Could it be londmathsoc.onlinelibrary.wiley.com/doi/10.1112/plms/…

This looks very much like the Legendre differential equation. Try to make $t=\tanh(x)$

@DmitriScheglov. This was nt a comment but an answer (joke). We did not publish the work and, as for as I know, the company I was working for does not want to release information since this cam when building a thermodynamic model for systems of polymers (in the Gibbs ensemble). Be sure that before answering I asked if I could give this result which is very simple. If fact, I am still working this specific case which is very interesting. If I find anything of interest, I shall post it.Cheers :-)

@fedja. I did not write to use five or six terms. Plot $y$ and the expansion for, say, fifty terms. Does it look bad ? Cheers :-)

@dohmatob. The polygamma functions and generalized harmonic numbers. I shall have a look at $\lambda_i=i^{-\beta}$. If anythig interesting, I shall edit. By the way, I faced very similar problems ... 50+ years ago ! You made me feeling youger. Cheers :-)