Calculation of Univariate Pade Approximants for solutions of the Michaelis-Menten equation with first order input using the Tau method

TL;DR

This work develops a Tau-method-based framework to construct diagonal univariate Padé approximants for the Michaelis–Menten equation with a first-order input, using a Jacobi-recursive scheme to generate A_n and B_n and an associated error structure. It reveals a rich algebraic organization of the Padé coefficients via monomial factorization and systematic cancellations, enabling explicit recursions for the approximation and error terms. While the univariate approach yields y_n(0) consistent approximants, achieving the boundary condition y_n(1)=0 requires extending to bivariate Padé approximants in x and x^β, paving the way for the broader bivariate analysis and potential generalizations to other Riccati-type equations. The methodology provides a detailed, cancellation-friendly pathway to compute Padé approximants and their residuals, supporting future extensions to bivariate formulations and more general coefficient structures.

Abstract

In this paper the Jacobi formula is used to recursively generate (diagonal) univariate Pade approximants using the Tau method for solutions Michaelis-Menten equation with first order input. In the algorithm the Jacobi coefficients and error terms in the Tau method are postulated to have a particular form, and this form is maintained by specific patterns of cancellations.