Prof. Aiden Bruen’s has recently contributed a paper https://arxiv.org/abs/2102.12692 in which he shows how the Shannon entropy function 
is expressible as a linear combination of other Shannon entropy functions involving quotients of polynomials in p, q of degree n for any n >= 2 .
Chapter 25 of the forthcoming 2nd Edition of the Handbook contains all the background information, additional details, and related material.
2020 MSC: 94A15 Information theory, 94A17 Measures of information, entropy, 94A60 Cryptography