Polynomial Connection Coefficient

Let and be two polynomial sequences. The polynomial connection coefficients are defined by the change of basis

(1)

The array forms a lower triangular matrix. If both sequences are monic, then .

If is orthogonal with respect to a regular linear functional , then orthogonality extracts the coefficients directly as

(2)

Suppose the monic orthogonal sequence satisfies

(3)

where , and the monic -orthogonal sequence satisfies

(4)

Then their connection coefficients obey

(5)

for and , with

(6)

Together with , this gives a recursive algorithm for the connection coefficients. Taking , equivalently , gives the coefficients in the canonical basis (Mesquita and da Rocha 2026).

If both sequences are Appell sequences, differentiation gives the simpler diagonal relation

(7)

Thus, every diagonal of the connection matrix is determined by its first nonzero entry.