A Riccati differential equation and free subgroup numbers for lifts of ₂() modulo powers of primes

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It is shown that the number fλ of free subgroups of index 6λ in the modular group ₂(), when considered modulo a prime power p with p≥5, is always (ultimately) periodic. In fact, an analogous result is established for a one-parameter family of lifts of the modular group (containing ₂() as a special case), and for a one-parameter family of lifts of the Hecke group H(4)=C₂*C₄. All this is achieved by explicitly determining Padé approximants to solutions of a certain multi-parameter family of Riccati differential equations. Our main results complement previous work by Kauers and the authors (arXiv:1107.2015 and ["A method for determining the mod-3ᵏ behaviour of recursive sequences", preprint]), where it is shown, among other things, that the free subgroup numbers of ₂() and its lifts display rather complex behaviour modulo powers of 2 and 3.