# Degenerater-Bell Polynomials Arising from Degenerate Normal Ordering

Authors: Taekyun Kim; Dae San Kim; Hye Kyung Kim
Journal: Journal of Mathematics
 (2022)

---

## Abstract

Recently, Kim-Kim introduced the degenerater-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 z2, 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.


---

## Body

## 1. Introduction and Preliminaries

It turns out that it is fascinating and fruitful to study various degenerate versions of some special polynomials and numbers (see [1–8] and the references therein), which has its origin in the work of Carlitz [9]. They have been explored by using various different methods and led to the introduction of degenerate gamma functions and degenerate umbral calculus (see [8, 10]).The aim of this paper is to study some properties of the degenerater-Bell polynomials and numbers via boson operators. In particular, we obtain two expressions for the generating function of the degenerate r-Bell polynomials in z2, and a recurrence relation and Dobinski-like formula for the degenerate r-Bell numbers. These are derived from the degenerate normal orderings 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.In more detail, the outline of this paper is as follows: in Section1, we recall the degenerate exponentials, the degenerate r-Stirling numbers of the second kind, and the degenerate r-Bell polynomials. We remind the reader of the boson operators, the number operators, the normal ordering of an integral power of the number operator in terms of boson operators, and the degenerate normal ordering of a degenerate integral power of the number operator in terms of boson operators. Section 2 is the main result of this paper. In Theorem 1, we state that the degenerate normal orderings of the ‘degenerate integral powers of the number operator’ n^+rn,λ and n^n−r,λa†rar in terms of the boson operators with the coefficients given by the degenerate r-Stirling numbers of the second kind. Let ft=zeλr+n^tz. We show that zn^+rn,λz is equal to ϕk,λrz2 in Theorem 2 and hence that ft is the generating function of that in Theorem 3, where ϕk,λrx is the degenerate r-Bell polynomial. We derive a differential equation for ft in Theorem 4 and thereby get another expression for ft in Theorem 5, which in turn yields an explicit generating function of ϕk,λrz2. In Theorem 6, we obtain a recurrence relation for the degenerate r-Bell numbers ϕn,λr=ϕn,λr1. Finally, we get a Dobinski-like formula for the degenerate r-Bell numbers from the representation of the coherent state in terms of the number operator in Theorem 8. We conclude the paper in Section 3. For the rest of this section, we recall the facts that are needed throughout this paper.For any nonzeroλ∈ℝ, the degenerate exponentials are defined by
(1)eλxt=1+λtx/λ=∑k=0∞xk,λtkk!,eλt=eλ1t,where (see [1–5, 8, 9])
(2)x0,λ=1,xk,λ=xx−λ⋯x−k−1λ,k≥1.Forr∈ℤ with r≥0, the degenerate r-Stirling numbers of the second kind are defined by Kim-Kim as (see [6–8]),
(3)x+rn,λ=∑k=0nn+rk+rr,λxk,n≥0,where x0=1,xn=xx−1⋯x−n+1,n≥1.From equation (3), we note that (see [7, 8])
(4)1k!eλt−1keλrt=∑n=k∞n+rk+rr,λtnn!,k≥0.Note thatlimλ⟶0n+rk+rr,λ=n+rk+rr are the r-Stirling numbers of the second kind. It is well known that the r-Stirling number of the second kind n+rk+rr counts the number of partitions of the set n=1,2,⋯,n into k nonempty disjoint subsets in such a way that the numbers 1,2,3,⋯,r are in distinct subsets.The degenerater-Bell polynomials ϕn,λrx are defined by
(5)eλrtexeλt−1=∑n=0∞ϕn,λrxtnn!.From equations (4) and (5), we note that (see [8])
(6)ϕn,λrx=∑k=0nn+rk+rr,λxk,n≥0.The boson operatorsa† and a satisfy the following commutation relation: (see [11–13])
(7)a,a†=aa†−a†a=1.The number statesm,m=0,1,2,⋯, are defined as (see [3–5, 11])
(8)am=mm−1,a†m=m+1m+1.The number operator is defined by (see [3–5, 12])
(9)n^k=kk,k≥0.By equations (8) and (9), we get n^=a†a.Thus, we note(10)a,n^=an^−n^a=a,n^,a†=n^a†−a†n^=a†.The normal ordering of an integral power of the number operatorn^=a†a in terms of boson operators a and a† can be written in the form (see [3, 4, 11–13])
(11)a†ak=∑l=0kS2k,la†lal,where S2k,l are the Stirling numbers of the second kind defined by xn=∑k=0nS2n,kxk, (see [10, 14, 15]).The degenerate normal ordering of a degenerate integral power of the number operatorn^ in terms of boson operators a and a† is given by (see [3–5])
(12)a†ak,λ=∑l=0kS2,λk,la†lal,where S2,λn,k are the degenerate Stirling numbers of the second kind defined by (see [1])
(13)xn,λ=∑k=0nS2,λn,kxk,n≥0.


