 A representation of the incomplete gamma function in terms of generalized Laguerre polynomials is
which converges for formula_72 and formula_73.Other important functional equations for the gamma function are Euler's reflection formula

which implies

and the Legendre duplication formula

Since formula_77

the gamma function can be represented as

Integrating by parts formula_33 times yields

which is equal to

This can be rewritten as

Then, using the functional equation of the gamma function, we get

It can be proved that

Then

Euler's reflection formula follows