# Questions tagged [hypergeometric-function]

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 (ODE). Every second-order linear ODE with three regular singular points can be transformed into this equation.

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### Pfaff formula in the degenerate case

The Pfaff transformation for hypergeometric functions is true under the assumption that the parameters are not negative integers. But, as far as I understand, it also holds sometimes in the degenerate ...
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### Unexpected (incorrect) solution to Lagrange Inversion solution to $x^4 - x^3 - x^2 - x - 1 = 0$ about the solution near $x = 2$

I am developing generalized hypergeometric solutions for a set of such polynomials. With this example we can write $x^4 - x^3 - x^2 - x - 1 = \frac{x^5 - 2 x^4 + 1}{x - 1}$. Lagrange Inversion ...
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### Behavior of the Gaussian Hypergeometric function when one of its arguments approaches $0$ or $1$

For two positive integers $a,b$, denote by $_2F_1(a,1-b;a+1;z)$ the Gaussian Hypergeometric function whose first three parameters are fixed at $a,1-b$ and $a+1$, respectively. such function is linked ...
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### Trying to simply an equation to find the limit of the hypergeometric distribution

My textbook states that, $\frac{((1-p)N)^{(n-x)}}{N^{(x)}(N-x)^{(n-x)}} = (1-p)^{(n-x)}$ where $a^{(b)} = aP_b$ I tried expanding the numerator and denominator, and then factoring out the $(1-p)*$ ...
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### Simplifying this hypergeometric function with variable b

The student-t CDf has a hypergeometric function as a component $$_2F_1\left(\frac{1}{2}, \frac{\nu + 1}{2}; \frac{3}{2}; -\frac{x^2}{\nu}\right)$$ where $\nu$ is the distributions degree of freedom. ...
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### Hypergeometric functions and modular forms

May I please ask if it is possible to write Hypergeometric functions in terms of Jacobi theta functions? I am trying to bring the following Hypergeometric expression (pg.9, eq 4.3) into a known ...
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### What is the amount of draws necessary to see all red cards from a standard deck of 52 cards if you draw 5 cards from the deck?

Problem abstraction A standard deck of $52$ cards has $26$ red cards: it has $13$ hearts, $13$ diamonds, as well as $26$ black cards ($13$ spades, as well as $13$ clubs). Let us draw $5$ cards from ...
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### Formula for ${}_2F_1(h,-n, 2h; 2)$.

Does anyone know a closed form for the following evaluations of the Hypergeometric function $${}_2F_1(h,-n, 2h; t^{-1})$$ with $h>0,n\geq 0$ both integers and $0\leq t\leq 1$ a real. For the most ...
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