## 2. Degenerater-Bell Polynomials Arising from Degenerate Normal Ordering

We recall that the coherent statesz,z∈ℂ, satisfy az=zz,zz=1. Note that za†=zz¯. For the coherent state z, we have (see [3–5, 11, 13])
(14)z=e−z2/2∑n=0∞znn!n.By equation (14), we get
(15)xy=e−1/2x2+y2+x¯y,x,y∈ℂ.It is easy to show that(16)ddxx=1+xddx.We recall that the standard boson commutation relationa,a†=aa†−a†a=1 can be considered, in a suitable space of functions f, by letting a=d/dx and a†=x.Now we observe that(17)xddx+rn,λfx=∑k=0nn+rk+rr,λxkddxkfx,(18)xddxn,λxrfx=∑k=0nn+rk+rr,λxk+rddxkfx,where n,r∈ℤ with n,r≥0.The equations (17) and (18) can be represented, respectively, by the degenerate normal orderings of the degenerate n-th powers in equations (19) and (20) of the number operator n^=a† in terms of boson operators a and a†.
(19)n^+rn,λ=a†a+rn,λ=∑k=0nn+rk+rr,λa†kak,(20)n^n,λa†r=a†an,λa†r=∑k=0nn+rk+rr,λa†k+rak.Therefore, from equations (19) and (20), we obtain the following theorem:Theorem 1.
Forn≥0, we have
(21)n^+rn,λ=a†a+rn,λ=∑k=0nn+rk+rr,λa†kak,
and, forn≥r, we have
(22)n^n−r,λa†rar=a†an−r,λa†rar=∑k=rnnkr,λa†kak.Letm=0,1,2,⋯. Then, by equation (8), we get
(23)a†a+rn,λm=n^+rn,λm=m+rn,λm,(24)n^+rn,λm=a†a+rn,λm=∑k=0nn+rk+rr,λa†kakm=∑k=0nn+rk+rr,λmkm.Thus, by equations (23) and (24), we get
(25)m+rn,λ=∑k=0nn+rk+rr,λmk,n≥0.This is the classical expression for the degeneraten-th power of m+r in terms of the falling factorial mk. This shows that equation (3) holds for all nonnegative integers x=m=0,1,2,⋯, which in turn implies equation (3) itself holds true.From equation (25), we note that
(26)zn^+rn,λz=za†a+rn,λz=∑k=0nn+rk+rr,λza†kakz=∑k=0nn+rk+rr,λz¯kzkzz=∑k=0nn+rk+rr,λz2k=ϕn,λrz2.Therefore, by equation (26), we obtain the following theorem:Theorem 2.
Forn≥0, we have
(27)zn^+rn,λz=za†a+rn,λz=ϕn,λrz2.Let us takeft=zeλr+n^tz. Then, by equation (26), we get
(28)ft=zeλr+n^tz=∑k=0∞tkk!zn^+rk,λz=∑k=0∞tkk!za†a+rk,λz=∑k=0∞tkk!∑l=0kk+rl+rr,λz2k=∑k=0∞ϕk,λrz2tkk!.Indeed, equation (28) says that ft is the generating function of the degenerate r-Bell polynomials which are considered by Kim-Kim.Therefore, by equation (28), we obtain the following theorem:Theorem 3.
The generating function of degenerater-Bell polynomials is given by
(29)zeλr+n^tz=zeλr+a†atz=∑n=0∞ϕn,λrz2tnn!.Now, we observe that(30)n^n^+r−λk,λ=a†aa†a+r−λk,λ=a†aa†a+r−λa†a+r−2λ⋯a†a+r−kλ=a†aa†+r−λaa†a+r−2λ⋯a†a+r−kλ=a†a†a+1+r−λaa†a+r−2λ⋯a†a+r−kλ=⋯=a†a†a+1+r−λa†a+1+r−2λ⋯a†a+1+r−kλa=a†a†a+1+r−λk,λa=a†n^+1+r−λk,λa,k≥0.From equation (30), we note that
(31)n^eλn^+r−λt=eλn^+r−λtn^=a†eλaa†+r−λta=a†eλn^+1+r−λta.By equations (28) and (31), we get
(32)∂ft∂t=∂∂tzeλa†a+rtz=zn^+reλn^+r−λtz=za†eλa†a+r+1−λtaz+rzeλa†a+r−λtz=eλ1−λtz¯zzeλa†a+rtz+rzeλa†a+rtzeλ−λt=eλ1−λtz2ft+reλ−λtft.Therefore, by equation (32), we obtain the following theorem:Theorem 4.
Letft=zeλr+n^tz. Then we have
(33)f′tft=z2eλ1−λt+reλ−λt.Note thatf0=1. From Theorem 4, we have
(34)logft=∫0tf′tftdt=z2∫0teλ1−λtdt+r∫0teλ−λtdt=z2eλt−1+r1λlog1+λt.Thus, by equation (34), we get
(35)ft=ez2eλt−1+r1/λlog1+λt=eλrtez2eλt−1.Therefore, by equation (35), we obtain the following theorem:Theorem 5.
Letft=zeλr+n^tz=zeλr+a†az. Then we have
