Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4655398 | Journal of Combinatorial Theory, Series A | 2013 | 25 Pages |
Abstract
It is shown that the number fλ of free subgroups of index 6λ in the modular group PSL2(Z), when considered modulo a prime power pα 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 PSL2(Z) as a special case), and for a one-parameter family of lifts of the Hecke group H(4)=C2âC4. 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 (2012) [12,15], where it is shown, among other things, that the free subgroup numbers of PSL2(Z) and its lifts display rather complex behaviour modulo powers of 2 and 3.
Related Topics
Physical Sciences and Engineering
Mathematics
Discrete Mathematics and Combinatorics
Authors
C. Krattenthaler, T.W. Müller,