# §16.2 Definition and Analytic Properties

## §16.2(i) Generalized Hypergeometric Series

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 .

## §16.2(ii) Case

When the series (16.2.1) converges for all finite values of and defines an entire function.

## §16.2(iii) Case

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 1, 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 1 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

## §16.2(iv) Case

### ¶ Polynomials

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

### ¶ Non-Polynomials

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.)

## §16.2(v) Behavior with Respect to Parameters

Let

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 .