According to (13.4) in Belkic: the trinomial roots, their powers and logarithms from the Lambert series, Bell polynomials and Fox–Wright function https://d-nb.info/1176090631/34
the solution of
Solve[x == 1 + t x^n, x]
is called confluent Fox-Wright function with four parameters(a particular case of Fox-H function). Solve is unaware of this. Is this special function implemented at all, under some name? The closest I came up with is Hypergeometric1F1, but that has only two parameters