What is the function whose Maclaurin series is given by $1 + \frac{3}{4} \, x^2 + \frac{75}{128} \, x^4 + \frac{245}{512} \, x^6 + \dots$?

Question

Let $$ f(x) = 1 + \frac{3}{4} \, x^2 + \frac{75}{128} \, x^4 + \frac{245}{512} \, x^6 + \frac{6615}{16384} \, x^8 + \frac{22869}{65536} \, x^{10} \\+ \frac{1288287}{4194304} \, x^{12} + \frac{4601025}{16777216} \, x^{14} + \dots $$

Does $f(x)$ correspond to any combination of known functions?

I understand that this question bears similarity to the one I posed yesterday; however, despite my diligent efforts, I have been unable to discover any combination of elliptic functions or their derivatives that align with the coefficients of this particular series.

Answer 1

It's the Maclaurin series of the function $$ f(x)=\frac{16}{\pi}\frac{(2-x^2)K(x^2)-2E(x^2)}{x^4}, $$ where $K$ and $E$ are the complete elliptic integrals of the first and second kind, respectively. See here.