Is there an automated way to express hypergeometric functions in series form using gamma functions, factorials, double factorials or rising factorials? For example using the formula (on the hypergeometric function wiki page):
$$_2 F_1(a,b;c;z) = \sum_{n=0}^\infty \frac{(a)_n (b)_n}{(c)_n} \frac{z^n}{n!}$$
or similar expressions for the general $_p F_q$ case. I work with the series/sum notation and occasionally Mathematica gives hypergeometric expressions which can be hard to read and annoying to convert manually.
FunctionExpand
will convert hypergeometric functions to simpler form, but don't count on it working too often. You have a better chance with numbers than with variables, and with variables you have a better chance withAssumptions
. Asking for theSeries
representation may work, but probably they will not be in terms ofGamma
functions. $\endgroup$ – Bill Watts Jul 15 '19 at 7:43InputForm
version of the definition and paste it into your notebook, replacingSum
withInactive[Sum]
to keep it in unevaluated form. Then you can replace individual values and later useActivate
on the expression to resume evaluation if you want. $\endgroup$ – Thies Heidecke Jul 15 '19 at 16:46