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,

then,

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,

where,

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.