Bateman polynomials

In mathematics, the Bateman polynomials are a family Fn 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 ( d d x ) sech ⁡ ( x ) = 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 Pn is a Legendre polynomial. In terms of generalized hypergeometric functions, they are given by

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

Pasternack (1939) generalized the Bateman polynomials to polynomials Fm
n with

F n m ( d d x ) sech m + 1 ⁡ ( x ) = 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

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

Carlitz (1957) showed that the polynomials Qn 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 ! ( 2 n n ) − 1 F n ( 2 x + 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.

Examples

The polynomials of small n read

F 0 ( x ) = 1 {\displaystyle F_{0}(x)=1} ;
F 1 ( x ) = − x {\displaystyle F_{1}(x)=-x} ;
F 2 ( x ) = 1 4 + 3 4 x 2 {\displaystyle F_{2}(x)={\frac {1}{4}}+{\frac {3}{4}}x^{2}} ;
F 3 ( x ) = − 7 12 x − 5 12 x 3 {\displaystyle F_{3}(x)=-{\frac {7}{12}}x-{\frac {5}{12}}x^{3}} ;
F 4 ( x ) = 9 64 + 65 96 x 2 + 35 192 x 4 {\displaystyle F_{4}(x)={\frac {9}{64}}+{\frac {65}{96}}x^{2}+{\frac {35}{192}}x^{4}} ;
F 5 ( x ) = − 407 960 x − 49 96 x 3 − 21 320 x 5 {\displaystyle 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]

∫ − ∞ ∞ F m ( i x ) F n ( i x ) sech 2 ⁡ ( π x 2 ) d x = 4 ( − 1 ) n π ( 2 n + 1 ) δ m n . {\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}.}

The factor ( − 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 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 B n ( x ) = i n F n ( i x ) {\displaystyle B_{n}(x)=i^{n}F_{n}(ix)} , for which it becomes

∫ − ∞ ∞ B m ( x ) B n ( x ) sech 2 ⁡ ( π x 2 ) d x = 4 π ( 2 n + 1 ) δ m n . {\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}.}

Recurrence relation

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

( n + 1 ) 2 F n + 1 ( z ) = − ( 2 n + 1 ) z F 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).}

Generating function

The Bateman polynomials also have the generating function

∑ n = 0 ∞ t n F n ( z ) = ( 1 − t ) z 2 F 1 ( 1 + z 2 , 1 + z 2 ; 1 ; t 2 ) , {\displaystyle \sum _{n=0}^{\infty }t^{n}F_{n}(z)=(1-t)^{z}\,_{2}F_{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)
  2. ^ Bateman, H. (1934), "The polynomial F n ( x ) {\displaystyle F_{n}(x)} ", Ann. Math. 35 (4): 767-775.
  3. ^ Bateman (1933), p. 28.
  4. ^ 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.
  • 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 Fn(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