The Ramanujan Journal

, Volume 24, Issue 1, pp 33-60

First online:

Multiple gamma functions, multiple sine functions, and Appell’s O-functions

  • Hidekazu TanakaAffiliated withDepartment of Mathematics, Tokyo Institute of Technology Email author 

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Kurokawa introduced q-multiple gamma functions and q-multiple sine functions. We show that the Appell’s O-function is expressed via the q-multiple gamma function. We also give some applications of this result. For example, we obtain a formula for the “Stirling modular form” and calculate special values of the q-multiple sine function. Moreover, we give some formulas of Eisenstein series and double cotangent functions and its generalization. Then the former gives an infinite product expression of the double sine function explicitly and a result of Kurokawa.


Appell’s O-function Multiple sine function Multiple gamma function

Mathematics Subject Classification (2000)