A method is presented, with standard examples, based on an elementary complex integral expression, for developing, in particular, series solutions for second-order linear homogeneous ordinary differential equations. Straightforward to apply, the method reduces the task of finding a series solution to the solution, instead, of a system of simple equations in a single variable. The method elimina...

Received 28 June 2007, in final form 13 November 2007 Published 12 December 2007 Online at stacks.iop.org/JPhysA/41/015207 Abstract Various functions, defined as infinite series of products of Bessel functions of the first kind, are studied. Integral representations are obtained, and then used to deduce asymptotic approximations. Although several methods have been investigated (including power ...

In this paper, we prove two theorems on ∣A∣k, 1 ≤ k ≤ 2, summability of orthogonal series. The first one gives a sufficient condition under which an orthogonal series is absolutely summable almost everywhere, and the second one, is a general theorem, which also gives a sufficient condition so that an orthogonal series is absolutely summable almost everywhere, but it involves a positive numerica...

There is a theory of continued fractions for formal power series in x−1 with coefficients in a field Fq. This theory bears a close analogy with classical continued fractions for real numbers with formal power series playing the role of real numbers and the sum of the terms of non-negative degree in x playing the role of the integral part. In this paper we give a family of formal power series in...