The hypergeometric function, ##{}_{2}F_1(a,b,c;z)## can be written in terms of a power series in ##z## as follows, $${}_{2}F_1(a,b,c;z) = \sum_{n=0}^{\infty} \frac{(a)_n (b)_n}{(c)_n} \frac{z^n}{n!}\,\,\,\,\,\text{provided}\,\,\,\,|z|<1$$(adsbygoogle = window.adsbygoogle || []).push({});

So we may reexpress any hypergeometric function as a power series like this as long as the last argument modulus is less than 1.

My question is, I also have an expansion of a hypergeometric of the form ##{}_2 F_1(\alpha,\beta+\epsilon, \gamma- \epsilon, z)##, where ##\alpha, \beta, \gamma## are real numbers, by using the HypExp package on Mathematica (expansion in ##\epsilon##) and was wondering if I use this expansion do I also require ##|z|<1##? When I write the code in Mathematica, the last argument is replaced by simply ##x## say and Mathematica gives me an expansion regardless of the size of ##x## so I am thinking ##{\it this}## expansion (and not the power series one) is maybe valid independent of ##|x|## but would be nice to confirm.

Thanks!

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Convergence of a hypergeometric

**Physics Forums | Science Articles, Homework Help, Discussion**