  In this paper we build on the work of \cite{kaber} where it was shown that
the one-parameter family of Gegenbauer Polynomials (GP) exhibit a Gibbs
Phenomenon at a jump discontinuity. We show that the one-parameter family of
Generalized Laguerre Polynomials (GLP) also exhibit a Gibbs Phenomenon. Among
many differences, a major one is that the GLP are orthogonal on a non-compact
subset of $\R$, while the GP are orthogonal on $[-1,1]$. Our strategy follows
that of \cite{kaber} and we use entirely elementary methods to arrive at our
result. As a special case we show that the Hermite Polynomials also possess a
Gibbs Phenomenon. We conclude with a numerical example exhibiting the rate of
convergence to the Gibbs constant and a conjectured identity for special values
of the GLP.
