Bessel Moment Polynomial

The Bessel moment polynomials are the polynomials defined by the even-power sums

(1)

for positive integers and real , where is a Bessel function of the first kind (Moya-Cessa 2026). Each has degree and positive integer coefficients for the powers through , with no constant term. These are distinct from the Bessel polynomials.

Writing

(2)

the coefficients are given by the finite sum

(3)

for (Moya-Cessa 2026). The first few polynomials are

			(4)
			(5)
			(6)
			(7)
			(8)

The linear coefficient is , and the leading coefficient is half the central binomial coefficient, . The leading coefficients for , 2, ... are 1, 3, 10, 35, 126, ... (OEIS A001700), with corresponding to term in the OEIS indexing.

The recurrence relation

(9)

for , together with , generates the family (Moya-Cessa 2026).

The classical Bessel function of the first kind addition identity

(10)

provides a generating function for the even-power sums. Differentiating at gives

(11)

where only powers of through can contribute (Moya-Cessa 2026).