answer
59 views
Asymptotic Expansion of Confluent Hypergeomtric Function U(a,b,x) for Large a and b such that b-a is finite [closed]
I am learning about Confluent Hypergeometric function U(a,b,x). In my work it appears as a solution to the equation of motion of a scalar field in a spacetime with Lifshitz symmetry or Hyperscaling ...
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1answer
54 views
Hypergeometric functions for $\sum_{k} {{a+b} \choose{a+k}}{{b+c} \choose{b+k}}{{c+a} \choose{c+k}}(-1)^k$ with $a,b,c\geq 0$
The sum is: $\sum_{k} {{a+b} \choose{a+k}}{{b+c} \choose{b+k}}{{c+a} \choose{c+k}}(-1)^k$ with $a,b,c \geq 0$.
The equivalent hypergeometric function can be found in page 214, Concrete Mathemtics, ...
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1answer
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A doubt about alternating combinatorial series [closed]
Which functions $f(k,n)$ satisfy
$\sum_{k=0}^{k=n}(-1)^kf(k,n)=0$ for every n?
One example is ${n\choose k}$, but I want to know more examples.
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1answer
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Continuation of Hypergeometric Function when a - b is natural number
I am currently implementing the 2F1 Gaussian hypergeometric function numerically, and need to know its continuation for $ |z| > 1 $.
I have researched this and found this nice formula in the ...
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1answer
23 views
Hypergeometric distribution, distribution function
I'm looking for a way to get a distribution function for having 1211 tokens in a bowl. I'm picking one by one, randomly out of the bowl without putting it back. And I want to have a 75% or 90% ...
3
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0answers
57 views
Infinite sum related to hypergeometric series
I was working on some integrals and I came across the following series: $$\sum_{k=1}^{\infty}\frac{1}{k(k+n)!}$$ Wolfram Alpha evaluates it to be $$\frac{_2F_2\left(1,1;2,n+2;1\right)}{(n+1)!}$$
Which ...
2
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1answer
111 views
Branch cuts of the Hypergeometric function with one complex parameter
I am trying to understand if the hypergeometric function with the following choice of parameters has any branch cut in the complex plane
$$
\ _{2}F_{1}\left(a,b - i \rho ,c ,1-z^{2}\right)
$$
here $a,...
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55 views
Estimating truncation error in hypergeometric series
If $\sum_{k=0}^\infty s_k$ is a hypergeometric series, $g(n)=\frac{s_{n}}{s_{n-1}}$ is a simple expression which can be written $g(n)=\sum_{k=0}^\infty \frac{b_k}{n^k}$.
I want to estimate the ...
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0answers
36 views
How to integrate this function involving Meijer G and Exponential?
I am stuck in solving this integral:
$$\int_0^{\infty} e^{-s\gamma}\cdot\gamma^{-1}\cdot G^{2,0}_{0,2}[\frac{\alpha^2\gamma}{\eta^2\bar{\gamma}}|\frac{-}{\alpha,\alpha}]d\gamma$$
Any help in this ...
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1answer
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Hpergeometric Reduction with Mathematica
We know that:
$$\,_2F_1(a,b,b,z)=(1-z)^{-a}$$
But putting $b<0$ with $b$ integer Mathematica does not use the previous formula but generates the hypergeometric polynomial. For instance:
$$\,_2F_1(a,...
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1answer
59 views
Evaluating $\sum_{k=1}^{a}\frac{-1-H_k}{k(1-e)^k}$
Question :
My attempt:
Let $a=17399172$
$$\begin{align}
&\sum_{k=1}^{a}\frac{-1-H_k}{\log_2\left(\sum_{j=0}^{k}\left(\ln\left(e^{C_j^k}\right)\right)\right)(1-e)^k} \\
&= \sum_{k=1}^{a}\frac{-...
5
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1answer
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Expansion of Confluent Hypergeometric Function in terms of $\operatorname{erfi}(x)$
I have the following confluent hypergeometric function: $_1F_1\left(2(m+1),\frac{1}{2},-x^2\right)$.
