Multiplicative Functions and Generalized Binomial Coefficients

The binomial coefficient $\binom{n+m}{n}$ can be thought of as $\dfrac{f(1) \cdots f(n+m)}{\left(f(1) \cdots f(n)\right) \left( f(1) \cdots f(m) \right)}$, where $f(j) = j$. We can generalize the binomial coefficient by considering functions $f$ other than the identity function. In particular, what kinds of functions $f$ give rise to generalized binomial coefficients $\binom{n+m}{n}_f$ that are integers for all values of $n$ and $m$? In this talk we describe research into generalized binomial coefficients for multiplicative functions $f$. Our main results are (1) a formula for $\binom{n+m}{n}_f$ when $f$ is multiplicative in terms of the values of $f$ at prime powers and carries when $n$ and $m$ are added in prime bases; and (2) a proof that if $f$ is both multiplicative and divisible then $\binom{n+m}{n}_f$ is an integer for all nonnegative integer values of $n$ and $m$.