Composition of linear fractional transformations in terms of tail sequences

Abstract:

We consider sequences $\left \{ {{s_n}} \right \}$ of linear fractional transformations. Connected to such a sequence is another sequence $\left \{ {{s_n}} \right \}$ of linear fractional transformations given by \[ {S_n} = {s_1} \circ {s_2} \circ \cdots \circ {s_n},\quad n = 1,2,3, \ldots .\] We introduce a new way of representing ${s_n}$ (in terms of so-called tail sequences). This representation is established to give nice expressions for ${S_n}$. It can be seen as a generalization of the canonical form for ${s_n}$, which gives nice expressions for \[ {T_k} = \underbrace {{s_n} \circ {s_n} \circ \cdots \circ {s_{n.}}}_{(k{\text { terms)}}}\]