(36)ft=eλrtez2eλt−1.From Theorems3 and 5, we have
(37)eλrtez2eλt−1=∑n=0∞ϕn,λrz2tnn!.That is, by equation (35), we get
(38)∑n=0∞zr+n^n,λztnn!=zeλr+n^tz=ft=∑k=0∞z2k1k!eλt−1keλrt=∑k=0∞z2k∑n=k∞n+rk+rr,λtnn!=∑n=0∞∑k=0nz2kn+rk+rr,λtnn!.Comparing the coefficients on both sides of equation (38), we have
(39)zr+n^n,λz=zr+a†an,λz=∑k=0nn+rk+rr,λz2k.By Theorems3 and 5, we get
(40)ft=zeλr+a†atz=∑n=0∞ϕn,λrz2tnn!.Differentiating equation (40) with respect to t, we have
(41)∂ft∂t=∑k=1∞tk−1k−1!ϕk,λrz2=∑k=0∞ϕk+1,λrz2tkk!.On the other hand, by equation (32), we get
(42)∂ft∂t=eλ1−λtz2ft+reλ−λtft=eλ1−λtz2∑k=0∞tkk!ϕk,λrz2+reλ−λt∑k=0∞ϕk,λrz2tkk!=z2∑l=0∞1−λl,λtll!∑k=0∞tkk!ϕk,λrz2+r∑l=0∞−λl,λtll!∑k=0∞ϕk,λrz2tkk!=∑n=0∞z2∑k=0nnk1−λn−k,λϕk,λrz2+r∑k=0nnk−λn−k,λϕk,λrz2tnn!.Therefore, by equations (41) and (42), we obtain the following theorem:Theorem 6.
Forn≥0, we have
(43)ϕn+1,λrz2=∑k=0nnkϕk,λrz2z21−λn−k,λ+r−λn−k,λ.Whenz=1 in Theorem 6, we have
(44)ϕn+1,λr=∑k=0nnkϕk,λr1−λn−k,λ+r−λn−k,λ,where ϕn,λr1=ϕn,λr are called the degenerate r-Bell numbers.Remark 7.
We can determine the degenerater-Bell numbers by using the recurrence relation in the above Theorem 6. The first few of them are as follows:
(45)ϕ0,λr=1,ϕ1,λr=1+r,ϕ2,λr=2+2r+r2−1+rλ,ϕ3,λr=5+r6+r3+r−6λ−3r2+rλ+21+rλ2,ϕ4,λr=15+r20+r12+r4+r−30λ−6r6+r3+rλ+112+r2+rλ2−61+rλ3,ϕ5,λr=52+r75+r50+r20+r5+r−150λ−10r20+r12+r4+rλ+355+r6+r3+rλ2−502+r2+rλ3+241+rλ4,ϕ6,λr=203+r312+r225+r100+r30+r6+r−780λ−15r75+r50+r20+r5+rλ+8515+r20+r12+r4+rλ2−2255+r6+r3+rλ3+2742+r2+rλ4−1201+rλ5.Evaluating the left hand side of Theorem2 by using the representation of the coherent state in terms of the number operator in equation (14), we have
(46)za†a+rk,λz=e−z2/2e−z2/2∑m,n=0∞z¯mznm!n!n+rk,λmn=e−z2∑n=0∞z2nn!n+rk,λ.By Theorem2 and equation (46), we get
(47)ϕk,λrz2=za†a+rk,λz=∑l=0kz2lk+rl+rr,λ=e−z2∑n=0∞z2nn!n+rk,λ,k≥0.In particular, by lettingz=1, we get the Dobinski-like formula.Theorem 8.
The following Dobinski-like formula holds true:(48)ϕk,λr=1e∑n=0∞1n!n+rk,λ,k,r∈ℕ∪0.


## 3. Conclusion

In recent years, various degenerate versions of many special numbers and polynomials have been explored by employing diverse tools such as generating functions, combinatorial methods, umbral calculus,p-adic analysis, differential equations, probability theory, operator theory, and analytic number theory.In this paper, we studied the degenerater-Bell polynomials and numbers via boson operators in quantum physics. To do that, we first recalled the degenerate normal ordering of a degenerate integral power of the number operator in terms of boson operators. Here the degenerate r-Stirling numbers of the second kind appear as the coefficients. From the degenerate normal ordering, we derived two expressions for the generating function of the degenerate r-Bell polynomials in z2, and a recurrence relation and Dobinski-like formula for the degenerate r-Bell numbers.It is one of our future research projects to continue to study many different versions of some special numbers and polynomials and to find their applications in physics, science and engineering as well as in mathematics.



---

*Source: 2626249-2022-10-12.xml*