Multiple Stirling Number Identities

H. Coskun Department of Mathematics, Texas A&M University–Commerce, Binnion Hall, Room 314, Commerce, TX 75429

Abstract A remarkable multiple analogue of the Stirling numbers of the ﬁrst and second kind was recently constructed by the author. Certain summation identities, and related properties of this family of multiple special numbers are investigated in the present paper. Keywords: multiple binomial coeﬃcients, multiple special numbers, qt-Stirling numbers, well-poised symmetric rational Macdonald functions 2000 MSC: 05A10, 11B65, 33D67

1. Introduction The Stirling numbers of the first and second kind are studied and their properties are investigated extensively in number theory and combinatorics. One dimensional generalizations of these numbers have also been subject of interest. An important class of generalizations is their one parameter q- extensions. Many have made significant contributions to such q-extensions investigating their properties and applications. We will give references to some of these important work in Section 3 below. In a recent paper, the author took a major step and constructed an el- egant multiple qt-generalization of Stirling numbers of the first and second kind, besides sequences of other special numbers including multiple binomial, Fibonacci, Bernoulli, Catalan, and Bell numbers [16]. In this paper, we focus on multiple Stirling numbers of both kinds, and give interesting new identities satisfied by them.

Email address: [email protected] (H. Coskun) URL: http://faculty.tamuc.edu/hcoskun (H. Coskun)

Preprint submitted to American Journal of Mathematics The multiple generalizations developed in [16] are given in terms of the qt-binomial coeﬃcients constructed in the same paper. Its deﬁnition may be written in the general form as

|µ| 2n(µ)+(1−n)|µ| j−i z q t Y (qt )µi−µj := w (qztδ(n); q, t) µ (qtn−1) (qtj−i−1) µ q,t µ 1≤i

n where µ is a partition of at most n parts, z ∈ C and q, t ∈ C. The wµ function that enters the deﬁnition is a limiting case of the BCn well–poised symmetric rational Macdonald function Wλ. Note that this deﬁnition makes sense even when µ is not an integer partition, but is a vector µ ∈ Cn.

2. Background

The basic q-Pochhammer symbol (a; q)α may be deﬁned formally for com- plex parameters q, α ∈ C as

(a; q)∞ (a)α = (a; q)α := α (1) (aq ; q)∞

Q∞ i where the infinite product (a; q)∞ is defined by (a; q)∞ := i=0(1−aq ). Note that when α = m is a positive integer, the definition reduces to the finite Qm−1 k product (a; q)m = k=0 (1 − aq ). An elliptic analogue is defined [17, 36] by

m−1 Y m (a; q, p)m := θ(aq ) (2) k=0 where a ∈ C, m is a positive integer, and the normalized elliptic function θ(x) is given by θ(x) = θ(x; p) := (x; p)∞(p/x; p)∞ (3) for x, p ∈ C with |p| < 1. The deﬁnition is extended to negative m by setting m (a; q, p)m = 1/(aq ; q, p)−m. It is clear that when p = 0, the elliptic (a; q, p)m reduces to the basic (trigonometric) q-Pochhammer symbol (1). For any partition λ = (λ1, . . . , λn) and t ∈ C, deﬁne [39]

n Y 1−i (a)λ = (a; q, p, t)λ := (at ; q, p)λi . (4) k=1

2 Note that when λ = (λ1) = λ1 is a single part partition, then (a; q, p, t)λ =

(a; q, p)λ1 = (a)λ1 . For brevity of notation, we also use

(a1, . . . , ak)λ = (a1, . . . , ak; q, p, t)λ := (a1)λ ... (ak)λ. (5)

