# Continued fractions and Bessel functions

## TL;DR

This paper explores transformations of differential equations, establishing invertibility conditions and applying these to derive continued fractions for Bessel functions and Chebyshev polynomials, revealing connections to Riccati equations.

## Contribution

It introduces new transformations and invertibility criteria for differential equations, linking elementary solutions of Bessel equations to fixed point Riccati transformations.

## Key findings

- Invertibility condition for differential equations established
- Transformations of Riccati equations constructed
- Continued fractions for Bessel functions derived

## Abstract

Elementary transformations of equations $A\psi=\lambda\psi$ are considered. The invertibility condition (Theorem 1) is established and similar transformations of Riccati equations in the case of second order differential operator $A$ are constructed (Theorem 2). Applications to continuous fractions for Bessel functions and Chebyshev polynomials are established. It is shown particularly that the elementary solutions of Bessel equations are related to a fixed point transformations of Riccati equations.