The multiple gamma functions and the multiple q-gamma functions

We give an asymptotic expansion (the higher Stirling formula) and an infinite product representation (the Weierstrass product representation) of the Vign\'{e}ras multiple gamma functions by considering the classical limit of the multiple q-gamma functions.Comment: AMS-Latex, 40 pages. Minor changes have been mad

The multiple gamma function and its q-analogue

We give an asymptotic expansion (the higher Stirling formula) and an infinite product representation (the Weierstrass product formula) of the Vign\'{e}ras multiple gamma function by considering the classical limit of the multiple q-gamma function.Comment: Latex, 15 page

On a q-analogue of the multiple gamma functions

A $q$-analogue of the multiple gamma functions is introduced, and is shown to satisfy the generalized Bohr-Morellup theorem. Furthermore we give some expressions of these function.Comment: 8 pages, AMS-Late