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Results tagged with hypergeometric-functions
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user 49208
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.
5
votes
Accepted
Asymptotic form of the Gauß Hypergeometric function 2F1 for three parameters approaching inf...
In fact your function is elementary and very simple, for its explicit form look at Brychkov,Marichev,Prudnikov, Integral and Series, vol.3 :
$$
F(a,a+1/2;2a;z)=\frac{1}{\sqrt{1-z}}\left(\frac{2}{1+\sq …
- 1,560
3
votes
Logarithm of the hypergeometric function
Please note the paper, may be it will be useful:
D. Karp, S.M. Sitnik,
Log-convexity and log-concavity of hypergeometric-like functions,
Journal of Mathematical Analysis and Applications, Volume 364, …
- 1,560
2
votes
Sharp upper bounds on hypergeometric function ${}_2F_1[a,b,c;z]$ when $|z|\geq1$
In this case series are not convergent. And the Gauss function is not defined by series outside the unit circle. But there are explicit analytical continuation formulas to return inside of the unit ci …
- 1,560
0
votes
A definite integral of hypergeometric function 2F1
May be use the series for $_{2}F_{1}$ and then integrate term by term? The result looks like a sort of Appell-type function of two arguments: $x, 1-x$...
- 1,560
1
vote
Conjectured bound on Kummer's function (confluent hypergeometric function)
1) Cf. inequalities in the paper:
D. Karp, S.M. Sitnik, Log-convexity and log-concavity of hypergeometric-like functions, Journal of Mathematical Analysis and Applications, Volume 364, Issue 2, P. 384 …
- 1,560
2
votes
Hypergeometric sum specific value
You may use the general formula from Brychkov, Prudnikov, Marichev, Integral and Series, Vol.3:
$$
_{2}F_{1}(1,1;\frac{1}{2};x)=(1-x)^{-1}\left(1+\frac{\sqrt{x}\arcsin{\sqrt{x}}}{\sqrt{1-x}}\right)
$$ …
- 1,560