Padé approximation for words generated by certain substitutions, and Hankel determinants (Q2707578)

Let \(\varepsilon=\varepsilon_0\varepsilon_1\varepsilon_2\cdots(\varepsilon_n=a\text{\;or\;}b)\) be the infinite word formed by two letters \(a,b\) which is invariant under the substitution \(\sigma(a)=a^kb,\sigma(b)=a\) \((k\geq 1)\). Let \(\varepsilon(z)=\varepsilon(z;a,b,k)=\sum_{n\geq 0}\varepsilon_nz^{-n-1}\) be the formal Laurent series with coefficients in \(\mathbb{Q}(a,b)\). Let \(f_n\) and \(g_n\) be two sequences recursively defined by \(x_n=kx_{n-1}+x_{n-2}\) with initial conditions \(f_{-2}=k-1,f_{-1}=1\) and \(g_{-2}=1,g_{-1}=0\), respectively. Set \(h_n=g_n a+g_{n-1}b\). NEWLINENEWLINENEWLINEThe author writes explicitly the Padé approximants of \(\varepsilon(z)\) by using \(f_n\) and \(h_n\) and gives some results about the distribution of the zeros of its denominators.NEWLINENEWLINEFor the entire collection see [Zbl 0932.00040].