Let , ,  and . The hypergeom(n, d, z) calling sequence is the generalized hypergeometric function . This function is frequently denoted by .

Formally,  is defined by the series

For the definition of the  symbol, see pochhammer.

If some  is a non-positive integer, the series is finite (that is,  is a polynomial in ).

If some  is a non-positive integer, the function is undefined for all non-zero , unless there is also a negative upper parameter of smaller absolute value, in which case the previous rule applies.

For the remainder of this description, assume no  or  is a non-positive integer.

When , this series converges for all complex , and hence defines  everywhere.

When , the series converges for .  is then defined for  by analytic continuation.  The point  is a branch point, and the interval (1,infinity) is the branch cut.

When  the series diverges for all .  In this case, the series is interpreted as the asymptotic expansion of  around .  The positive real axis is the branch cut.

Hypergeom is the unevaluated form of hypergeom (that is, it returns unevaluated because it is the inert form of this function). Use value to evaluate a call to Hypergeom, or evalf to compute a floating-point approximate value.  See also simplify and convert[StandardFunctions].