Recall that we use V to denote [15] the space of infinite lower–triangular matrices whose entries are rational functions over the field F = C(q, p, t, r, a, b) which are indexed by partitions with respect to the partial inclusion ordering ⊆ defined by µ ⊆ λ ⇔ µi ≤ λi, ∀i ≥ 1. (6) The condition that a matrix u ∈ V is lower triangular with respect to ⊆ can be stated in the form uλµ = 0, when µ 6⊆ λ. (7) The multiplication operation defined by X (uv)λµ := uλνvνµ (8) µ⊆ν⊆λ for matrices u, v ∈ V makes V into an algebra over F. 2.1. Well–poised Macdonald functions The construction of our multiple Stirling numbers involves the elliptic well–poised Macdonald functions Wλ/µ on BCn [15]. These remarkable families of symmetric rational functions are first introduced in the author’s Ph.D. thesis [13] in the basic (trigonometric) case, and later in [14] in the more general elliptic form. Let λ = (λ1, . . . , λn) and µ = (µ1, . . . , µn) be partitions of at most n parts for a positive integer n such that the skew partition λ/µ is a horizontal strip; i.e. λ1 ≥ µ1 ≥ λ2 ≥ µ2 ≥ . . . λn ≥ µn ≥ λn+1 = µn+1 = 0. Following [15], define

µi−µj−1 j−i λi+λj 3−j−i Y (q t )µj−1−λj (q t b)µj−1−λj Hλ/µ(q, p, t, b) := (qµi−µj−1+1tj−i−1) (qλi+λj +1t2−j−ib) 1≤i

λi−µj−1+1 j−i−1 µi+λj +1 1−j−i (q t )µj−1−λj Y (q t b)µj−1−λj · λ −µ j−i · µ +λ 2−j−i (9) (q i j−1 t )µ −λ (q i j t b)µ −λ j−1 j 1≤i<(j−1)≤n j−1 j

3 and

−1 (x , ax)λ(qbx/t, qb/(axt))µ Wλ/µ(x; q, p, t, a, b) := Hλ/µ(q, p, t, b) · −1 (x , ax)µ(qbx, qb/(ax))λ n 1−2i 2µi 1−2i Y θ(bt q ) (bt )µi+λi+1 · · ti(µi−λi+1) (10) θ(bt1−2i) (bqt−2i) i=1 µi+λi+1 where q, p, t, x, a, b ∈ C. Note that Wλ/µ(x; q, p, t, a, b) vanishes unless λ/µ is a horizontal strip. The function Wλ/µ(y, z1, . . . , z`; q, p, t, a, b) is extended to ` + 1 variables y, z1, . . . , z` ∈ C through the following recursion formula

Wλ/µ(y, z1, z2, . . . , z`; q, p, t, a, b) X −` 2` ` = Wλ/ν(yt ; q, p, t, at , bt ) Wν/µ(z1, . . . , z`; q, p, t, a, b). (11) ν≺λ 2.2. The Limiting Cases

The Macdonald functions Wλ are essentially equivalent to BCn abelian functions constructed independently in [35]. The limiting cases deﬁned above are closely related to the Macdonald polynomials [31], and interpolation Mac- donald polynomials [34]. The following limiting case of the basic (the p = 0 case of the elliptic) W functions will be used in our constructions below. The existence of these limits can be seen from (p = 0 case of) the deﬁnition (10), the recursion formula (11) and the limit rule

|µ| |µ| |µ| −n(µ) n(µ0) lim a (x/a)µ = (−1) x t q (12) a→0

Pn Pn 0 Pn µi where |µ| = i=1 µi and n(µ) = i=1(i − 1)µi, and n(µ ) = i=1 2 . We denote Hλ/µ(q, t) = Hλ/µ(q, 0, t, 0), and for x ∈ C deﬁne

|λ|−|µ| wλ/µ(x; q, t) := lim s lim Wλ/µ(x; q, t, a, as) s→∞ a→0 −1 −|λ|+|µ| −n(λ0)+n(µ0) (x )λ = (−q/x) q Hλ/µ(q, t) −1 (13) (x )µ

