# Bateman polynomials

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Bateman_polynomials
> Markdown URL: https://mediated.wiki/source/Bateman_polynomials.md
> Source: https://en.wikipedia.org/wiki/Bateman_polynomials
> Source revision: 1295206206
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

In mathematics, the **Bateman polynomials** are a family *F**n* of [orthogonal polynomials](/source/Orthogonal_polynomials) introduced by txt. The **Bateman–Pasternack polynomials** are a generalization introduced by txt.

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](/source/Legendre_polynomial). In terms of [generalized hypergeometric functions](/source/Generalized_hypergeometric_function), 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).

Pasternack (1939) 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).

Carlitz (1957) showed that the polynomials *Q**n* studied by Touchard (1956) , see [Touchard polynomials](/source/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](/source/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[1][2]

- \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[3]

- (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.[4]

## References

1. Koelink (1996)

1. Bateman, H. (1934), ["The polynomial F_n(x)"](https://www.jstor.org/stable/1968493), *Ann. Math.* **35** (4): 767-775.

1. Bateman (1933), p. 28.

1. Bateman (1933), p. 23.

- Al-Salam, Nadhla A. (1967). "A class of hypergeometric polynomials". *Ann. Mat. Pura Appl.*. **75** (1): 95–120. [doi:10.1007/BF02416800](https://doi.org/10.1007/BF02416800)
- Bateman, H. (1933), ["Some properties of a certain set of polynomials."](https://www.jstage.jst.go.jp/article/tmj1911/37/0/37_0_23/_article/-char/ja/), *Tôhoku Mathematical Journal*. **37**: 23–38
- Carlitz, Leonard (1957), "Some polynomials of Touchard connected with the Bernoulli numbers", *[Canadian Journal of Mathematics](/source/Canadian_Journal_of_Mathematics)*. **9**: 188–190, [doi:10.4153/CJM-1957-021-9](https://doi.org/10.4153/CJM-1957-021-9). [ISSN 0008-414X](https://www.worldcat.org/issn/0008-414X). MR 0085361
- Koelink, H. T. (1996), "On Jacobi and continuous Hahn polynomials", *[Proceedings of the American Mathematical Society](/source/Proceedings_of_the_American_Mathematical_Society)*. **124** (3): 887–898, [arXiv:math/9409230](https://arxiv.org/abs/math/9409230). [doi:10.1090/S0002-9939-96-03190-5](https://doi.org/10.1090/S0002-9939-96-03190-5). [ISSN 0002-9939](https://www.worldcat.org/issn/0002-9939). MR 1307541
- Pasternack, Simon (1939), "A generalization of the polynomial Fn(x)", *London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science*. **28** (187): 209–226, [doi:10.1080/14786443908521175](https://doi.org/10.1080/14786443908521175). MR 0000698
- Touchard, Jacques (1956), "Nombres exponentiels et nombres de Bernoulli", *[Canadian Journal of Mathematics](/source/Canadian_Journal_of_Mathematics)*. **8**: 305–320, [doi:10.4153/cjm-1956-034-1](https://doi.org/10.4153/cjm-1956-034-1). [ISSN 0008-414X](https://www.worldcat.org/issn/0008-414X). MR 0079021

---
Adapted from the Wikipedia article [Bateman polynomials](https://en.wikipedia.org/wiki/Bateman_polynomials) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Bateman_polynomials?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.