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
:$F_n\left(\frac\right)\operatorname(x) = \operatorname(x)P_n(\tanh(x)).$
where ''P''_''n'' is a Legendre polynomial. In terms of generalized hypergeometric functions, they are given by
:$F_n(x)=_3F_2\left(\begin-n,~n+1,~\tfrac12(x+1)\\ 1,~1 \end; 1\right).$

generalized the Bateman polynomials to polynomials ''F'' with

:$F_n^m\left(\frac\right)\operatorname^(x) = \operatorname^(x)P_n(\tanh(x))$

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

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
:$Q_n(x)=(-1)^n2^nn!\binom^F_n(2x+1)$

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

Examples

The polynomials of small ''n'' read
:$F_0(x)=1$;
:$F_1(x)=-x$;
:$F_2(x)=\frac+\fracx^2$;
:$F_3(x)=-\fracx-\fracx^3$;
:$F_4(x)=\frac+\fracx^2+\fracx^4$;
:$F_5(x)=-\fracx-\fracx^3-\fracx^5$;

Properties

Orthogonality

The Bateman polynomials satisfy the orthogonality relation
:$\int_^F_m(ix)F_n(ix)\operatorname^2\left(\frac\right)\,dx = \frac\delta_.$
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
:$\int_^B_m(x)B_n(x)\operatorname^2\left(\frac\right)\,dx = \frac\delta_.$

Recurrence relation

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

Generating function

The Bateman polynomials also have the generating function
:$\sum_^t^nF_n(z)=(1-t)^z\,_2F_1\left(\frac,\frac;1;t^2\right),$
which is sometimes used to define them.Bateman (1933), p. 23.

References

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*{{Citation , last1=Touchard , first1=Jacques , title=Nombres exponentiels et nombres de Bernoulli , mr=0079021 , year=1956 , journal=Canadian Journal of Mathematics , issn=0008-414X , volume=8 , pages=305–320 , doi=10.4153/cjm-1956-034-1 , doi-access=free

Orthogonal polynomials