By using Mathematica, I know that for values of $m=0,1,2,...$ this function expands into a power ...
9
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0answers
161 views
When exactly is the principal AGM equal to the optimal AGM?
Definitions
Following the terminology of SageMath, let the principal arithmetic-geometric mean, $\operatorname{AGP}$, of $(a,b)\in\mathbb{C}^2$ for $a\ne 0$, $b\ne 0$, $a\ne\pm b$ be defined as ...
3
votes
1answer
54 views
Trying to find a 3-variable generating function
For months, I've been trying to find a closed form for a generating function for this three-variable sequence
$$a(G,M,T)=(-1)^{G+T+M} 2^{M-1} \frac{(M-1)!}{(2M-2)!} \binom{M}{T} {}_3F_2(M+1, -\frac{G}{...
1
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1answer
61 views
Looking for a hypergeometric 2F1 identity
I'm trying to express
$$_2F_1\left(\frac{1}{2} , 1;\,\frac{3}{2}+m;\,z\right), \quad m\in\mathbb{Z}\quad\text{and}\quad m\geq0$$
in terms of $\tanh^{-1}(\sqrt{z})$ and $\sqrt{z}$, basically ...
4
votes
2answers
115 views
Binomial identity involving central binomial coefficient
I came across this nice binomial identity
$${\sum}_{k=0}^{2n} \frac{(-1)^k {2n \choose k} {2k \choose k}}{n+k \choose k} = 1$$
This is equivalent to the hypergeometric function ${}_2 F_1(-2n, 1/2; n+...
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How is this beautiful function obtained? $2^s 3^{\frac{s+1}{2}} \pi\; {}_2{F}_1\left(\frac{1}{2},-\frac{s+1}{2};1;1-\frac{K^2}{12}\right)$
From this integral (where $K$ and $s$ are constants):
$$\int_0^{\frac{\pi}{2}} {\left(K^2 {\mathrm{sin}}^2 \;t+12\;{\mathrm{cos}}^2 \;t\right)}^{\frac{s+1}{2}} \mathrm{dt}=$$
This beautiful result is ...
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Why do the elements have to be distinct within hypergeometric distribution [duplicate]
I am trying to understand the hypergeometric distribution
I looked at this example, I can see that it makes sense when we force each object to be distinct, though I am wondering why this is correct ...
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1answer
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Conjecture: $-\partial_b F(b-a,b-a;b;z)$ is increasing in $z$ (proof of special case provided in answers)
See answers below for a proof of a special case to the following conjecture.
I am looking to prove the following conjecture:
Conjecture:
Let $0<b<a<b+1$. Then,
$$
f(z):=-\frac{\mathrm d}{\...
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1answer
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Simplifications to $\sum\limits_{k=1}^{\infty} \frac{(-z)^k}{k!} F_{2,1}\Big(-j,-k,\frac{1}{2},-y\Big)$ [closed]
I want to ask if anyone knows a simplified form to the sum:
$\displaystyle\sum_{k=1}^{\infty} \frac{(-z)^k}{k!} F_{2,1}\Big(-j,-k,\frac{1}{2},-y\Big)$, where $F_{2,1}$ is the Gauss Hypergeometric ...
4
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1answer
151 views
Integral involving two hypergeometric functions of multiple variables.
Let us define an auxiliary functions:
$$f_i(x)=x^{c_i-1}\Phi^{(2)}_2(b_i,b'_i;c_i;v_i x,w_i x)$$
with $\Phi^{(2)}_2(\cdot)$ - standard Humbert function of two variables and $i=1,2$.
The question is: ...
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2answers
82 views
Hypergeometric differential equation for $c=1$ and $a+b+1=0$
I would like to find the base of solutions for the following differential equation,
$$z(1-z)f''(z)+f'(z)+ \alpha \cdot f(z)=0$$
where $\alpha$ is a parameter and the prime indicates derivative w.r.t z....
