A remark on holomorphic functions rational in some variables

Considering the holomorphic functions in a polycylindrical domain which are rational in some of the variables for the arbitrary fixed values of other variables, we prove that the functions are representable as a ratio of polynomials in these variables whose coefficients are holomorphic in the remaining variables. We use the Kronecker method for establishing some criterion for the rationality of a holomorphic function of one complex variable in a neighborhood of zero by using the properties of Hankel matrices.