  Based on the technique previously developed by the author, we present a
conjecture which claims that the reciprocal of the n-th largest (in absolute
value) eigenvalue of the Gauss-Kuzmin-Wirsing operator is equal to the sum of a
certain infinite series. This series is constructed recurrently. It consists of
rational functions with integer coefficients in two variables X, Y, specialized
at X=n and Y=2^n. This gives a strong evidence to the conjecture of Mayer and
Roepstorff that eigenvalues have alternating sign. Further, a very similar
recursion yields a series for the dominant eigenvalue of the Mayer-Ruelle
operator.
