We introduce a multiple analogue of the gamma function which differs from the one defined by Barnes [B]. Using this function, we give expressions of the multiple sine and cosine functions in terms of zeta regularized products. The expression of the multiple sine function can be interpreted as a reflection formula of this new multiple analogue of the gamma function.
"Zeta Regularized Product Expressions for Multiple Trigonometric Functions." Tokyo J. Math. 27 (2) 469 - 480, December 2004. https://doi.org/10.3836/tjm/1244208402