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Results tagged with hypergeometric-functions
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user 66131
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.
4
votes
how to reduce the integral into hypergeometric function?
Letting $\beta=\frac{2\Pi}{1+\Pi^2}$, write the integrand as
$$\sqrt{1+\Pi^2}\left(\sqrt{1+\beta \sqrt{1-y^{2}}}-\sqrt{1-\beta \sqrt{1-y^{2}}}\right).\tag1$$
Use the binomial expansion to express (1), …
- 43.1k
1
vote
Product of two hypergeometric functions
If you allow some relation between $m$ and $n$, then it is possible.
Let $m+n=2\alpha+2$. Then
\begin{align}
&_2 F_1(\alpha, \alpha+1 ; m ; z)\, _2 F_1(\alpha, \alpha+1 ; n ; z) \\
=&\,\,_4F_3(\alpha …
- 43.1k
5
votes
Accepted
A contiguous ${}_3F_2(1)$ hypergeometric identity?
First, we convert the desired identity into equivalent forms:
\begin{align}
\sum_{i=0}^{\lfloor n/3\rfloor}(-1)^i \left\{ \prod_{j=1}^i{{n-3j\choose3}\over{n\choose3}-{n-3j\choose3}}
-\frac{n}3\, \pro …
- 43.1k
9
votes
Accepted
Closed expression for hypergeometric sum
You may argue as GH from MO from your other post.
the coefficient of $y^m$ in $(1-xy)^{-s}$ equals $\binom{s+m-1}{s-1}x^m$;
the coefficient of $y^{\ell-m}$ in $(1-y)^{\ell-s-1}$ equals $\binom{s-m}{ …
- 43.1k
4
votes
speeding up Gosper and WZ algorithms
The group at RISC has been working aggressively towards developing improved algorithms for many computer algebra problems, including the Zeilberger's. So, it is a good place to ask.
Meantime, I just …
- 43.1k