# Questions tagged [hypergeometric-functions]

Hypergeometric functions are the analytic functions defined by Taylor expansions of the shape $\sum_{n \geq 0} a_n x^n$, where $a_{n+1}/a_n$ is a rational function of $n$. This general family of functions encompasses many classical functions. The hypergeometric functions play an important role in many parts of mathematics.

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### The infinite product and the Polya-Laguerre class

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### Are hypergeometric series not taught often at universities nowadays, and if so, why?

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### Distribution of iid hypergeometric random variables conditioned on the sum

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### Hypergeometric function with changed argument [closed]

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### When exactly is the principal AGM equal to the optimal AGM?

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### Solving a differential equation related to the hypergeometric differential equation

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### Applying Tannery's theorem to generalised hypergeometric functions

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### Totally rational hypergeometric evaluations

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### High-order associated Legendre polynomials

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### Log of the confluent hypergeometric function of the second kind

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### Higher-order asymptotics of generalized hypergeometric function

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### References for generalized confluent hypergeometric differential equation

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### Two questions on asymptotic expansion of confluent hypergeometric functions for real variable $x, |x| \to \infty$

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### On expressions of hypergeometric functions of two variables

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### Integral representation of product of two Whittaker functions

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### Estimating the alternating sum $\sum_{j \ge 1} (-1)^j e^{-j^2} j^k$

**2**

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### Asymptotic expansion of hypergeometric function ${}_3F_2$ for large parameters

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### Integral of Legendre's function

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### Proving Clausen hypergeometric identity

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### What is the asymptotics of the Fourier transform of $\exp(-x^4)$ for large wave numbers?

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### Upper bounding the sum with hypergeometric and binomial probabilities

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### Tightness of matrix hypergeometric bound

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### Conformal mapping between two right-angled triangles

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### Proof of certain $q$-identity for $q$-Catalan numbers

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### A recurrence formula for the Legendre function $P_\mu^\nu(x)$

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### Proving that a quotient of hypergeometric functions is smaller than a certain function

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### Integral expressions for Bessel-like power series

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### Asymptotic expansion of hypergeometric 2F2

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### Trying to bound the generalized hypergeometric function ${}_2F_3(x+1,x+1;1,1,1;\alpha)$ as $x\to \infty$?

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### Sturm Liouville differential equation and hypergeometric functions

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### Yet another real-rooted polynomial

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### Gegenbauer's addition theorem for Jacobi polynomials

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### Can the integral $ \int_0^R\quad J_{m-n}(a r)J_m(b r) dr$ be explicitly represented in a closed form?

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### An integral of $|\sin(x)\cos(nx)^{-2/n}|$ from $-\pi$ to $\pi$

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### Asymptotic approximations or upper bounds for ${}_{2}F_{1}(x+1,x+1,1,z)$ when $x \gg 1$?

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### Estimation of Hypergeometric function ${_3F_2}$ [closed]

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### Reflection formula for the ${}_2\!F_1$ hypergeometric function of a matrix argument

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### Why do we define the hypergeometric function of a matrix argument for symmetric matrices only?

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### Does the hypergeometric function of a matrix argument depend on $\alpha$ for a $1\times 1$ matrix?

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### Evaluation of hypergeometric type continued fraction

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### Evaluate Gaussian Hypergeometric Function $_2F_1(1;1;c;z)$?

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### A hypergeometric identity related to Bessel functions

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### Hypergeometric representation of Eisenstein series

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### Hypergeometric equation in a particular case

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### question about equality series containing hypergeometric term and a simple term

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### A special solution to the Hermite Differential Equation

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### “Find a representation [using Mellin transform] of 𝑓(𝜇,𝛽) as Gauss hypergeometric function in variable 𝜇”

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### Asymptotic expansion of hypergeometric function near $z=1$

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**1**answer

### Perform a univariate integral, involving a Gauss hypergeometric function

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