In mathematics, the Bateman polynomials are a family Fn of orthogonal polynomials introduced by . The Bateman–Pasternack polynomials are a generalization introduced by . Bateman polynomials can be defined by the relation where Pn 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 Qn 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 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 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. References Orthogonal polynomials