A new approach to fully degenerate Bernoulli numbers and polynomials

Abstract

In this paper, we consider the doubly indexed sequence a(r) ? (n,m), (n,m ? 0), defined by a recurrence relation and an initial sequence a(r) ? (0,m), (m ? 0). We derive with the help of some differential operator an explicit expression for a(r) ? (n, 0), in term of the degenerate r-Stirling numbers of the second kind and the initial sequence. We observe that a(r) ? (n, 0) = ?n,?(r), for a(r) ? (0,m) = 1/m+1 , and a(r) ? (n, 0) = En,?(r), for a(r) ? (0,m) = (1/2)m . Here ?n,?(x) and En,?(x) are the fully degenerate Bernoulli polynomials and the degenerate Euler polynomials, respectively.