ACADEMIA Letters A new identity relating a polynomial to infnite series of the hyperbolic secant functions Salar Saadatian, Louisiana State University Harris Wong, Louisiana State University Abstract Hyperbolic functions do not form a complete set and in general it is not possible to expand a given function as an infnite series of hyperbolic functions. Here, we take the classical problem of steady laminar fow in a rectangular duct and turn the duct 90°. The maximum velocity in the duct should remain unchanged if the fow is driven by the same pressure gradient. This leads to a new identity that relates a second-order polynomial to infnite series of the hyperbolic secant functions. We discuss the mathematical properties of this identity and verify it by two diferent methods. 1. Introduction The Dirac delta function is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one [1]. The delta function is useful in modeling many physical processes, such as a point charge, point mass, or point heat source [1]. In some cases, multiple delta functions are needed in a single model, such as the modeling of multiple facet planes in a single crystal [2]. There- fore, it is of general interest to know if a given function can be expanded as an infnite series of the delta function. In numerical simulations, the Dirac delta function is commonly repre- sented by a bump function, such as the hyperbolic secant function [3]. Hence, as a frst step, Academia Letters, November 2021 Corresponding Author: Harris Wong, hwong@lsu.edu Citation: Saadatian, S., Wong, H. (2021). A new identity relating a polynomial to infnite series of the hyperbolic secant functions. Academia Letters, Article 3943. https://doi.org/10.20935/AL3943. 1 ©2021 by the authors — Open Access — Distributed under CC BY 4.0