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.

Filter by
Sorted by
Tagged with
1
vote
1answer
19 views

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
votes
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}{...
1
vote
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) $$\...
0
votes
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
votes
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
votes
0answers
70 views

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
vote
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 ...
3
votes
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, ...
-1
votes
1answer
47 views

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.
1
vote
1answer
23 views

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 ...
0
votes
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
votes
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
votes
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,...
1
vote
0answers
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 ...
0
votes
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 ...
1
vote
1answer
43 views

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,...
1
vote
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
votes
1answer
62 views

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
votes
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
vote
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+...
1
vote
0answers
81 views

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 ...
0
votes
0answers
29 views

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 ...
1
vote
1answer
124 views

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}{\...
0
votes
1answer
38 views

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
votes
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: ...
1
vote
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....
0
votes
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 $$ (-1)^{u+1}\beta\left(\frac{1}{2},u,1-u\right)=\log 2-\mathfrak{X}_{u-1}\exp(-\psi_{u-1}) \label{a}\tag{1} $$...
1
vote
0answers
33 views

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+\...
6
votes
0answers
106 views

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} \...
1
vote
0answers
53 views

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''' - [...
0
votes
2answers
43 views

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
vote
1answer
72 views

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}{...
0
votes
0answers
17 views

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 ...
0
votes
1answer
29 views

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^...
1
vote
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
votes
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 ...
1
vote
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
vote
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
vote
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
vote
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
votes
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
vote
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 ...

1
2 3 4 5
23