I think I fixed it. I haven't looked at this problem in a while, so I had to figure out what it was. The coefficients $a,b$ in the argument of $H$ are dependent on both $i$ and $j$. The main difference in the two expressions is that the top is the sum of the individual $H$ over $j$ where the bottom is not, only over some (any) fixed $k$

$f(n)$ would be a factor in $n$ dependent on the sum. Note the Euler / Bernoulli Polynomial equation in my post, the $f(n)=2^n/n$. I'm about to fix the sums as well too.

I think i agree with you in terms of the $\beta_k$. Part of the reason i need this is i am relating function expansions to identify identities, so having it in the partial fraction summation form helps, i believe, for what i need.

Hello. I noticed the comments and edits were taken down. I wanted to get more in depth with the edits you made, but now I dont see them Do you know whathappened?

I gave you credit for answering my question. I have another question though. Is there a way to discern the $P_k(x)$ and if so, what method could i employ to find them specifically?