The bilateral hypergeometric function is defined for 
by the two-sided series
 |
where 
is the Pochhammer symbol and 
for positive 
. For noninteger upper and lower parameters,
both tails can converge only when 
and 
.
In that case the series is absolutely
convergent iff the real part
of 
exceeds 1. On specializing term by term to 
,
the terms with negative 
vanish by continuity of 
at 
.
The remaining series defines the generalized
hypergeometric function 
wherever that series converges.
Dougall's bilateral sum is (Dougall 1907; Slater 1966, Ch. 6; NIST DLMF, Eqn. 16.4.17)
 |
for noninteger 
, 
, 
,
and 
with ![Re[c+d-a-b]>1](/images/equations/BilateralHypergeometricFunction/Inline17.svg)
. Here 
is the gamma function.
The bilateral hypergeometric function is implemented in the Wolfram Language as BilateralHypergeometricPFQ[{a1, ..., ap}, {b1, ..., bq}, z].
