# Pad\'{e}-type approximations to the resolvent of fractional powers of operators

## TL;DR

This paper develops a reliable pole selection method for Padé-type rational approximations of the resolvent of fractional powers of operators, providing accurate error estimates and demonstrating effectiveness through numerical examples.

## Contribution

It introduces a new pole selection strategy based on hypergeometric functions for better rational approximation of fractional operator resolvents.

## Key findings

- Accurate error estimates for Padé approximation of fractional powers
- Numerical validation of theoretical error bounds
- Enhanced rational Krylov methods based on the new approximation approach

## Abstract

We study a reliable pole selection for the rational approximation of the resolvent of fractional powers of operators in both the finite and infinite dimensional setting. The analysis exploits the representation in terms of hypergeometric functions of the error of the Pad\'{e} approximation of the fractional power. We provide quantitatively accurate error estimates that can be used fruitfully for practical computations. We present some numerical examples to corroborate the theoretical results. The behavior of the rational Krylov methods based on this theory is also presented.