# 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.

1,281 questions
Filter by
Sorted by
Tagged with
61 views

• 107
1 vote
64 views

• 53
93 views

### Laplace transforms of Appell functions F1, F2, F3, F4 relative to one of the two variables

Appell's two-variable functions $F_1, F_2, F_3$ and $F_4$ are known to have numerous uses in applied mathematics, notably mathematical Physics. I am looking for generalized Laplace transforms (if they ...
• 362
44 views

### Generalization of Hermite polynomials for negative, fractional degree.

After solving a differential equation I've gotten a solution given as a linear combination of Hermite polynomials and confluent hypergeometric functions. The caveat is that the Hermite polynomials ...
• 31
45 views

### Does this second order linear ODE admits a solution?

I am considering the following ODE in $x$: $$x^2v''(x) + \alpha x v'(x) + \beta v(x)=0,$$ where $\alpha,\beta \in\mathbb{R}$ and one of the boundary conditions says $v(0)=\gamma>0$. If I consider ...
• 359
1 vote
35 views

### Can one express hypergeometric function with 1 as the first and -1 as the last argument by some factorials or gamma functions [closed]

As a result of integration in Mathematica I got: \begin{equation} _{2}F_{1}(1,\frac{1}{2}(2+n-r),\frac{1}{2}(2-n-r),-1)\quad n,r\in\mathbf{N} \end{equation} I try to express it as some factorials (as ...
1 vote
38 views

• 82
1 vote
60 views

### Evaluate $\sum^{\infty}_{n=0} (-1)^n \frac{z^{4n+1}}{(4n+1)!}$

I want to evaluate $\sum^{\infty}_{n=0} (-1)^n \frac{z^{4n+1}}{(4n+1)!}$ but while doing my research, I noticed that \begin{align*} &\sin(z) = \sum^{\infty}_{n=1} (-1)^n \frac{z^{2n+1}}{(2n+1)!} \\...
• 1,596
1 vote
69 views

### Ideas/Methods for solving an non linear DE with non constant coefficients

I have to solve the following equation. $2 A x^2 u''(x)-2 A x u''(x)+2 A x u'(x)-A u'(x)+B u(x)+\frac{C u(x)}{x}$ I have been thinking in a way of solving this since it is pretty complicated. I tried ...
• 53
167 views

### Transformation Identities of the $_2F_1$ function

From Wolfram Functions we have the following identities for the hypergeometric function $_2F_1$: \begin{align} _2F_1\left(a,c-b;c;\tfrac{z}{z-1}\right)&=(1-z)^a\,_2F_1(a,b;c;z)\tag1\\ _2F_1\left(...
• 15.9k
1 vote
50 views

### Solution of a hypergeometric equation

I am thinking the differential equation $(1+e^x)\frac{d^2 y}{dx^2}+(2+e^x-\delta)\frac{dy}{dx}+y=0$. Here $\delta$ is a parameter I want it to approaches to zero. I think this can be transformed into ...
• 41
I found the following hypergeometric function as the antiderivative of some function in the interval where x is between 0 and 1. ${}_2F_1(-\frac{1}{k}, \frac{1}{k}; 1 + \frac{1}{k}; x^k)x$ Or by ...
I'm interested in a closed-form (special functions included) for the integer moments of the Poisson distribution. Let $X\sim\operatorname{Poisson}(\lambda)$. Then we have for the moment generating ...