Cameron’s operator in terms of determinants and hypergeometric numbers

Abstract

By studying Cameron’s operator in terms of determinants, two kinds of “integer” sequences of incomplete numbers were introduced. One was the sequence of restricted numbers, including s-step Fibonacci sequences. Another was the sequence of associated numbers, including Lamé sequences of higher order. By the classical Trudi’s formula and the inverse relation, more expressions were able to be obtained. These relations and identities can be extended to those of sequence of negative integers or rational numbers. As applications, we consider hypergeometric Bernoulli, Cauchy and Euler numbers with some modifications.