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

The Bateman polynomials satisfy the orthogonality relation
The factor occurs on the right-hand side of this equation because the Bateman polynomials as defined here must be scaled by a factor 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, for which it becomes

The sequence of Bateman polynomials satisfies the recurrence relation

The Bateman polynomials also have the generating function
which is sometimes used to define them.