Degenerater‐Bell Polynomials Arising from Degenerate Normal Ordering

Abstract

Recently, Kim-Kim introduced the degenerate r -Bell polynomials and investigated some results which are derived from umbral calculus. The aim of this paper is to study some properties of the degenerate r -Bell polynomials and numbers via boson operators. In particular, we obtain two expressions for the generating function of the degenerate r -Bell polynomials in z 2 , and a recurrence relation and Dobinski-like formula for the degenerate r -Bell numbers. These are derived from the degenerate normal ordering of a degenerate integral power of the number operator in terms of boson operators where the degenerate r -Stirling numbers of the second kind appear as the coefficients.

Some identities involving degenerate Stirling numbers arising from normal ordering

“…We recall that φ n,λ and S 2,λ (n, k) are related to special quantum states, called coherent states, defined as linear combinations of the eigenstates of the harmonic oscillator, H = a † a, H|n = n|n , n|m = δ m,n and defined as |z = e − |z| 2 2 ∑ ∞ n=0 z n √ n |n , with z|z = 1, for complex z, (see [3,7,[11][12][13][14]).…”

“…t n n! , (see [8,9,11,12]). ( 9) By ( 9), we get (1), (n ≥ 0), are called the degenerate Bell numbers.…”

“…Recently, the degenerate Stirling numbers of the second kind are defined by [7,8,10,11,12]), (6) where…”

“…Recall that a and a † are the boson annihilation and creation operators such that [a, a † ] = aa † − a † a = 1, (see [2,3,[6][7][8][11][12][13]15]). (11) The number states |m , m = 0, 1, • • • , are given by…”

Normal ordering associated with λ-Stirling numbers inλ-Shift algebra

“…The Weyl algebra is the unital algebra generated by letters a and a † satisfying the commutation aa † − a † a = 1, (see [1,2,[7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]). (10) Katriel proved that the normal ordering in Weyl algebra is given by (see ( 6))…”