## Evelyn Boyd

### 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.