On application of Euler's differential method to a continued fraction depending on parameter

Abstract

In this paper we apply Euler's differential method, which was not used by mathematicians for a long time, to derive a new formula for a certain kind irregular continued fraction depending on a parameter. This formula is in the form of the ratio of two integrals. In case of integer values of the parameter, the formula reduces to the ratio of two finite sums. Asymptotic behavior of this continued fraction is investigated numerically and it is shown that it increases in the same rate as the root function. © 2014 The Indian National Science Academy. © 2020 Elsevier B.V., All rights reserved.