In mathematics, the Bateman polynomials are a family F<sub>n</sub> of orthogonal polynomials introduced by . The BatemanâÂÂPasternack polynomials are a generalization introduced by .
Bateman polynomials can be defined by the relation
where P<sub>n</sub> 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<sub>n</sub> 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.
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.