# A
Riccati differential equation and free subgroup numbers
for lifts of *PSL*_{2}(**Z**) modulo prime powers

### (30 pages)

**Abstract.**
It is shown that the sequence of numbers *f*_{\lambda} of free subgroups
of index 6*\lambda* in the
modular group *PSL*_{2}(**Z**), when considered
modulo a prime power *p*^{\alpha} 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 *PSL*_{2}(**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
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
([*Electron. J. Combin.* **18**(2) (2012), Article P37]
and [*"A
method for determining the mod*-3^{k} behaviour of
recursive sequences," preprint]),
where it is shown, among other things, that the
free subgroup numbers of *PSL*_{2}(**Z**) and
its lifts display rather complex behaviour modulo powers of 2 and 3.

The following versions are available:

The paper is accompanied by the *Mathematica* input files:
Both *Mathematica* files provide a function, `FreeGF`, which produces
automatically the rational function form of the generating function for
the numbers of free subgroups in *PSL*_{2}(**Z**)
(file `psl2zmod.m`), respectively in **H**(4)
(file `h4lift.m`) of given
index, when reduced modulo a prime power *p*^{e},
as predicted by Theorem 11. For using it, you have to first load the file:
`In[1]:= <<psl2zmod.m`

and then call the function:

`In[2]:= FreeGF[p,e]`

(where, clearly, `p` has to be a specific prime number and
`e` a specific exponent). For the file `h4lift.m`
this works analogously.

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