Cameron's operator in terms of determinants, and hypergeometric numbers

TL;DR

This paper explores Cameron's operator through determinants, introducing new integer sequences related to Fibonacci and Lamé sequences, and extends classical formulas to rational and negative indices, with applications to hypergeometric special numbers.

Contribution

It introduces new integer sequences via Cameron's operator, extends classical identities, and applies these to hypergeometric Bernoulli, Cauchy, and Euler numbers.

Findings

Introduced sequences of restricted and associated numbers including Fibonacci and Lamé types.

Extended classical formulas like Trudi's to negative and rational indices.

Applied the developed theory to hypergeometric special 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 Lam\'e 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.