# 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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### How can I show that this product is equal to a product of Gamma functions? [migrated]

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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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### Series expansion for Gaussian hypergeometric function valid for branch points?

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

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### Compute the two-fold partial integral, where the three-fold full integral is known

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### What partial sum formulae exist for this basic hypergeometric series?

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### Correction terms in the asymptotic expansion of hypergeometric function

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### On the convergence of MIller's Algorithm for special function evaluation (hypergeometric 1F1)

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### Asymptotic expansion of an integral involving hypergeometric function

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### Euclidean volumes under the matrix norm of one-parameter subsets of unit balls of $2 \times 2$ matrices

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### Conjectural nonvanishing of some combinatorial sums (6j symbols)

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### best-possible inequalities for hypergeometric functions

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### Integration of hypergeometric product for legendre polynomials

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### lower bound for absolute value of a hypergeometric function

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### Uniform Asymptotic Approximation of the Whittaker function

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### Limit of a hypergeometric function(1F2)

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### Understanding the branch cut and discontinuity of the hypergeometric function

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### Value of the hypergeometric function

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### A minimizing problem involving Gauss hypergeometric functions

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### How to use this generalised 'generating function' for the Gegenbauer polynomials

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### Compute Confluent Hypergeometric Function 1F1

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### Polya urn: Mean number of draws to get a specific sequence of colors?

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### integral involving hypergeometric function of matrix argument

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### Basis for solutions hypergeometric differential equation

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### Does the Riemann characterization of the hypergeometric function have a q-analog?

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### Is there a name for the hypergeometric function with more parameters than Lauricella but more variables than Kampé de Fériet?

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