Definition.
Orthogonality.
Recurrence relation.
Normalized recurrence relation.
where
Differential equation.
Forward shift operator.
Backward shift operator.
or equivalently
Rodrigues-type formula.
Generating functions.
where 
denotes the largest integer smaller than or equal to 
.
Remarks. The Hermite polynomials can also be written as :
where 
denotes the largest integer smaller than or equal to 
.
The Laguerre polynomials defined by (1.11.1) and the Hermite polynomials defined by (1.13.1) are connected by the following quadratic transformations :
and
References. [2], [10], [13], [18], [19], [31], [34], [39], [43], [49], [64], [74], [82], [87], [89], [91], [92], [112], [123], [128], [131], [137], [138], [154], [158], [195], [196], [200], [201], [202], [214], [239], [274], [285], [288], [301], [302], [306], [314], [316], [323], [329], [332], [360], [367], [376], [381], [388], [390], [394], [397], [406].
