The total positivity properties of products of totally positive (TP) functions are well-known to follow from the Binet-Cauchy formula. By contrast, the total positivity properties of sums or arbitrary powers of TP functions are more difficult to ascertain, and existing results apply only to functions that are TP$_2$. In this talk, we present conditions under which a sum or product of TP$_k$ functions is TP$_k$ for $k > 2$.

This talk is based on joint work with Satoshi Kuriki (ISM, Tokyo) and Ishan Muzumdar (Penn State University).