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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{d}{dx}\right)\operatorname{sech}(x) = \operatorname{sech}(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{array}{c}-n,n+1,\tfrac12(x+1)\ 1,~1 \end{array}; 1\right).

generalized the Bateman polynomials to polynomials F 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 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{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;

Properties

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.

References

  • {{cite journal|first1= Nadhla A. |last1=Al-Salam |doi-access=free}}

References

  1. Koelink (1996)
  2. Bateman, H. (1934), [https://www.jstor.org/stable/1968493 "The polynomial F_n(x)"], ''Ann. Math.'' '''35''' (4): 767-775.
  3. Bateman (1933), p. 28.
  4. Bateman (1933), p. 23.
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