Bateman polynomials

In mathematics, the Bateman polynomials are a family "F""n" of orthogonal polynomials introduced by . The Bateman–Pasternack polynomials are a generalization introduced by .
Bateman polynomials can be defined by the relation
where "P""n" is a Legendre polynomial. In terms of generalized hypergeometric functions, they are given by 
 generalized the Bateman polynomials to polynomials "F" with
These generalized polynomials also have a representation in terms of generalized hypergeometric functions, namely
 showed that the polynomials "Q""n" studied by , see Touchard polynomials, are the same as Bateman polynomials up to a change of variable: more precisely
Bateman and Pasternack's polynomials are special cases of the symmetric continuous Hahn polynomials.
Examples.
The polynomials of small "n" read
Properties.
Orthogonality.
The Bateman polynomials satisfy the orthogonality relation
The factor formula_13 occurs on the right-hand side of this equation because the Bateman polynomials as defined here must be scaled by a factor formula_14 to make them remain real-valued for imaginary argument. The orthogonality relation is simpler when expressed in terms of a modified set of polynomials defined by formula_15, for which it becomes
Recurrence relation.
The sequence of Bateman polynomials satisfies the recurrence relation
Generating function.
The Bateman polynomials also have the generating function
which is sometimes used to define them.