Bateman polynomials

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
 * $$F_n\left(\frac{d}{dx}\right)\operatorname{sech}(x) = \operatorname{sech}(x)P_n(\tanh(x)).$$

where Pn is a Legendre polynomial. In terms of generalized hypergeometric functions, they are given by
 * $$F_n(x)={}_3F_2\left(\begin{array}{c}-n,~n+1,~\tfrac12(x+1)\\ 1,~1 \end{array}; 1\right).$$

generalized the Bateman polynomials to polynomials F$m n$ with


 * $$F_n^m\left(\frac{d}{dx}\right)\operatorname{sech}^{m+1}(x) = \operatorname{sech}^{m+1}(x)P_n(\tanh(x))$$

These generalized polynomials also have a representation in terms of generalized hypergeometric functions, namely
 * $$F_n^m(x)={}_3F_2\left(\begin{array}{c}-n,~n+1,~\tfrac12(x+m+1)\\ 1,~m+1 \end{array}; 1\right).$$

showed that the polynomials Qn studied by, see Touchard polynomials, are the same as Bateman polynomials up to a change of variable: more precisely
 * $$ Q_n(x)=(-1)^n2^nn!\binom{2n}{n}^{-1}F_n(2x+1)$$

Bateman and Pasternack's polynomials are special cases of the symmetric continuous Hahn polynomials.

Examples
The polynomials of small n read
 * $$F_0(x)=1$$;
 * $$F_1(x)=-x$$;
 * $$F_2(x)=\frac{1}{4}+\frac{3}{4}x^2$$;
 * $$F_3(x)=-\frac{7}{12}x-\frac{5}{12}x^3$$;
 * $$F_4(x)=\frac{9}{64}+\frac{65}{96}x^2+\frac{35}{192}x^4$$;
 * $$F_5(x)=-\frac{407}{960}x-\frac{49}{96}x^3-\frac{21}{320}x^5$$;

Orthogonality
The Bateman polynomials satisfy the orthogonality relation
 * $$\int_{-\infty}^{\infty}F_m(ix)F_n(ix)\operatorname{sech}^2\left(\frac{\pi x}{2}\right)\,dx = \frac{4(-1)^n}{\pi(2n+1)}\delta_{mn}.$$

The factor $$(-1)^n$$ occurs on the right-hand side of this equation because the Bateman polynomials as defined here must be scaled by a factor $$i^n$$ 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 $$B_n(x)=i^nF_n(ix)$$, for which it becomes
 * $$\int_{-\infty}^{\infty}B_m(x)B_n(x)\operatorname{sech}^2\left(\frac{\pi x}{2}\right)\,dx = \frac{4}{\pi(2n+1)}\delta_{mn}.$$

Recurrence relation
The sequence of Bateman polynomials satisfies the recurrence relation
 * $$(n+1)^2F_{n+1}(z)=-(2n+1)zF_n(z) + n^2F_{n-1}(z).$$

Generating function
The Bateman polynomials also have the generating function
 * $$\sum_{n=0}^{\infty}t^nF_n(z)=(1-t)^z\,_2F_1\left(\frac{1+z}{2},\frac{1+z}{2};1;t^2\right),$$

which is sometimes used to define them.