*The function
has derivatives of any order
, and these derivatives are obtained by term-by-term differentiation, namely:
*

We integrate the function:

Integrating the series term-by-term and equating both sides to 0 for , we have:

*Then the series
is absolutely convergent for
and its sum is equal to
..*

and |

The product of these series is given by:

i.e. for even and for odd . We have:

This could have been obtained either by a direct computation or by noting that

Noah Dana-Picard 2007-12-28