On incidences of ϕ and σ in the function field setting

Erdős first conjectured that infinitely often we have ϕ(n) = σ(m), where ϕ is the Euler totient function and σ is the sum of divisors function. This was proven true by Ford, Luca and Pomerance in 2010. We ask the analogous question of whether infinitely often we have ϕ(F) = σ(G) where F and G are polynomials over some finite field F_q. We find that when q ≠ 2 or 3, then this can only trivially happen when F = G = 1. Moreover, we give a complete characterisation of the solutions in the case q = 2 or 3. In particular, we show that ϕ(F) = σ(G) infinitely often when q = 2 or 3.