I find Chebyshev polynomials useful in situations when I need to fit some function in a general way, imposing as little pre-concieved ideas about the form as possible but still allowing an analytic form of the fitted function. This can be acheived by numerically searching for the values of co-efficients in front of Chebyshev polynomials of increasing order.

However, I want to extend this to some unknown function of multiple dimensions. Does anyone know what the generating functions of Chebyshev polynomials in multiple dimensions are?