A Riccati differential equation and free subgroup numbers for lifts of PSL2(Z) modulo prime powers

作者：Krattenthaler C^*; Mueller T W

来源：Journal of Combinatorial Theory - Series A, 2013, 120(8): 2039-2063.

DOI：10.1016/j.jcta.2013.08.002

• Polynomial recurrences
• Free subgroup numbers
• Inhomogeneous modular group
• Pade approximant
• Hypergeometric series

It is shown that the number f(lambda) of free subgroups of index 6 lambda in the modular group PSL2(Z), when considered modulo a prime power p(alpha) with p %26gt;= 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) = C-2 * C-4. All this is achieved by explicitly determining Pade 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.

• 出版日期2013-11