d-Orthogonal Polynomial

Let be a polynomial sequence, and let be linear functionals on the space of polynomials. The sequence is -orthogonal with respect to if

(1)

and

(2)

for and , 2, ..., . The first conditions are the -orthogonality conditions, while the second are regularity conditions. Ordinary orthogonal polynomials are recovered when .

A generalized Favard theorem states that a monic polynomial sequence is -orthogonal iff it satisfies the initial conditions , , and, for ,

(3)

and the recurrence relation

(4)

where for (Maroni 1989). For , this becomes the usual three-term recurrence for orthogonal polynomials.

A polynomial sequence is called -symmetric if

(5)

where is a root of unity. For a -orthogonal monic sequence, -symmetry is equivalent to

			(6)
			(7)

It also gives the decomposition

(8)

Consequently, the nonzero roots occur in orbits under multiplication by the st roots of unity, and the coefficients in the canonical basis vanish unless their exponents are congruent to the degree modulo (Douak and Maroni 1992, Mesquita and da Rocha 2026).