Applications of Continuous Fractions in Orthogonal Polynomials

Several applications of continuous fractions are restricted to theoretical studies, such as problems associated with the approximation of functions, determination of rational and irrational numbers, applications in physics in determining the resistance of electric circuits and integral equations and in several other areas of mathematics. This work aimed to study the results that open the way for the connection of continuous fractions with the orthogonal polynomials. As support, we will study the general case, where the applications of the Wallis formulas in a monolithic orthogonal polynomial, which generates a continuous fraction of the Jacobi type. It will be allowed applications with relations of recurrence of three terms in the polynomials of Tchebyshev and Legendre, through the results found, establishing connection between them with the continuous fractions. And finally, will be presented the "Number of gold", that is an application of this theory.