Multiplicative Functions Generalized Binomial Coefficients, and Generalized Catalan Numbers

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 clue to Ward.