Binomial Type - Characterization By A Convolution Identity

Characterization By A Convolution Identity

For sequences a_n, b_n, n = 0, 1, 2, ..., define a sort of convolution by

Let be the nth term of the sequence

Then for any sequence a_i, i = 0, 1, 2, ..., with a₀ = 0, the sequence defined by p₀(x) = 1 and

for n ≥ 1, is of binomial type, and every sequence of binomial type is of this form. This result is due to Alessandro di Bucchianico (see References below).