Bessel moment polynomial

In mathematics and quantum optics, Bessel moment polynomials are a family of polynomials that express the even moments of squared Bessel functions of the first kind. These polynomials arise naturally when computing number-operator moments and statistical properties, such as the Mandel Q parameter, for nonlinear coherent states.

Unlike odd moments, which yield combinations of Bessel functions, the even moments evaluate precisely to finite polynomial expressions in u = x2.

Definition

For a positive integer n ≥ 1, the n-th Bessel moment polynomial pn(u) is defined by the discrete sum of the squared Bessel functions of the first kind, Jk(2x): $\mathcal{P}_n(x)=\sum_{k=0}^{\infty} k^{2n} J_k^2(2x)=p_n(u),\quad \text{where } u=x^2.$

Each pn(u) is a polynomial in u of degree n with positive rational coefficients:

$p_n(u)=\sum_{m=1}^{n} a_{n,m} u^m.$

First polynomials

The first few Bessel moment polynomials are:

  • p1(u) = u
  • p2(u) = 3u2 + u
  • p3(u) = 10u3 + 15u2 + u
  • p4(u) = 35u4 + 140u3 + 63u2 + u
  • p5(u) = 126u5 + 1050u4 + 1470u3 + 255u2 + u

Explicit coefficient formula

The coefficients an, m for 1 ≤ m ≤ n are given explicitly by the finite alternating sum:

$a_{n,m}=\frac{1}{(m!)^2}\sum_{j=0}^{m-1}(-1)^j \binom{2m}{j}(m-j)^{2n}.$

For m > n, an, m = 0.

Specific coefficients

  • Leading coefficient: the highest-degree coefficient an, n is one-half of the n-th central binomial coefficient:

$a_{n,n}=\frac12 \binom{2n}{n}.$

  • Linear coefficient: the coefficient of u1 is always unity:

an, 1 = 1  for all n ≥ 1.

  • Second term: for n ≥ 2, the sub-leading coefficient an, 2 satisfies:

an, 2 = 4n − 1 − 1.

Table of coefficients

The coefficient matrix an, m for 1 ≤ n, m ≤ 5:

n \ m

1

2

3

4

5

1

1

—

—

—

—

2

1

3

—

—

—

3

1

15

10

—

—

4

1

63

140

35

—

5

1

255

1470

1050

126

Differential recurrence relation

The sequence of polynomials satisfies a three-term differential-integral recurrence relation linking pn + 1(u) to pn(u): $p_{n+1}(u)=\left(u\frac{d}{du}\right)^2 p_n(u)+4u\,p_n(u)-2\int_0^u p_n(t)\,dt,$

with initial condition p1(u) = u.

Generating function representation

The polynomial evaluation can be derived using the Jacobi–Anger-type identity for squared Bessel functions:

$\sum_{k=-\infty}^{\infty} J_k^2(2x)e^{iky}=J_0\left(4x\sin\frac{y}{2}\right).$

Differentiating 2n times with respect to y at y = 0 yields the compact derivative form:

$p_n(x^2)=\frac{(-1)^n}{2}\left.\frac{d^{2n}}{dy^{2n}}J_0\left(4x\sin\frac{y}{2}\right)\right|_{y=0}.$

Applications

Bessel moment polynomials are applied in quantum optics and optical physics:

  • Quantum statistics: calculating higher-order photon-number moments and Mandel Q-parameters for nonlinear coherent states generated by Susskind–Glogower phase operators.
  • Waveguide lattices: characterizing intensity dynamics and mode distributions in array couplers and photonic lattices governed by discrete Schrödinger-like equations.

See also

  • Bessel functions
  • Bessel polynomials
  • Lommel polynomial
  • Coherent state