  We study $L^q$-spectra of planar self-affine measures generated by diagonal
systems with an emphasis on providing closed form expressions. We answer a
question posed by Fraser in 2016 in the negative by proving that a certain
natural closed form expression does not generally give the $L^q$-spectrum and,
using a similar approach, find counterexamples to a statement of Falconer-Miao
from 2007 and a conjecture of Miao from 2008 concerning a closed form
expression for the generalised dimensions of generic self-affine measures. In
the positive direction we provide new non-trivial closed form bounds in both of
the above settings, which in certain cases yield sharp results. We also provide
examples of self-affine measures whose $L^q$-spectra exhibit new types of phase
transitions. Our examples depend on a combinatorial estimate for the
exponential growth of certain split binomial sums.
