HBpolynomials

HBpolynomials are a family of polynomials in one variable, defined by a parameter sequence and a second-order recurrence. They arise in mathematical contexts where families of polynomials are studied via recurrence relations and are used to generalize several classical polynomial families.

Definition and construction

Let H0(x) = 1 and H1(x) = x. For a fixed sequence a = (a0, a1, a2, …) of nonzero real numbers, the HBpolynomials are defined for n ≥ 2 by

Hn(x; a) = x · Hn−1(x; a) − a(n−1) · Hn−2(x; a).

The HBpolynomials thus depend on the chosen parameter sequence a, and for each such sequence they form a distinct family with deg Hn = n.

Properties

For a fixed a, the HBpolynomials satisfy a three-term recurrence, which implies linear independence of successive polynomials and that the set {H0, H1, …, Hn} spans the space of polynomials of degree at most n. Under suitable choices of the sequence a, the HBpolynomials can be orthogonal with respect to a weight function on an interval and related to moment problems. Closed forms and explicit expressions are available for small n, and efficient generation follows from the recurrence.

Special cases and relationships

Special choices of the parameter sequence a can recover or approximate known families of polynomials, while other sequences yield novel families with different zero patterns and symmetry properties.

Applications

HBpolynomials are used in theoretical studies of polynomial systems, in algebraic combinatorics, and as a flexible framework in numerical methods and approximation theory where a tunable recurrence relation is advantageous.

See also

Orthogonal polynomials, three-term recurrence, Chebyshev polynomials, Hermite polynomials, moment problem.

References

Literature on polynomial recurrence relations and parameterized polynomial families provides context for HBpolynomials and their properties.