Bateman polynomials

In mathematics, the Bateman polynomials are a family F_n of orthogonal polynomials introduced by Bateman (1933). The Bateman–Pasternack polynomials are a generalization introduced by (Pasternack 1939). Bateman polynomials can be defined by the relation

[math]\displaystyle{ F_n\left(\frac{d}{dx}\right)\operatorname{sech}(x) = \operatorname{sech}(x)P_n(\tanh(x)). }[/math]

where P_n is a Legendre polynomial. In terms of generalized hypergeometric functions, they are given by

[math]\displaystyle{ F_n(x)={}_3F_2\left(\begin{array}{c}-n,~n+1,~\tfrac12(x+1)\\ 1,~1 \end{array}; 1\right). }[/math]

(Pasternack 1939) generalized the Bateman polynomials to polynomials Fmn with

[math]\displaystyle{ F_n^m\left(\frac{d}{dx}\right)\operatorname{sech}^{m+1}(x) = \operatorname{sech}^{m+1}(x)P_n(\tanh(x)) }[/math]

These generalized polynomials also have a representation in terms of generalized hypergeometric functions, namely

[math]\displaystyle{ F_n^m(x)={}_3F_2\left(\begin{array}{c}-n,~n+1,~\tfrac12(x+m+1)\\ 1,~m+1 \end{array}; 1\right). }[/math]

(Carlitz 1957) showed that the polynomials Q_n studied by (Touchard 1956) , see Touchard polynomials, are the same as Bateman polynomials up to a change of variable: more precisely

[math]\displaystyle{ Q_n(x)=(-1)^n2^nn!\binom{2n}{n}^{-1}F_n(2x+1) }[/math]

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

Examples

The polynomials of small n read

[math]\displaystyle{ F_0(x)=1 }[/math];

[math]\displaystyle{ F_1(x)=-x }[/math];

[math]\displaystyle{ F_2(x)=\frac{1}{4}+\frac{3}{4}x^2 }[/math];

[math]\displaystyle{ F_3(x)=-\frac{7}{12}x-\frac{5}{12}x^3 }[/math];

[math]\displaystyle{ F_4(x)=\frac{9}{64}+\frac{65}{96}x^2+\frac{35}{192}x^4 }[/math];

[math]\displaystyle{ F_5(x)=-\frac{407}{960}x-\frac{49}{96}x^3-\frac{21}{320}x^5 }[/math];

Properties

Orthogonality

The Bateman polynomials satisfy the orthogonality relation

[math]\displaystyle{ \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}. }[/math]

The factor [math]\displaystyle{ (-1)^n }[/math] occurs on the right-hand side of this equation because the Bateman polynomials as defined here must be scaled by a factor [math]\displaystyle{ i^n }[/math] 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 [math]\displaystyle{ B_n(x)=i^nF_n(ix) }[/math], for which it becomes

[math]\displaystyle{ \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}. }[/math]

Recurrence relation

The sequence of Bateman polynomials satisfies the recurrence relation

[math]\displaystyle{ (n+1)^2F_{n+1}(z)=-(2n+1)zF_n(z) + n^2F_{n-1}(z). }[/math]

Generating function

The Bateman polynomials also have the generating function

[math]\displaystyle{ \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), }[/math]

which is sometimes used to define them.

References

• Al-Salam, Nadhla A. (1967). "A class of hypergeometric polynomials". Ann. Mat. Pura Appl. 75 (1): 95–120. doi:10.1007/BF02416800. 
• Bateman, H. (1933), "Some properties of a certain set of polynomials.", Tôhoku Mathematical Journal 37: 23–38, https://www.jstage.jst.go.jp/article/tmj1911/37/0/37_0_23/_article/-char/ja/ 
• Carlitz, Leonard (1957), "Some polynomials of Touchard connected with the Bernoulli numbers", Canadian Journal of Mathematics 9: 188–190, doi:10.4153/CJM-1957-021-9, ISSN 0008-414X 
• Koelink, H. T. (1996), "On Jacobi and continuous Hahn polynomials", Proceedings of the American Mathematical Society 124 (3): 887–898, doi:10.1090/S0002-9939-96-03190-5, ISSN 0002-9939 
• Pasternack, Simon (1939), "A generalization of the polynomial F_n(x)", London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 28 (187): 209–226, doi:10.1080/14786443908521175 
• Touchard, Jacques (1956), "Nombres exponentiels et nombres de Bernoulli", Canadian Journal of Mathematics 8: 305–320, doi:10.4153/cjm-1956-034-1, ISSN 0008-414X