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Hermite polynomials
Chihara starts with the general recurrence relation
{$P_{n+1}(x)=[x - (dn + f)]P_n(x) - [n(gn + h)]P_{n-1}(x)$} where {$g>0, g+h>0, d, f\in\mathbb{R}$}
Hermite polynomials arise when {$d=0, g=0$}.
Chihara furthermore sets {$f=0$}. This yields
{$P_{n+1}(x)= xP_n(x) - hnP_{n-1}(x)$}
In my notation
{$\alpha=0,\beta=0$}
Note that