Matrix continued fractions and expansions of the error function (Q6585120)

Let \(\mathcal{M}_m\) be the set of \(m\times m\) real (complex) matrices endowed with the subordinate matrix infinity norm defined by\N\[\N\|A\|=\max_{1\leq i \leq m} \sum_{j=1}^m |a_{i,j}| \qquad (A=(a_{i,j})\in \mathcal{M}_m ).\N\]\NFor any \(A, B \in \mathcal{M}_m\) with \(A\) invertible, \(B/A\) is written to mean \(A^{-1}B\).\N\NLet \((A_n)_{n\geq 0}, (B)_{n\geq 1}\)be two nonzero sequences of \(\mathcal{M}_m\). The continued fraction of \((A_n)\) and \((B_n)\), denoted by \(K(B_n/A_n)\), is the quantity\N\[\NA_0+\cfrac{B_1}{A_1+\cfrac{B_2}{A_2+\dots}}=\bigg[A_0; \cfrac{B_1}{A_1}, \cfrac{B_2}{A_2}, \dots \bigg].\N\]\NThe definitions and statements for matrix continued fractions are similar to those for continued fractions. Let \(A\) be a matrix in \(\mathcal{M}_m\). The error function is defined by the expression\N\[\N\mathrm{erf}(A)=\frac{2}{\sqrt{\pi}} \sum_{n=0}^{+\infty} \cfrac{(-1)^n}{(2n+1) n!} A^{2n+1}.\N\]\NThe following main result is proven in the paper.\N\NTheorem. Let \(A\) be a matrix in \(\mathcal{M}_m\), such that \(\|A\|=\alpha\), where \(0<\alpha < \tfrac{1}{2}\). The continued fraction\N\[\N\bigg[0; \cfrac{(2/\sqrt{\pi})A}{I}, \cfrac{A^2}{3I-A^2}, \cfrac{-(n-1)(2n-1)^2 A^2}{(-1)^{n-1} (n(2n+1)I-(2n-1)A^2)}\bigg]_{n=2}^{+\infty}\N\]\Nconverges in \(\mathcal{M}_m\). Furthermore, this continued fraction represents \(\mathrm{erf}(A)\). So\N\[\N\mathrm{erf}(A)=\bigg[0; \cfrac{(2/\sqrt{\pi})A}{I}, \cfrac{A^2}{3I-A^2}, \cfrac{-(n-1)(2n-1)^2 A^2}{(-1)^{n-1} (n(2n+1)I-(2n-1)A^2)}\bigg]_{n=2}^{+\infty}.\N\]\NExamples are included in the article.