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0answers
28 views
The special function $\beta(1/2,u,1-u).$
If $u\in\mathbb{N}$ then following identify is established as an exercise in substitution
$$
(-1)^{u+1}\beta\left(\frac{1}{2},u,1-u\right)=\log 2-\mathfrak{X}_{u-1}\exp(-\psi_{u-1}) \label{a}\tag{1}
$$...
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Applying Tannery's theorem to generalised hypergeometric functions
I am thinking about applying Tannery's theorem to some generalised hypergeometric functions, which seems to be a standard method to derive various formulæ. For example,
\begin{eqnarray}
\lim_{n\to+\...
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Justifying: $\frac{\mathrm d}{\mathrm d c}{_2F_1}(a,b;c;x)=\sum_{k=1}^\infty\frac{\mathrm d}{\mathrm d c}\frac{(a)_k(b)_k}{(c)_k}\frac{z^k}{k!}$
I was reading this paper on derivatives of the hypergeometric function $F(a,b;c;x)$ w.r.t. the parameters $a$, $b$, and $c$. In the paper the authors simply state without justification
$$
\tag{1}
\...
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Solving a third order ODE (hypergeometric differential equation)
Consider the following ODE:
$(x^2 + ax + b)y''' + (cx^2 + dx+e)y'' + (fx+g)y'+hy=0$
where $y$ is a function of $x$ and $a,b,c,d,e,f,g,h$ are constants. A special case of this general ODE,
$x^2 y''' - [...
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2answers
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How to evaluate $_2F_1(\frac{1}{2},-\frac{1}{2};\frac{1}{2}; \sin^{2}(x))=\cos(x)$
$$_2F_1(\frac{1}{2},-\frac{1}{2};\frac{1}{2}; \sin^{2}(x))=\cos(x)$$
I plug these values into the definition of the hypergeometric function:
$$_2F_1(a,b,c,x)=\displaystyle\sum_{n=0}^{+\infty}\dfrac{a^{...
1
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1answer
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Does $\sin(x)$ and $\cos(x)$ have representations as hypergeometric series?
Through wolfram and wiki, I've learnt that these elementary functions have a representation as hypergeometric series:
$$_2F_1(\color{red}{1,1;2;-x})=\ln(x+1)$$
$$_2F_1(\color{red}{\color{red}{\frac{1}{...
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hypergeometric distribution of choosing cards to compete a set
A certain brand of tea has a picture card in every packet. The cards form a set of 50 different pictures and are distributed among the packets so that any packet purchased is equally likely to contain ...
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1answer
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Need some assistance on the expression $\sum_{n=0}^\infty \frac{n!x^n}{\prod_{j=1}^n(1+jx)}$
In this thread, Markus Scheuer attempted to prove that:
$$\sum_{n=0}^\infty \frac{n!x^n}{\prod_{j=1}^n(1+jx)}=\frac{1}{1-x}\tag{1}$$
He wrote as followed:
$$\sum_{n=0}^\infty \frac{n!\color{red}{x^...
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0answers
157 views
Infinite series with binomials and a hypergeometric function
I've been struggling for days with Mathematica to sum this, but no luck, any ideas?
$$ \sum_{k=0}^\infty \frac{(-1)^k}{2k+1} \sum_{j=0}^\infty (-x)^j \sum_{r=0}^\infty (xw)^r \binom{k+2}{r} \binom{j+...
3
votes
1answer
59 views
Specific values or asymptotic behaviour of ${}_{1}F_{1}$
I'm interested in analysing the behaviour of the hypergeometric function ${}_{1}F_{1}$ for the following cases:
$${}_{1}F_{1}\Big[\frac{1}{2} - \frac{k}{2}, \frac{1}{2}; -z^2\Big]$$
and
$${}_{1}F_{1}\...
