Bateman polynomials explained

In mathematics, the Bateman polynomials are a family F_n of orthogonal polynomials introduced by . The Bateman–Pasternack polynomials are a generalization introduced by .

Bateman polynomials can be defined by the relation

n\left(	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

F_n(x)={}_3F_{2\left(\begin{array}{c}-n,~n+1,~\tfrac12(x+1)\ 1,~1}\end{array};1\right).

generalized the Bateman polynomials to polynomials F with

m\left(	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

	m(x)={}
F
	3F

_{2\left(\begin{array}{c}-n,~n+1,~\tfrac12(x+m+1)\ 1,~m+1}\end{array};1\right).

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

	n2
Q
	n(x)=(-1)

^{nn!\binom{2n}{n}}^-1F_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

2(x)=	1
	4

	3
	4

x²

3(x)=-	7
	12

	5
	12

x³

4(x)=	9
	64

	65
	96

2+	35
	192

x⁴

5(x)=-	407
	960

	49
	96

3-	21
	320

x⁵

Properties

Orthogonality

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

	infty
\int
	-infty

F_m(ix)F

2\left(	\pix
	2

n(ix)\operatorname{sech}

\right)dx=

	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

iⁿ

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

	nF
B
	n(ix)

, for which it becomes

	infty
\int
	-infty

B_m(x)B

2\left(	\pix
	2

n(x)\operatorname{sech}

\right)dx=

	4
	\pi(2n+1)

\delta_mn.

Recurrence relation

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

	2F
(n+1)
	n+1

(z)=-(2n+1)zF_n(z)+

	2F
n
	n-1

(z).

Generating function

The Bateman polynomials also have the generating function

	infty
\sum
	n=0

	z
t
	2F

1\left(	1+z
	2

	1+z
	2

;1;t^2\right),

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

References

Nadhla A. . Al-Salam. A class of hypergeometric polynomials. Ann. Mat. Pura Appl.. 1967 . 75. 1 . 95–120 . 10.1007/BF02416800. free.

Notes and References

Koelink (1996)
Bateman, H. (1934), "The polynomial
F_n(x)

", Ann. Math. 35 (4): 767-775.
Bateman (1933), p. 28.
Bateman (1933), p. 23.