Definition.
Orthogonality.
Recurrence relation.
Normalized recurrence relation.
where
Difference equation.
where
Forward shift operator.
or equivalently
Backward shift operator.
or equivalently
Rodrigues-type formula.
Generating functions. For 
we have
Remarks. The Krawtchouk polynomials are self-dual, which means that
By using this relation we easily obtain the so-called dual orthogonality relation from the orthogonality relation (1.10.2) :
where 0 < p < 1 and 
.
The Krawtchouk polynomials are related to the Meixner polynomials defined by (1.9.1) in the following way :
References. [13], [31], [32], [39], [43], [50], [64], [67], [104], [119], [123], [136], [142], [145], [146], [154], [159], [181], [183], [212], [250], [272], [274], [286], [287], [294], [296], [298], [301], [307], [323], [338], [340], [385], [386], [388], [407], [409].
