Abstract
Abstract In this paper, with the aid of the Faà di Bruno formula and by virtue of properties of the Bell polynomials of the second kind, the authors define a kind of notion of degenerate Narumi numbers and polynomials, establish explicit formulas for degenerate Narumi numbers and polynomials, and derive explicit formulas for the Narumi numbers and polynomials and for (degenerate) Cauchy numbers.
Highlights
In [10], several Sheffer sequences and many relations of several polynomials arising from umbral calculus were dealt with
We introduce degenerate Narumi polynomials Nn(α)(x, λ) by λ[e[(1 + t ) λ − 1] / λ (1 + t)λ −
In [2, p. 136, equation [3n]], it was given that the Bell polynomials of the second kind Bn,k satisfy
Summary
The Stirling numbers of the first kind s(n, k) for n ≥ k ≥ 0 can be generated [1,2] by [ln(1 + x)]k. The Stirling numbers of the second kind S(n, k) for n ≥ k ≥ 0 can be generated [1,2] by 303, equation (1.2)], see [6,7,8], the r-associate Stirling numbers of the second kind, denoted by S(n, k; r), were defined by. 306, (3.11)] and [6, Theorem 3.1]) For k ≥ 1, the 1-associate Stirling numbers of the second kind S(n, k; 1) satisfy S(0, 0; 1) = 1, S(n, 0; 1) = 0 for n ≥ 1, and.
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