Summary: We investigate generalized binomial coefficients of multiplicative functions. We provide a formula for these coefficients and use this formula to prove that the coefficients are always integral if the function is also a divisible function. Furthermore, we prove that multiplicative and divisible functions have integral generalized Fuss-Catalan numbers. Along the way, we include some results about specific multiplicative functions such as $gcd_{k}$ and $\phi $. We finish by connecting these results to a classical result due to Ward.

11B65, 05A10, 11A05

multiplicative function, divisible function, generalized binomial coefficient, Kummer's theorem, fuss-Catalan number