Recursive construction of a series converging to the eigenvalues of the Gauss-Kuzmin-Wirsing operator

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.