The great success mathematicians had studying hypergeometric functions 
for the convergent cases (
) prompted attempts to provide
interpretations for such functions in divergent cases when 
. An interesting approach to interpreting sums of these
divergent series was suggested by Emile Borel in 1899. By definition the generalized
Borel sum of an arbitrary series
 |
(1)
|
is the value of the integral
 |
(2)
|
with 
.
This definition allows interpretation of the sums of divergent hypergeometric series as generalized Borel sums, where these Borel sums always coincide with other convergent hypergeometric series.
Consider an example related to the asymptotic formula for the function 
, beginning with the relation
 |
(3)
|
The right-hand side of the above formula can be interpreted as a Borel sum (with 
)
of the classically divergent series 
. Choose 
so that
 |
(4)
|
Then
 |
(5)
|
so the Borel-regularized sum 
for the divergent series 
is equal to
 |  |  |
(6)
|
 |  |  |
(7)
|
Borel's approach to the summation of divergent series was not investigated deeply for hypergeometric series, however; the most effective results for hypergeometric series were found with the Mellin-Barnes integral later on.