2
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1answer
40 views
How to express error function by confluent hypergeometric function
I am struggling to express error function by confluent hypergeometric function
Which I found on wikipedia: $\displaystyle \operatorname{erf}(x)=\frac{2x}{\sqrt \pi} \Phi \left(\frac{1}{2};\frac{3}{2};-...
4
votes
2answers
52 views
Showing integral $\int_0^k \text{sinh}^{-2/3}(x)\mathrm{d}x$, has a hypergeometric solution.
For my undergraduate physics research I encountered the integral,
\begin{align}
I=\int_0^k\frac{1}{\text{sinh}^{2/3}x}\mathrm dx, \ \ \ k=\text{arsinh}(\alpha),\ \ \ \alpha\in(0,\infty)\tag{1}
\end{...
1
vote
1answer
76 views
Continuity of hypergeometric function $_{1}F_1(a;2;k\pi i)$ in the argument $a$
For $0\leq a \leq 2$, are the hypergeometric functions $_{1}F_1(a;2;k\pi i)$, $_{1}F_1(a;2;-k\pi i)$ continuous in the argument $a$? where $k\in \{1,2,3 \}$. How to prove it?
Are the following ...
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0answers
53 views
Limit of the confluent Lauricella function of three variable
In my research (in communication theory) I ran across the limit of the confluent Lauricella function of three variable, i.e.
$$\lim_{x\to\infty}\Phi_2^{(3)}\left(1,a+x,-x;c+1;t,t\frac{1}{1+b_1},t\frac{...
1
vote
1answer
81 views
Closed formula for sum of exponentials
I'd like to have a closed formula for $\sum\limits_{i=0}^n e^{\sqrt{i}x}=1+e^{\sqrt{1}x}+...+e^{\sqrt{n}x}$, something similar to the formula $\sum\limits_{i=0}^ne^{ix}=\frac{e^{(n+1)x}-1}{e^x-1}$. ...
1
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0answers
63 views
How to see this approximation for hypergeometric function?
I came across this formula in a physics paper(equation 2.28). For $k \gg \omega$, it was stated that the following is true:
${}_2 F_1\hspace{-4pt}\left[\frac{\Delta+i (k+\omega)}{2},
\frac{\Delta-...
1
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0answers
35 views
1F2 Hypergeometric and Bessel function
$$ _{1}F_{2}\left(n-\frac{1}{2};n+1,2n+1;-x^{2}\right)=\frac{2^{2n+1}\Gamma^{2}(n+1)}{x^{2n}(2n+1)}\left(\left(xJ^{'}_{n}(x)+\frac{J_{n}(x)}{2}\right)^{2}+\left(x^{2}-n^{2}+\frac{1}{4}\right)J_{n}^{2}(...
1
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2answers
48 views
How can I derive the Legendre function of first kind in terms of the hypergeometric function?
I was reading in Wikipedia about Legendre's differential equation. I was particularly interested in the simple case of the equation given by
$$
\left(1-x^2\right)y'' -2xy' + \lambda(\lambda+1)y = 0 \...
0
votes
0answers
13 views
Simplification using confluent hyper-geometric function
While trying to simplify an expression of the form $\frac{{M\left( {a + n, b + n, c} \right)}}{{M\left( {a,b,c} \right)}}$...(1)
where
$M\left( {a,b,c} \right) = \sum\limits_{n = 0}^\infty {\frac{{{{\...
0
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0answers
26 views
Confluent hyper-geometric differential equation
I have attempt to solve a differential equation of the form $\left( {1 - z} \right)G''\left( z \right) - \frac{1}{{{c_1}}}\left\{ {{c_1} + {c_2}\left( {1 - z} \right) + {c_3}} \right\}G'\left( z \...
1
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1answer
22 views
Confluent hyper-geometric function
Is there any identity in the confluent hypergeometric function that connects the following?
${}_1{F_1}\left( {a + n + 1,b + n + 1, - c} \right)$ and
${}_1{F_1}\left( {a + 1,b + 1, - c} \right)$ where ...
