Normal ordering of degenerate integral powers of number operator and its applications

Degenerate r-associated Stirling numbers

“…Explorations for the degenerate versions of some special numbers and polynomials have become lively interests for some mathematicians in recent years, which began from the pioneering work of Carlitz (see [1,2]). These have been done by employing various methods, such as generating functions, combinatorial methods, p-adic analysis, umbral calculus, operator theory, differential equations, special functions, probability theory and analytic number theory (see [5,[9][10][11][12][13]16,17] and the references therein).…”

Degenerate Lah–Bell polynomials arising from degenerate Sheffer sequences

“…It turns out that it is fascinating and fruitful to study varioius degenerate versions of some special polynomials and numbers (see [6][7][8][9][10][11][12] and the references therein), which has its origin in the work of Carlitz in [2]. They have been explored by using various different methods and led to the introduction of degenerate gamma functions and degenerate umbral calculus.…”

“…(see [2,[6][7][8][9]12]). (2) For r ∈ Z with r ≥ 0, the degenerate r-Stirling numbers of the second kind are defined by Kim-Kim as…”

“…The number operator is defined by n|k = k|k , (k ≥ 0), (see [5,8,9]). ( 9) By ( 8) and ( 9), we get n = a † a.…”

“…, (see [8,9]), (11) where S 2,λ (n, k) are the degenerate Stirling numbers of the second kind defined by…”