The recurrence formula for wλ/µ function turns out to be

X `(|λ|−|ν|) −` wλ/µ(y, z; q, t) = t wλ/ν(yt ; q, t) wν/µ(z; q, t) (14) ν≺λ

4 Similarly, for x ∈ C deﬁne the dual function wˆλ/µ(x; q, t) := lim lim Wλ/µ(x; q, t, a, as) s→0 a→0 −1 −n(λ)+|µ|+n(µ) (x )λ = t Hλ/µ(q, t) −1 (15) (x )µ

The recurrence formula for the dualw ˆλ/µ(x; q, t) may be written as

X −` wˆλ/µ(y, z; q, t) = wˆλ/ν(yt ; q, t)w ˆλ/µ(z; q, t) (16) ν≺λ for y ∈ C and z ∈ C`. We now recall some old, and derive some new basic properties of the w function and its dual, and their connections.

n Corollary 1. Let µ be an n-part partition, and x = (x1, . . . , xn) ∈ C . (1) The wµ and its dual wˆµ are ﬂipped versions of one another. That is,

−|µ| −2n(µ)−(n−1)|µ| wˆµ(x, q, t) = q t wµ(1/x, 1/q, 1/t)

δ(n) (2) The limit limq→1 wµ(xt ; q, t) exists when denominators do not vanish. For the particular case when x = λ is a partition, we use the notation

λ δ(n) −µ1 λ δ(n) w¯µ(q t ; 1, t) := lim(1 − q) wµ(q t ; q, t) (17) q→1

Proof. Both properties follow, by direct calculation, from the deﬁnition (13) of wλ/µ, the recurrence relation (14) for wλ/µ, and limit formula (12), and the ﬂip formula

|µ| −1 |µ| n(µ0) −n(µ) −1 −1 x (x , q, t)µ = (−1) q t (x; q , t )µ (18)

The proof also uses the result that, in the limit

(k)m lim Hλ/µ(t, q) = (19) q→1 m! where k is the maximum of the list k = max{λ1 −λ2, λ2 −λ3, . . . , λn−1 −λn}, and m is the maximum of m = max{λ1 −µ1, µ1 −λ2, λ2 −µ2, . . . , µn−1 −λn}. If the second list has a negative number (which means that λ/µ is not a horizontal strip), then Hλ/µ(t, q) = 0.

5 Remark 1. We will need the following properties from [16] in what follows.

(3) If z = xtδ(n), for some x ∈ C, we get

j−i+1 0 Y (t )µi−µj w (xtδ(n); q, t) = (−1)|µ|x|µ|tn(µ)q−|µ|−n(µ )(x−1) (20) µ µ (tj−i) 1≤i

j−i+1 Y (t )µi−µj w (xtδ(n); q, t) = q−|µ|(x; q−1, t−1) (21) µ µ (tj−i) 1≤i

(4) The vanishing property of W functions implies that

λ δ wµ(q t ; q, t) = 0 (22) when µ 6⊆ λ, where ⊆ denotes the partial inclusion ordering.

(5) Let λ be an n-part partition with λn 6= 0 and 0 ≤ k ≤ λn for some integer n k, and let x = (x1, . . . , xn) ∈ C . It was shown in [16] that

n −nk Y 1−k wk¯(x; q, t) = q (q xi)k (23) i=1 where k¯ = (k, . . . , k) ∈ Cn. 2.3. qt-Binomial Coefficients The multiple Stirling numbers we develop in this paper are closely con- nected with binomial coefficients just as in the one dimensional case. Recall that qt-binomial coefficient is defined by [16]

n Definition 1. Let z = (x1, . . . , xn) ∈ C and µ be an n-part partition. Then the qt-binomial coefficient is defined by

|µ| 2n(µ)+(1−n)|µ| j−i z q t Y (qt )µi−µj := w (qztδ(n); q, t) (24) µ (qtn−1) (qtj−i−1) µ q,t µ 1≤i

6 Setting t = qα and sending q → 1 yields the multiple ordinary α-binomial coefficients. For n = 1, the definition reduces to that of the one dimensional q-binomial coefficients n (q) := n (25) k q (q)n−k(q)k which are also known as the Gaussian polynomials. These are studied extensively in the literature including but not limited to the works in [2, 3, 23, 4, 25, 32, 24, 12]. For the special case z =x ¯ = (x, . . . , x) ∈ Cn, we get

