Multiple gamma functions, multiple sine functions, and Appell’s O-functions
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- Tanaka, H. Ramanujan J (2011) 24: 33. doi:10.1007/s11139-010-9260-3
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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.