The generalised ellipsoidal wave equation \([0,3,1_ 1]\) (Q1915558)

The author considers the ordinary differential equation with three regular singularities and one irregular singularity, namely: \[ x(1 - x) (p - x) y'' + \biggl[ (a + b + 1) x^2 - \bigl\{ p(a + b - d + 1) + c + d \bigr\} x + pc \biggr] y' + (kx^2 + abx + q) y = 0 \] and gives its explicit solutions in the following form: \[ y = \sum^\infty_{r = 0} (-k)^r y_r (x). \tag{*} \] The variables and parameters are supposed to be complex. The explicit expression of \(y_r (x)\) in terms of infinite series are given. It is shown that this series converges absolutely and uniformly for all values of the parameters if \(|x |< 1\). Similarly, the series in \((*)\) is also absolutely and uniformly convergent. The behaviour at the point at infinity which is an irregular singular point is also discussed.