2n(µ)+(1−n)|µ| x¯ t x −1 −1 = n−1 (q ; q , t )µ µ q,t (qt )µ j−i j−i+1 Y (qt )µi−µj (t )µi−µj · (26) (qtj−i−1) (tj−i) 1≤i

n Deﬁnition 2. Let z = (x1, . . . , xn) ∈ C . Then

[z, s]µ := [z, s, n, q, t]µ n j−i |µ| Y 1 Y (t )µi−µj z δ(n) = q wµ(sq t ; q, t) (28) (1 − qtn−i)µi (tj−i+1) i=1 1≤i

7 The deﬁnition [z]µ may also be written as

n n−i −2n(µ)−(1−n)|µ| Y (qt )µi [z]µ = t (1 − qtn−i)µi i=1 j−i j−i−1 Y (t )µi−µj (qt )µi−µj z · (30) (tj−i+1) (qtj−i) µ 1≤i

n n x i−1 −1 Y 1 x −1 −1 Y (q t ; q )µi [¯x]µ = (q ; q , t )µ = (31) (1 − qtn−i)µi (1 − qtn−i)µi i=1 i=1 This deﬁnition reduces to the classical q-bracket in the one variable case. Note that (x; 1/q, 1/t)µ, with the reciprocals of q and t, corresponds to a multiple basic qt-analogue of the falling factorial xn := x(x−1) ··· (x−(n−1)) as opposed to the rising factorial or the Pochhammer symbol.

3. Multiple basic and ordinary qt-Stirling numbers In this section we review the definition and fundamental properties of the multiple Stirling numbers of the first and second kind indexed by partitions [16]. The classical Stirling numbers of the first kind are defined to be the coefficients of the falling factorial xn in the expansion

n x X x = n! = s (n, k) xk (32) n n 1 k=0 The q-analogue of these numbers are defined in [5] and their properties are studies in [33, 26, 29, 41] and the works of others. First, we recall [16] the definition of the multiple qt-Stirling numbers generalizing the one dimensional q-analogues. Definition 3. For an n-part partition λ, the qt-Stirling numbers of the first kind s1(λ, µ) are defined by

µ1 X −n(λ0) 2n(µ)−(n−1)|µ| Y i [x]λ = q t s1(λ, µ) [¯x ] (33) µ⊆λ i=1

8 i n 0 where x ∈ C, and x¯ = {x, . . . , x, q, . . . , q} ∈ C with µi copies of x for the dual partition µ0. That is,

µ1 n Y Y (1 − qxtn−i)µi [¯xi] = (34) (1 − qtn−i)µi i=1 i=1

Likewise, the qt-Stirling numbers of the second kind s2(λ, µ) are deﬁned by

n λ x n−i λi 1 Y (1 − q t ) Y i X n(µ0) −2n(ν)+(n−1)|ν| = [¯x ] = q t s2(λ, µ)[x]µ (35) (1 − qtn−i)λi i=1 i=1 µ⊆λ

We now revise an explicit formula for the qt-Stirling numbers given in [16], generalizing one dimensional q-analogues as follows.

Theorem 2. For n-part partitions ν and µ, an explicit formula for the qt- Stirling numbers of ﬁrst and second kind s1(λ, µ) and s2(λ, µ) are given by

qn(ν0)t−2n(µ)+2(n−1)|µ| s1(ν, µ) = s1(ν, µ, q, t) = n fµ(q, t) Q n−i νi−µi i=1(1 − qt ) X (1−n)|λ| −λ −δ(n) · u(ν, λ) t w¯µ(q t ; 1, 1/t) (36) µ⊆λ⊆ν and

