 These identities can be proven in several ways, for example by expanding out the series and comparing coefficients, or by taking the derivative in several ways and eliminating it from the equations generated.The simplest case involves

Starting with the identity

we may take

giving

or

This expansion converges to the meromorphic function defined by the ratio of the two convergent series (provided, of course, that "a" is neither zero nor a negative integer).The next case involves
for which the two identities
are used alternately.