On a q-analogue of the multiple gamma functions

Abstract

Abstract. 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.1995 Mathematical Subject Classification. 33D05, 81R50

On Barnes' Multiple Zeta and Gamma Functions

“…From Prop. 2.3 it follows that 9 N (w) has a holomorphic extension to C & (2.21), and that the remainder in (3.13) satisfies To proceed, we introduce the multiple gamma function 19) where \ N is Barnes' modular constant. Our definition is in accord with most of the later literature.)…”

A note on q-Bernoulli numbers and polynomials

“…The identities for Barnes' multiple Bernoulli polynomials are now intensively studied by mathematicians and physicists, cf. [7,10,11,12,13,15,16,17,19,20]. As a side remark, we note that special values of Barnes' multiple zeta function at positive integers have come to the foreground in the recent years, both in connection with theoretical physics (Feynman diagrams) and the theory of mixed Tate motives, cf.…”

“…These Barnes' multiple zeta function interpolates Barnes' multiple Bernoulli numbers at negative integers, cf. [1,10,11,12,13,14,20 ]. In [13], K Ota have studied Kummer-type congruences for derivatives of these' Barnes' multiple Bernoulli polynomials.…”

“…The computation of Feynman diagrams has conforented physicists with classes of integrals that are usually hard to be evaluated, both analytically and numerically, cf. [10,11,12,14,15,16,17].…”

Multiple Gamma Functions and Multiple $q$-Gamma Functions

“…Recently, one of the authors [19] constructed the function G n (z\ q) which satisfies a ^-analogue of the generalized Bohr-Morellup theorem: Theorem 2 0 2 0 There exists a unique hierarchy of functions which satisfy…”