Definition.
Orthogonality.
Recurrence relation.
Normalized recurrence relation.
where
Differential equation.
Forward shift operator.
Backward shift operator.
or equivalently
Rodrigues-type formula.
Generating functions.
Remarks. The definition (1.11.1) of the Laguerre polynomials can also be written as :
In this way the Laguerre polynomials can be defined for all 
. Then we
have the following connection with the Charlier polynomials defined by
(1.12.1) :
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
In combinatorics the Laguerre polynomials with 
are often called Rook
polynomials.
References. [1], [2], [3], [6], [9], [10], [12], [13], [18], [19], [31], [34], [39], [43], [49], [50], [52], [56], [64], [77], [79], [89], [91], [92], [95], [102], [103], [106], [107], [108], [109], [111], [114], [121], [123], [128], [130], [137], [138], [149], [154], [155], [158], [182], [184], [195], [196], [198], [199], [201], [202], [210], [214], [215], [222], [227], [231], [233], [239], [241], [244], [250], [253], [255], [268], [270], [273], [274], [284], [286], [287], [288], [291], [292], [301], [302], [306], [314], [316], [323], [329], [330], [332], [360], [367], [372], [373], [374], [375], [376], [388], [390], [394].
