Butson Hadamard Matrices from Pairs of Gauss Sums

Let $q$ be a prime power. We give a construction of Butson Hadamard matrices $BH((F_q\times F_q,+),lcm(6,d))$ for any divisor $d>1$ of $q-1$. Most character values of the group ring elements corresponding to these matrices involve products of two Gauss sums over $F_q$ (with non-quadratic Gauss sums occurring) and this property distinguishes our result from all previously known constructions of group invariant Butson Hadamard matrices. In fact, our $BH((F_q\times F_q,+),lcm(6,d))$ matrices arise as a special case of a more general construction based on what we call ``Butson-Jacobsthal functions’’ that resembles the construction of Hadamard matrices from Jacobsthal matrices.