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Results tagged with hypergeometric-functions
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user 82588
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.
6
votes
Hypergeometric function evaluation 4F3
From https://dlmf.nist.gov/15.4 one has
$$
\sum_{k=0}^n\frac{(1/3)_k(-1/3)_k}{k!(1/2)_k}(-z^2)^k=\frac{1}{2}\left(\left(\sqrt{1+z^{2}}+%
z\right)^{2/3}+\left(\sqrt{1+z^{2}}-z\right)^{2/3}\right)
$$
an …
- 5,624
3
votes
Accepted
On $x^k+y^k=1$ and the Dixonian elliptic functions
Define the generalized trigonometic functions (discussion of these functions is given in this answer)
$$
z=\int_0^{\sin_{pr}z}\frac{dt}{\sqrt[p]{1-t^r}},\qquad \cos_{pr}z=\sqrt[r]{1-(\sin_{pr}z)^r},\q …
- 5,624
4
votes
Accepted
Double Series involving Gamma function
This problem can be reduced at least formally to a compact double integral, which might be easier to solve.
Starting with the integral representation for the Gamma function, we write the double sum a …
- 5,624
13
votes
Accepted
Yet another real-rooted polynomial
First, we write the polynomial in hypergeometric form
$$
Q_{m,n}(t)=\frac{(-1)^{m+n+1} \Gamma (m+n-t+1) }{\Gamma (-t) \Gamma (m+n+2)}\, _3F_2\left({-m,-n,m+n-t+1\atop 1,-t};1\right).
$$
Applying Thoma …
- 5,624