Is it possible to find all roots of the family of polynomials
$1+(-1)^df\cdot t^d+\sum\limits_{j=1}^{d-1}(-1)^{j+d}{d\choose j}t^{j-1}=0$
for $d\ge2, d\in \mathbb{N}$, $f\in \mathbb{C}$ and $|f-1|\le d-2$ in an algebraically closed form, and if so what is the explicit expression? With this I mean in terms of a function of $f$, which includes generalized hypergeometric functions. The brackets denote the binomial coefficients.
These equations are all quite simple to solve numerically, but Mathematica fails to solve them analytically for d>4, as it is expected for quintic polynomials. But if one plots the numerical solutions as functions of f, one finds remarkable similarities between the solutions of different degree.
It is definitely possible to give a formula for the roots for $f=1$ and any $d$, as they all lie on a circle of radius 1 around 1, i.e. for $f=1$
$t=1+e^{i\phi}\;,$
with the phases depending on the degree.
Is there a way to express these roots in the general case? Obviously for $d\in \{2,3,4\}$ this is easy, so i focused on the quintic case. I tried solving the quintic case by finding a Tschirnhaus transformation to the Bring–Jerrard normal form and then use the Glasner formula to express the solutions in terms of the Hypergeometric $_4F_3$, but the expressions for the transformations become incredible cumbersome, Mathematica was unable to express these in any useful terms. Bringing the quintic only to the principal form keeps the coefficients much simpler. Then applying the Klein approach from https://library.wolfram.com/infocenter/TechNotes/158/ worked and gave an expression in terms of $_2F_1$, but this expression is still much too complicated to be of any use.
I would expect that rewriting the solutions for $d\in \{2,3,4\}$ in terms of hypergeometrics could give a hint on the form of the solutions for any d like it is done in https://arxiv.org/pdf/math/9411224.pdf, but i am not able to do this as Mathematica has the habit of rewriting the hypergeometrics in elementary functions whenever possible.