q−n(µ0)t2n(ν)+(1−n)|ν| s2(ν, µ) = s2(ν, µ, q, t) := n Q n−i νi−µi i=1(1 − qt ) X |λ| n(λ) ν δ(n) · (−1) t w¯λ(q t ; 1, t) fλ(q, t) v(λ, µ, q, t) (37) µ⊆λ⊆ν where wˆµ and w¯µ are as deﬁned above in Corollary 1, and f(µ), u(λ, µ) and v(λ, µ) are given by

n−1 j−i n−1 Y (t)µi−µi+1 Y (t )µi−µj Y f(µ, q, t) := µ ! (µ − µ )! (38) (tn−i) (tj−i−1) n i i+1 i=1 µi 1≤i

|µ| 2n(µ) j−i q t Y (qt )µi−µj u(λ, µ, q, t) := wˆ (qλtδ(n); q, t) (39) (qtn−1) (qtj−i−1) µ µ 1≤i

9 and

|µ| |µ|+n(µ0) n(µ)+(1−n)|µ| j−i (−1) q t Y (qt )µi−µj v(λ, µ, q, t) := (qtn−1) (qtj−i−1) µ 1≤i

Proof. We write the W –Jackson sum [15] in the form

−2n −n Wλ(x; q, p, t, at , bt ) −1 −n−1 j−i+1 −i−j+1 (s)λ(as t )λ Y (t )λi−λj (qbt )λi+λj = · (qbs−1t−1) (qbtns/a) (tj−i) (qbt−i−j) λ λ 1≤i

Set b = ar in this identity and send a → 0, multiply both sides by (rtn)|λ| and send r → ∞ using the limit rule (12) to get

j−i+1 0 Y (t )λi−λj w (x; q, t) = (−1)|λ| q−|λ|−n(λ )tn(λ)s−|λ|(s) · λ λ (tj−i) 1≤i

Simplify this in the one variable case z = xtδ(n) using (21), and send s → ∞ to get −1 −1 X |µ| (x; q , t )λ = u(λ, µ) x (43) µ⊆λ with the deﬁnition of u(λ, µ) above. Similarly, apply the shifts a → as2, b → bs and x → x/s in (41) at the beginning, and follow the same steps

10 except send s to 0 to get

|λ| X −1 −1 x = v(λ, µ)(x; q , t )µ (44) µ⊆λ where v(λ, µ) is deﬁned as in the theorem. It is clear, by a change of basis argument, that X δνλ = u(ν, λ) v(λ, µ) (45) µ⊆λ⊆ν Note that the left hand side (44) does not depend on q or t. Now, ﬂip the parameters q → 1/q, t → 1/t, take limit q → 1, and multiply and divide the Qn n−i µi summand by i=1 1/(1 − qt ) to get

n µ1 X Y Y x|λ| = (1 − qtn−i)µi lim v(λ, µ, 1/q, 1/t) hx¯ii (46) q→1 µ⊆λ i=1 i=1 Multiply and divide the summand in (43) by t(n−1)|µ|, substitute (46) into (43) n−1 |µ| Qn n−i νi for (xt ) , multiply both sides of this latter identity by i=1 1/(1−qt ) to get

n n n−i µi Y 1 −1 −1 X Y (1 − qt ) (x; q , t )ν = (1 − qtn−i)νi (1 − qtn−i)νi i=1 µ⊆ν i=1

! µ1 X Y · u(ν, λ) t−(n−1)|λ| lim v(λ, µ, 1/q, 1/t) hx¯ii (47) q→1 µ⊆λ⊆ν i=1 Multiplying and dividing the summand now by qn(ν0)t−2n(µ)+(n−1)|µ| gives n n−i µi X −n(ν0) 2n(µ)−(n−1)|µ| Y (1 − xt ) hxiν = q t s1(ν, µ) · (48) (1 − qtn−i)µi µ⊆ν i=1 where s1(ν, µ) is as deﬁned in the theorem. A similar sequence of calculations show that

n n Y (1 − xtn−i)νi X Y (1 − qtn−i)µi = (1 − qtn−i)νi (1 − qtn−i)νi i=1 µ⊆ν i=1 ! X (n−1)|λ| · lim u(ν, λ, 1/q, 1/t) t v(λ, µ, q, t) hxiµ (49) q→1 µ⊆λ⊆ν

