Binomial Type - Characterization By Delta Operators

Characterization By Delta Operators

It can be shown that a polynomial sequence { p_n(x) : n = 0, 1, 2, ... } is of binomial type if and only if all three of the following conditions hold:

• The linear transformation on the space of polynomials in x that is characterized by

is shift-equivariant, and

• p₀(x) = 1 for all x, and

• p_n(0) = 0 for n > 0.

(The statement that this operator is shift-equivariant is the same as saying that the polynomial sequence is a Sheffer sequence; the set of sequences of binomial type is properly included within the set of Sheffer sequences.)