# Tagged Questions

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 ...

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### Exponential approximation for 3F2 hypergeometric function with repeated indices

In my research I have run across the hypergeometric function $${}_3F_2(d,d,d;d+1,d+1;z)$$ where d is a positive integer and 0≤z≤1. When I plot this as a function of d on a semilog plot, it appears to ...
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### How to prove ${}_2\phi_1(q^{-n},b;c;q,q)=\frac{(c/b;q)_n}{(c;q)_n}b^n.$

Firstly, we have already known the one of $q$-analogues of Vandermonde's formula, which is $${}_2\phi_1(q^{-n},b;c;q,cq^n/b)=\frac{(c/b;q)_n}{(c;q)_n}.$$ And there is a hint, when we change the order ...
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### Approximation of $_2F_1((b-1)a,b;ba;x)$

Is there any (simple) approximation of this Hypergeometric function: $_2F_1((b-1)a,b;ba;x)$, where $0<x<1$ and $b>a>1$. Thanks!
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### How to prove that $(1-x)^b$ $_2F_1(a,b;c;x)$ can be approximated to $1-\alpha x$ (with $\alpha \approx 1$) for $x\ll 1$ in this specific case

After multiple plots I noticed that function $h(x)= (1-x)^b$ $_2F_1(a,b;c;x)$ can be approximated to $1-\alpha x$ (with $\alpha \approx 1$), for $x\ll 1$ (specifically $0<x<0.1$) and $a=(K-1)d$,...
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Let $n,N,T$ be positive integers, with $N=\binom{n}{2}$, and $3\leq n\leq T\leq N$. Define: $$P(z):=z^{N+1-T}\int_0^1\frac{(1-t^2)^{n-2}}{(1-(1-z)t)^{N+1}}{}_2F_1\left[-T,N+1,N+2-T; \frac{tz}{1-(1-z)... 4answers 293 views ### Exponential of a specific hypergeometric series This is motivated by this question. Let f be the hypergeometric series  f(x) = 2 x \, _{4}F_3([1, 1, 4/3, 5/3], [2, 2, 2], 27 x)  which is explictly given by  f(x) = \sum_{n \geq 1} \frac{(3n-... 2answers 2k views ### Binomial supercongruences: is there any reason for them? One of the recent questions, in fact the answer to it, reminded me about the binomial sequence$$ a_n=\sum_{k=0}^n{\binom{n}{k}}^2{\binom{n+k}{k}}^2, \qquad n=0,1,2,\dots, $$of the Apéry ... 8answers 2k views ### A good reference to grok hypergeometric functions? When I was introduced during my degree to special functions, I made friends with a number of nice functions - Laguerre, Legendre, Hermite, Bessel, and whatnot - but I made only a passing acquaintance ... 0answers 73 views ### Identities for {~}_3\phi_1? I am looking for some source of summation formulas for the q-hypergeometric function {~}_3\phi_1 in the sense of Gasper-Rahman book. For some reason, the book focuses heavily on {~}_{r+1}\phi_r ... 1answer 168 views ### Are binomial coefficients F_1 analogs of q-binomial coefficients? This is a mostly philosophical question. Is it fair to think of usual binomial coefficients and their identities as an F_1 case of q-binomial coefficients and identities? Here F_1 is the field ... 2answers 293 views ### Sharp upper bounds on hypergeometric function {}_2F_1[a,b,c;z] when |z|\geq1 Generally, hypergeometric function {}_2F_1[a,b,c;z] is defined using Gauss series {}_2F_1[a,b,c;z]=\sum_{n=0}^\infty\frac{(a)_n(b)_n}{(c)_nn!}z^n with |z|<1, and there seems to be a lot of ... 5answers 3k views ### Groups, quantum groups and (fill in the blank) In the study of special functions there are three levels of objects, classical, basic and elliptic. These correspond to classical hypergeometric functions, basic (q-) hypergeometric functions, and ... 0answers 160 views ### System of linear ODEs with hypergeometric coefficients For quite some time I have been trying to solve the following system of differential equations for the two functions G and H defined on the interval [0,1]:$$ \begin{align}x G''(x)=&\mathscr{...
The generating function $f(z)$ of the Catalan numbers which is characterized by $f(z)=1+zf(z)^2$ is D-finite, or holonomic, i.e. it satisfies a linear differential equation with polynomial ...