We study the algebraic properties of Generalized Laguerre Polynomials for negative integral values of the parameter. For integers $r,n\ge 0$, we conjecture that $L_{n}^{(-1-n-r)}(x)=\Sigma _{j=0}^{n}\left( _{n-j}^{n-j+r} \right){{x}^{j}}/j!$ is a $\mathbb{Q}$-irreducible polynomial whose Galois group contains the alternating group on $n$ letters. That this is so for $r=n$ was conjectured in the 1950's by Grosswald and proven recently by Filaseta and Trifonov. It follows fromrecent work of Hajir and Wong that the conjecture is true when $r$ is large with respect to $n\ge 5$. Here we verify it in three situations: (i) when $n$ is large with respect to $r$, (ii) when $r\le 8$, and (iii) when $n\le 4$. The main tool is the theory of $p$-adic Newton Polygons.

The concept of semi-bounded generalized hypergroups (SBG hypergroups) is developed. These hypergroups are more special than generalized hypergroups introduced by Obata and Wildberger and more general than discrete hypergroups or even discrete signed hypergroups. The convolution of measures and functions is studied. In the case of commutativity we define the dual objects and prove some basic theorems of Fourier analysis. Furthermore, we investigate the relationship between orthogonal polynomials and generalized hypergroups. We discuss the Jacobi polynomials as an example.