Normal ordering associated with {\lambda}-Stirling numbers in{\lambda}-Shift algebra

The Stirling numbers of the second kind are related to normal orderings in the Weyl algebra, while the unsigned Stirling numbers of the first kind are related to normal orderings in the shift algebra. Kim-Kim introduced a {\lambda}-analogue of the unsigned Stirling numbers of the first kind and that of the r-Stirling numbers of the first kind. In this paper, we introduce a {\lambda}-analogue of the shift algebra (called {\lambda}-shift algebra) and investigate normal orderings in the {\lambda}-shift algebra. From the normal orderings in the {\lambda}-shift algebra, we derive some identities about the {\lambda}-analogue of the unsigned Stirling numbers of the first kind .


INTRODUCTION
The Stirling number of the first kind S 1 (n, k) is defined in such a way that the unsigned Stirling number of the first kind n k = (−1) n−k S 1 (n, k) enumerates the number of permutations of the set [n] = {1, 2, 3, . . . , n} which are products of r disjoint cycles. The unsigned r-Stirling number of the first kind n k r is the number of permutations of [n] with exactly k disjoint cycles in such a way that the numbers 1, 2, . . . , r are in distinct cycles.
In [16], introduced are a λ -analogue of the unsigned Stirling numbers of the first kind n k λ and that of the unsigned r-Stirling numbers of the first kind n k r,λ respectively as a λ -analogue of n k and that of n k r , (see (8), (9)). The Stirling numbers of the second kind appear as the coefficients in normal orderings in the Weyl algebra (see (10), (11)), while the unsigned Stirling numbers of the first kind appear as those in normal orderings in the shift algebra S (see (12), (13)).
The aim of this paper is to introduce the λ -shift algebra S λ (for any λ ∈ C), which is a λanalogue of S (see (14)), and to investigate normal orderings in the λ -shift algebra. In addition, from the normal orderings in the λ -shift algebra S λ , we derive some identities about the unsigned λ -Stirling numbers of the first kind.
The outline of this paper is as follows. In Section 1, we recall the λ -falling factorial numbers, the falling factorial numbers, the λ -rising factorial numbers and rising factorial numbers. We remind the reader of the unsigned λ -Stirling numbers of the first kind and the λ -r-Stirling numbers of the first kind. We recall the Weyl algebra and the normal ordering result in that algebra. We remind the reader of the shift algebra and the normal ordering result in that algebra. Finally, we define the λ -shift algebra as a λ -analogue of the shift algebra. Section 2 is the main result of this paper. We derive normal ordering results in S λ in Theorem 1 and Theorem 2 where n k λ and n+r k+r r,λ appear respectively as their coefficients. We obtain three other normal ordering results in Theorem 3. In Theorem 4, we get a recurrence relation for the unsigned λ -Stirling numbers of the first kind. In Theorem 6, we get another expression of the defining equation in (8) in terms of the λ -shift operator (see (30)). In Theorem 7, we show a λ -analogue of the dual to Spivey's identity (see Remark 8).
Finally, we conclude this paper in Section 3. For the rest of section, we recall what are needed throughout this paper.
Especially, the rising factorial sequence is given by Observe that lim λ →1 x n,λ = x n .
In addition, the unsigned Stirling numbers of the first kind are given by n k = (−1) n−k S 1 (n, k), (n, k ≥ 0). The Stirling numbers of the second kind are defined by Recently, with the notation in (1) the λ -Stirling numbers of the first kind, which are λ -analogues of the Stirling numbers of the first kind, are defined by [11]).
In addition, with the notation in (3) the unsigned λ -Stirling numbers of the first kind are defined by For r ∈ N∪{0}, the λ -r-Stirling numbers of the first kind, which are λ -analogues of the r-Stirling numbers of the first kind, are defined by Note that lim λ →1 n+r k+r r,λ = n+r k+r , where n+r k+r are the r-Stirling numbers of the first kind which are introduced by Broder (see [3]) and given by (see (4)) 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)) From the definition of the Stirling numbers of the second kind and (11), we note that The shift algebra S is defined as the complex unital algebra generated by a and a † satisfying the commutation relation aa † − a † a = a, (see [21]). (12) A word in S is said to be in normal ordered form if all letters a stand to the right of all letters a † .
From (12), we note that the normal ordering in the shift algebra S is given by (see (11)) (a † a) n = n ∑ k=0 n k (a † ) k a n , (see [21]).
For any λ ∈ C, we consider a λ -analogue of the shift algebra S which is defined as the complex unital algebra generated by a and a † satisfying the commutation relation (see (12)) The λ -analogue of the shift algebra S is called the λ -shift algebra and denoted by S λ .

λ -ANALOGUES OF NORMAL ORDERING IN THE λ -SHIFT ALGEBRA.
Let S λ be the λ -shift algebra defined in (14). A word in S λ is said to be in normal ordered form if all letters a stand to the right of all letters a † .
In S λ , by (14), we get Continuing this process, we have (a † a) n = a † n,λ a n , (n ≥ 1).
Therefore, by (16), we obtain the following theorem. Theorem 1. In S λ , the unsigned λ -Stirling numbers of the first kind appear as the coefficients of (a † a) n in normal ordered form, as it is given by For r ≥ 0, by (14), we get . Continuing this process, we have ((a † + r))a n = a † + r n,λ a n , (n ≥ 1).
Therefore, by (18), we obtain the following theorem.
Theorem 2. Let r be a nonnegative integer. In S λ , the λ -r-Stirling numbers of the first kind appear as the coefficients of ((a † + r)a) n in normal ordered form, as it is given by From (14), we note that and a(a † ) n = (aa † )(a † ) n−1 = (λ + a † )a(a † ) n−1 By (19), we get Therefore, by (19), (20) and (21), we obtain the following theorem.
Theorem 3. For m, n ∈ N and λ = 0, we have in S λ the normal orderings given by Now, we observe from Theorem 3 that On the other hand, by Theorem 1, we get Therefore, by (22) and (23), we obtain the following theorem.
Theorem 4. Let n, j ∈ Z with n ≥ 0 and j ≥ 1. In S λ , the unsigned λ -Stirling numbers of the first kind satisfy the following recurrence relation: For n ≥ 1, by (15) and (17), we have the λ -analogues of Boole's relations in the λ -Shift algebra given by (a † a) n = a † n,λ a n , ((a † + r)a) n = a † + r n,λ a n . Now, we define the λ -analogues of n! as (see (8)) Note that lim λ →1 (n) λ ! = n!.
From (8) and (9), we note that where k is nonnegative integer.
Thus, by (25) and (26), we get Comparing the coefficients on both sides of (27), we get From (9) Therefore, by (29), we obtain the following theorem.