Padé approximants are rational functions which interpolate a power series at its center to the "highest possible" order. They can be constructed in an algorithmic manner; their coefficients satisfy linear equations whose coefficients are moments of the approximated function. While their construction does not fix the location of their poles, it doesn't take long to find examples where the poles do behave in a highly structured manner (with the exception of a few wandering poles). For large classes of approximated functions the behavior of the poles of the Padé approximants, as well as their convergence properties, can be explained in detail by exploiting a connection with orthogonal polynomials.

In this talk, I will discuss this connection with orthogonal polynomials and explain the behavior of the poles and convergence properties of Padé approximants while focusing on functions with four branch points as a running example. In doing so, I hope to highlight the breadth of analytical methods used in this study. This work is joint with Maxim Yattselev.