Many mathematicians studied “poly” as a generalization of the well-known special polynomials such as Bernoulli polynomials, Euler polynomials, Cauchy polynomials, and Genocchi polynomials. In this paper, we define the degenerate poly-Lah-Bell polynomials arising from the degenerate polyexponential functions which are reduced to degenerate Lah-Bell polynomials when k = 1 . In particular, we call these polynomials the “poly-Lah-Bell polynomials” when λ ⟶ 0 . We give their explicit expression, Dobinski-like formulas, and recurrence relation. In addition, we obtain various algebraic identities including Lah numbers, the degenerate Stirling numbers of the first and second kind, the degenerate poly-Bell polynomials, the degenerate poly-Bernoulli numbers, and the degenerate poly-Genocchi numbers.

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.

Many important special numbers appear in the expansions of some polynomials in terms of central factorials and vice versa, for example, central factorial numbers, degenerate central factorial numbers, and central Lah numbers which are recently introduced. Here we generalize this to any sequence of polynomials P = {p n (x)} ∞ n=0 such that deg p n (x) = n, p 0 (x) = 1. The aim of this paper is to study the central factorial numbers of the second kind associated with any sequence of polynomials and of the first kind associated with any sequence of polynomials, in a unified and systematic way with the help of umbral calculus technique. The central factorial numbers associated with any sequence of polynomials enjoy orthogonality and inverse relations. We illustrate our results with many examples and obtain interesting orthogonality and inverse relations by applying such relations for the central factorial numbers associated with any sequence of polynomials to each of our examples.

The aim of this study is to represent any polynomial in terms of the degenerate Genocchi polynomials and more generally of the higher-order degenerate Genocchi polynomials. We derive explicit formulas with the help of umbral calculus and illustrate our results with some examples.

In this paper, we consider the problem of representing any polynomial in terms of the degenerate Bernoulli polynomials arising from Volkenborn integral and more generally of the higher-order degenerate Bernoulli polynomials arising from iterated Volkenborn integral. We derive explicit formulas with the help of umbral calculus and illustrate our results with some examples.