# 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.

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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 ...
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### 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 ...
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### 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 ...
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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 ...
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### 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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### 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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### 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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### 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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### 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% ...
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### 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 ...
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### 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{...
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### 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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### 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 \...
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 $M\left( {a,b,c} \right) = \sum\limits_{n = 0}^\infty {\frac{{{{\... 0answers 26 views ### Confluent hyper-geometric differential equation I have attempt to solve a differential equation of the form$\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 \...
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 ...