### On Laguerre Series in the Complex Domain

Yale University

Summary

The purpose of this paper is to study series of the form\[f(z) = \sum_{n=0}^\infty f_nL_n^{(\alpha)}(z) \quad \alpha > -1\]

where the \(L_n^{(\alpha)}(z)\) are Laguerre polynomials of the complex variable *z* and may be defined by\[e^{-z}z^{\alpha}L_n^{(\alpha)} = \frac{1}{n!}\frac{d^n}{dz^n}\left(e^{-z}z^{n+\alpha}\right)\]

It is shown that the domain of convergence of the series is the interior of a parabola and that the series represents an analytic function in this domain. We introduce the differential operator\[\delta_z = -z\frac{d^\alpha}{dz^\alpha} + (z-\alpha-1)\frac{d}{dz}\]

and establish some properties of the operator \(\delta_z\) and functions of \(\delta_z\) which we find useful later in the discussion of singularities of the function \(f(z)\) defined by the Laguerre series on the boundary of the domain of convergence of the series under various assumptions on the coefficients \(f_n\). We give several conditions which will make the domain of convergence of the series the natural domain of existence of \(f(z)\). Also, we give two examples of factor sequences \(\{a_n\}\) which are such that if\[F(z) = \sum_{n=0}^\infty a_nf_nL_n^{(\alpha)}(z)\]

then this series converges in a domain at least as large as the domain of convergence of the series\[\sum_{n=0}^\infty f_nL_n^{(\alpha)}(z)\]

and the function \(F(z)\) can be continued analytically along any finite path along which \(f(z)\) can be so continued.