a a^2 a^3
1 + - + ------- + -------------- + ....
z 2z(z+1) 2.3.z(z+1)(z+2)
{370} Let z be positive: this series is convergent for all values of a, and
approaches without limit to unity as z increases without limit. Change z
into z + 1, and form [phi]z - [phi](z+1): the following equation will
result--
a
[phi]z-[phi](z+1) = ------([phi](z+2))
z(z+1)
a [phi](z+1) a [phi](z+1) a [phi](z+2)
or a = - ---------- . z + - ---------- . --- ----------
z [phi]z z [phi]z z+1 [phi](z+1)
a = [psi]z(z+[psi](z+1))
[psi]z being (a/z)([phi](z+1)/[phi]z); of which observe that it diminishes
without limit as z increases without limit. Accordingly, we have
[psi]z = a/z+ [psi](z+1) = a/z+ a/(z+1)+ [psi](z+2)
= a/z+ a/(z+1)+ a/(z+2)+ [psi](z+3), etc.