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

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

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

$F_n(x)={}_3{F}_2(\begin{array}{c}-n,n+1,\frac{1}{2}(x+1)\\ 1,1\end{array};1).$

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

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

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

$F_n^m(x)={}_3{F}_2(\begin{array}{c}-n,n+1,\frac{1}{2}(x+m+1)\\ 1,m+1\end{array};1).$

(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

$Q_n(x)=(-1{)}^n{2}^{n}n!{(\genfrac{}{}{0ex}{}{2n}{n})}^{-1}{F}_n(2x+1)$

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

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{F}_m(ix)F_n(ix){\mathrm{sech}}^2\left(\frac{\pi x}{2}\right)dx=\frac{4(-1{)}^n}{\pi (2n+1)}{\delta }_{mn}.$

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 $B_n(x)=i^n{F}_n(ix)$, for which it becomes

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{B}_m(x)B_n(x){\mathrm{sech}}^2\left(\frac{\pi x}{2}\right)dx=\frac{4}{\pi (2n+1)}{\delta }_{mn}.$

Recurrence relation

The sequence of Bateman polynomials satisfies the recurrence relation

$(n+1{)}^2{F}_{n+1}(z)=-(2n+1)z{F}_n(z)+n^2{F}_{n-1}(z).$

Generating function

The Bateman polynomials also have the generating function

$\sum _{n=0}^{\mathrm{\infty }}{t}^n{F}_n(z)=(1-t{)}^z{}_2{F}_1(\frac{1+z}{2},\frac{1+z}{2};1;t^2),$

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