when this product converges.
provided that .
In the limit as , (17.2.35) reduces to the standard binomial theorem
provided that . When , where is a nonnegative integer, (17.2.37) reduces to the -binomial series
The -derivatives of are defined by
When the -derivatives converge to the corresponding ordinary derivatives.
-differential equations are considered in §17.6(iv).
If is continuous at , then
and more generally,
If is continuous on , then
provided that converges.
These identities are the first in a large collection of similar results. See §17.14.