Thanks to Robert Israel who answered my question in mathstackexchange, we have a generalization of the binomial sums of the previous post. Interestingly, it turns out roots of unity are involved. Given,
where k is an even integer then,
for appropriate z such that the sum converges. For the special case when,
Note that the terms are complex, but the sum is a real number so they must come in conjugate pairs. The arcsin of a complex root of unity can be given as,
With this transformation, it is now possible to have an expression all in real terms. The case k = 2, 4 was given in the previous post. For k = 6, we have the counterpart to Sprugnoli’s equality as,
Note that the prime factors of 65 are 5 and 13, and the square root of both appear above. However, for k = 8, while the expression contains the fraction as expected, the argument of the log and arcsin do not factor over the quadratic extension , but rather only over . Furthermore, the argument of the log for both k = 6, 8 are no longer simply expressible in terms of the Dedekind eta function, so observations for lower k do not generalize to higher ones.