Throughout this chapter it is assumed that none of the bottom parameters , , , is a nonpositive integer, unless stated otherwise. Then formally
Equivalently, the function is denoted by or , and sometimes, for brevity, by .
When the series (16.2.1) converges for all finite values of and defines an entire function.
Suppose first one or more of the top parameters is a nonpositive integer. Then the series (16.2.1) terminates and the generalized hypergeometric function is a polynomial in .
If none of the is a nonpositive integer, then the radius of convergence of the series (16.2.1) is , and outside the open disk the generalized hypergeometric function is defined by analytic continuation with respect to . The branch obtained by introducing a cut from to on the real axis, that is, the branch in the sector , is the principal branch (or principal value) of ; compare §4.2(i). Elsewhere the generalized hypergeometric function is a multivalued function that is analytic except for possible branch points at , and . Unless indicated otherwise it is assumed that in the DLMF generalized hypergeometric functions assume their principal values.
On the circle the series (16.2.1) is absolutely convergent if , convergent except at if , and divergent if , where
In general the series (16.2.1) diverges for all nonzero values of . However, when one or more of the top parameters is a nonpositive integer the series terminates and the generalized hypergeometric function is a polynomial in . Note that if is the value of the numerically largest that is a nonpositive integer, then the identity
can be used to interchange and .
Note also that any partial sum of the generalized hypergeometric series can be represented as a generalized hypergeometric function via
See §16.5 for the definition of as a contour integral when and none of the is a nonpositive integer. (However, except where indicated otherwise in the DLMF we assume that when at least one of the is a nonpositive integer.)
compare (15.2.2) in the case , . When and is fixed and not a branch point, any branch of is an entire function of each of the parameters .