A resolution of Kellner's conjectures on Wilson quotients

Kellner expressed higher congruences for the Wilson quotient $W_p$ of an odd prime $p$ in terms of power sums of Fermat quotients, and recursively constructed certain $p$-independent polynomials $\psi_\nu$ occurring in these congruences. In this note, we give a simple $p$-adic proof of these congruences using the $p$-adic logarithm and $p$-adic exponential function. We also provide an explicit generating function for the polynomials $\psi_\nu$ in terms of complete Bell polynomials. This gives a direct derivation of Kellner's polynomials and allows us to resolve two conjectures stated by Kellner.