A Kac-Murdock-Szegö (KMS) matrix is a matrix of the form $A_{ij}=\rho^{|i-j|}$ for $i,j=1,2,\ldots,n$ and $\rho\neq1$.
The inverse of $A^{-1}$ is well known, see e.g. https://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/493.
My question is regarding the inverse of a perturbed KMS matrix $(\alpha I+A)$ where $I$ is the identity matrix and $\alpha\neq0\in\mathbb{R}$.
Does anyone know an explicit formula ( if there exists an explicit formula) for $(\alpha I+A)^{-1}$ ?
With "explicit formula" I am referring to a formula as for $A^{-1}$, no formula like the Woodbury identity. At least a formula which involves only the computation of $A^{-1}$ or $A$ as $(\alpha I+A)^{-1}=\sum_{k=0}^{\infty} (-1)^k A^k \cdot (\alpha)^{-k-1}$, but with finite number of terms.
Probably the answer is negative...
Thanks!