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

${\displaystyle F_{n}\left({\frac {d}{dx}}\right)\operatorname {sech} (x)=\operatorname {sech} (x)P_{n}(\tanh(x)).}${\displaystyle 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

${\displaystyle F_{n}(x)={}_{3}F_{2}\left({\begin{array}{c}-n,~n+1,~{\tfrac {1}{2}}(x+1)\1,円~1\end{array}};1\right).}${\displaystyle F_{n}(x)={}_{3}F_{2}\left({\begin{array}{c}-n,~n+1,~{\tfrac {1}{2}}(x+1)\1,円~1\end{array}};1\right).}

Pasternack (1939) generalized the Bateman polynomials to polynomials F^m
_n with

${\displaystyle F_{n}^{m}\left({\frac {d}{dx}}\right)\operatorname {sech} ^{m+1}(x)=\operatorname {sech} ^{m+1}(x)P_{n}(\tanh(x))}${\displaystyle 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

${\displaystyle F_{n}^{m}(x)={}_{3}F_{2}\left({\begin{array}{c}-n,~n+1,~{\tfrac {1}{2}}(x+m+1)\1,円~m+1\end{array}};1\right).}${\displaystyle F_{n}^{m}(x)={}_{3}F_{2}\left({\begin{array}{c}-n,~n+1,~{\tfrac {1}{2}}(x+m+1)\1,円~m+1\end{array}};1\right).}

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

${\displaystyle Q_{n}(x)=(-1)^{n}2^{n}n!{\binom {2n}{n}}^{-1}F_{n}(2x+1)}${\displaystyle Q_{n}(x)=(-1)^{n}2^{n}n!{\binom {2n}{n}}^{-1}F_{n}(2x+1)}

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

The polynomials of small n read

${\displaystyle F_{0}(x)=1}${\displaystyle F_{0}(x)=1};

${\displaystyle F_{1}(x)=-x}${\displaystyle F_{1}(x)=-x};

${\displaystyle F_{2}(x)={\frac {1}{4}}+{\frac {3}{4}}x^{2}}${\displaystyle F_{2}(x)={\frac {1}{4}}+{\frac {3}{4}}x^{2}};

${\displaystyle F_{3}(x)=-{\frac {7}{12}}x-{\frac {5}{12}}x^{3}}${\displaystyle F_{3}(x)=-{\frac {7}{12}}x-{\frac {5}{12}}x^{3}};

${\displaystyle F_{4}(x)={\frac {9}{64}}+{\frac {65}{96}}x^{2}+{\frac {35}{192}}x^{4}}${\displaystyle F_{4}(x)={\frac {9}{64}}+{\frac {65}{96}}x^{2}+{\frac {35}{192}}x^{4}};

${\displaystyle F_{5}(x)=-{\frac {407}{960}}x-{\frac {49}{96}}x^{3}-{\frac {21}{320}}x^{5}}${\displaystyle F_{5}(x)=-{\frac {407}{960}}x-{\frac {49}{96}}x^{3}-{\frac {21}{320}}x^{5}};

The Bateman polynomials satisfy the orthogonality relation^{[1] [2]}

${\displaystyle \int _{-\infty }^{\infty }F_{m}(ix)F_{n}(ix)\operatorname {sech} ^{2}\left({\frac {\pi x}{2}}\right),円dx={\frac {4(-1)^{n}}{\pi (2n+1)}}\delta _{mn}.}${\displaystyle \int _{-\infty }^{\infty }F_{m}(ix)F_{n}(ix)\operatorname {sech} ^{2}\left({\frac {\pi x}{2}}\right),円dx={\frac {4(-1)^{n}}{\pi (2n+1)}}\delta _{mn}.}

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

${\displaystyle \int _{-\infty }^{\infty }B_{m}(x)B_{n}(x)\operatorname {sech} ^{2}\left({\frac {\pi x}{2}}\right),円dx={\frac {4}{\pi (2n+1)}}\delta _{mn}.}${\displaystyle \int _{-\infty }^{\infty }B_{m}(x)B_{n}(x)\operatorname {sech} ^{2}\left({\frac {\pi x}{2}}\right),円dx={\frac {4}{\pi (2n+1)}}\delta _{mn}.}

The sequence of Bateman polynomials satisfies the recurrence relation^[3]

${\displaystyle (n+1)^{2}F_{n+1}(z)=-(2n+1)zF_{n}(z)+n^{2}F_{n-1}(z).}${\displaystyle (n+1)^{2}F_{n+1}(z)=-(2n+1)zF_{n}(z)+n^{2}F_{n-1}(z).}

The Bateman polynomials also have the generating function

${\displaystyle \sum _{n=0}^{\infty }t^{n}F_{n}(z)=(1-t)^{z},円_{2}F_{1}\left({\frac {1+z}{2}},{\frac {1+z}{2}};1;t^{2}\right),}${\displaystyle \sum _{n=0}^{\infty }t^{n}F_{n}(z)=(1-t)^{z},円_{2}F_{1}\left({\frac {1+z}{2}},{\frac {1+z}{2}};1;t^{2}\right),}

which is sometimes used to define them.^[4]

• 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, JFM 59.0364.02
• 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, MR 0085361
• Koelink, H. T. (1996), "On Jacobi and continuous Hahn polynomials", Proceedings of the American Mathematical Society , 124 (3): 887–898, arXiv:math/9409230 , doi:10.1090/S0002-9939-96-03190-5 , ISSN 0002-9939, MR 1307541
• 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, MR 0000698
• 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, MR 0079021

• Orthogonal polynomials