  The class of differential-equation eigenvalue problems
$-y''(x)+x^{2N+2}y(x)=x^N Ey(x)$ ($N=-1,0,1,2,3,...$) on the interval
$-\infty<x<\infty$ can be solved in closed form for all the eigenvalues $E$ and
the corresponding eigenfunctions $y(x)$. The eigenvalues are all integers and
the eigenfunctions are all confluent hypergeometric functions. The
eigenfunctions can be rewritten as products of polynomials and functions that
decay exponentially as $x\to\pm \infty$. For odd $N$ the polynomials that are
obtained in this way are new and interesting classes of orthogonal polynomials.
For example, when N=1, the eigenfunctions are orthogonal polynomials in $x^3$
multiplying Airy functions of $x^2$. The properties of the polynomials for all
$N$ are described in detail.
