The Second Question

That took longer than expected, but now we know all about which matrices represent continued fractions, and we can easily answer the original question: which generalized Cohn matrices represent continued fractions?

• u < 0. None, since denominators can't be negative.
• u = 0. None, since the identity matrix isn't a generalized Cohn matrix.
• u = 1. For every value of a₀, there are two generalized Cohn matrices that represent continued fractions, P(a₀+1) with determinant −1 and P(a₀−1,1) with determinant 1.
• u = 2 and the normal case u ≥ 3. For every value of a₀ and every value of the determinant (±1), earlier we said the following.

  … for every distinct square root [of  ∓1] in the group U_u, we get an infinite family of parameter values, a primary one in the range (0,u) and an infinite number of others that are congruent to it mod u.

  From the second row of the matrix parametrization we can see that only the primary ones lead to matrices that represent continued fractions. Also, from the first column of the matrix parametrization we can see that in this case the arbitrary integer a₀ really is the first coefficient of the expansion.