Specifically,
$$H_m^{(2n)} \approx\ ?$$
and
$$H_m^{(4n)} \approx\ ?$$
where $(m, n)$ $\in \mathbb N_{>1}$
I would not like to use special functions like the (Riemann zeta function) unless they themselves have polynomial expressions approximating them with shrinking error terms (see https://math.stackexchange.com/a/1583465/322359)
You may wish to reference : http://mathworld.wolfram.com/HarmonicNumber.html
This is motivated by my work involving summation of generalized harmonic numbers and series/alternate representations there in. Summation is not too difficult, series/alternate representations are a bit rough for me.
Take a look at work here: https://math.stackexchange.com/a/1583465/322359 which seems to solve the issue entirely using the Euler-Maclaurin formula, only we have that pesky (Riemann) zeta function as one of the terms...