Research Article On the Regular Integral Solutions of a Generalized Bessel Differential Equation L. M. B. C. Campos, 1 F. Moleiro , 1 M. J. S. Silva, 2 and J. Paquim 2 1 Center for Aeronautical and Space Science and Technology (CCTAE), IDMEC, LAETA, Instituto Superior T´ ecnico, Universidade de Lisboa, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal 2 Instituto Superior T´ ecnico, Universidade de Lisboa, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal Correspondence should be addressed to F. Moleiro; filipa.moleiro@tecnico.ulisboa.pt Received 15 July 2018; Accepted 28 October 2018; Published 4 November 2018 Academic Editor: Emilio Turco Copyright © 2018 L. M. B. C. Campos et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e original Bessel differential equation that describes, among many others, cylindrical acoustic or vortical waves, is a particular case of zero degree of the generalized Bessel differential equation that describes coupled acoustic-vortical waves. e solutions of the generalized Bessel differential equation are obtained for all possible combinations of the two complex parameters, order and degree, and finite complex variable, as Frobenius-Fuchs series around the regular singularity at the origin; the series converge in the whole complex plane of the variable, except for the point-at-infinity, that is, the only other singularity of the differential equation. e regular integral solutions of the first and second kinds lead, respectively, to the generalized Bessel and Neumann functions; these reduce to the original Bessel and Neumann functions for zero degree and have alternative expressions for nonzero degree. 1. Introduction e Bessel differential equation was first considered in connexion with the oscillations of a heavy chain [1] and vibrations of a circular membrane [2] and has had since [3] a vast number of applications supported by an extensive theory [4]. A substantial number of applications arise from the separation of variables in the Laplace operator in cylindrical and spherical coordinates that leads, respectively, to the cylin- drical and spherical Bessel functions. e cases of specific interest include two types of cylindrical waves: (i) sound waves as compressible perturbations of a uniform flow; (ii) vortical waves as incompressible perturbations of a uniform flow with superimposed rigid body rotation. Whereas (i) and (ii) separately lead to the original Bessel differential equation, their coupling leads to a generalization. us, the consideration of coupled acoustic-vortical waves as rotational compressible perturbations of a uniform mean flow with rigid body rotation leads to the generalized Bessel equation that differs from the original in having an extra term involving a second parameter, namely, the degree , in addition to the order ]. e generalized Bessel differential equation of order ] and degree  may have other applications and deserves separate study as it leads to generalizations of the Bessel and Neumann functions. e generalized Bessel differential equation may also be obtained, aside from any physical or engineering motivations, by a purely mathematical argument, starting from the original Bessel differential equation and replacing the coefficients of the dependent variable and its derivative by polynomials of the independent variable; in this case the ori- gin remains a regular singularity of the differential equation and the only other singularity is the point-at-infinity. us, solutions exist as Frobenius-Fuchs series [5, 6] with recur- rence formula for the coefficients reducing to two terms only in the case of the generalized Bessel differential equations. e generalized Bessel differential equation has singular- ities only at the origin and infinity. Since the singularity at the origin is regular, the Frobenius-Fuchs method leads to solutions valid for finite values of the variable. e solutions of the generalized Bessel differential equation around the regular singularity at the origin has (i) indices that are exponents of the leading power depending only on the order; (ii) recurrence relation for the coefficients of the power series Hindawi Advances in Mathematical Physics Volume 2018, Article ID 8919516, 9 pages https://doi.org/10.1155/2018/8919516