11 Multiplying and dividing the summand now by q−n(µ0)t2n(ν)+(1−n)|ν| gives the explicit formula for the qt-Stirling numbers of the second kind n n−i νi Y (1 − xt ) X n(µ0) −2n(ν)+(n−1)|ν| = q t s2(ν, µ) · hxiµ (50) (1 − qtn−i)νi i=1 µ⊆ν Finally, substituting x → qx completes the proof. We conclude by simplifying the ﬂips and limits that entered the formulas above. It follows immediately from (18) that, if j−i Y (qt )µi−µj h(µ, q, t) := and g(µ, q, t) := (qtn−1) (51) (qtj−i−1) µ 1≤i

12 wherew ¯µ is deﬁned as above. Using the limit rule (12), direct calculations give that

n−1 ! n−1 1 1 µ1 Y Y lim (1 − q) = µn! (µi − µi+1)! (57) q→1 (q) (q) µn i=1 µi−µi+1 i=1 Hence

|µ| n(µ)−(n−1)|µ| λ δ(n) lim u(λ, µ, 1/q, 1/t) = (−1) t w¯µ(q t ; 1, t) f(µ, q, t) (58) q→1 Similarly,

(n−1)|µ| −λ −δ(n) lim v(λ, µ, 1/q, 1/t) = t w¯µ(q t ; 1, 1/t) f(µ, q, t) (59) q→1 which completes the proof. Remark 2. The immediate properties of the Stirling numbers established in [16] are listed as follows:

(a) The multiple Stirling numbers s1(ν, µ) and s2(ν, µ) admit explicit combinatorial formulas which are derived in the Theorem above.

(b) These explicit formulas reduce to those for the q-Stirling numbers given by Kim in [29] for n = 1. Moreover, sending q → 1 in that case yields classical Stirling numbers of both types.

(c) The matrix m with entries mλµ = s1(λ, µ) is invertible in the sense of V −1 algebra, and its inverse is given by mλµ = s2(λ, µ). More precisely, we have X X δνλ = s1(ν, λ) s2(λ, µ) = s2(ν, λ) s1(λ, µ) (60) µ⊆λ⊆ν µ⊆λ⊆ν which follows immediately from the inversion relation (45).

(d) Similar to the one dimensional case for the q-Stirling numbers, we have

s1(λ, λ) = s2(λ, λ) = 1 (61) for an arbitrary n-part partition λ.

(e) Setting t = qα and sending q → 1 gives the multiple ordinary α-Stirling numbers of the ﬁrst and second kind.

13 4. Additional multiple Stirling number identities We derive some additional new properties of the multiple Stirling numbers in this section.

Qn n−i µi (1) First note that limx→0hxiµ = i=1 1/(1 − qt ) follows readily from the formula (31) written in the qt-angle brackets hx¯iµ as

n n i−1 −1 Y 1 −1 −1 Y (xt ; q )µi hx¯iµ = (x; q , t )µ = (62) (1 − qtn−i)µi (1 − qtn−i)µi i=1 i=1 In the multiple case, setting x = 0 in (48) and (50) respectively gives

n n Y 1 X −n(ν0) 2n(µ)+(n−1)|µ| Y 1 = q t s1(λ, µ) (63) (1 − qtn−i)νi (1 − qtn−i)µi i=1 µ⊆ν i=1 and

n n Y 1 X n(µ0) −2n(ν)+(n−1)|ν| Y 1 = q t s2(ν, µ) (64) (1 − qtn−i)νi (1 − qtn−i)µi i=1 µ⊆ν i=1

These appear to be new identities, even in the one dimensional case n = 1.

(2) Note that, for an n-part partition ν with νn 6= 0, the bracket hxiν has 1−j mj roots at x = t q for j = 1, . . . , n, and mj = 0, . . . , νj − 1. The limit Qn n−i µi n−i µi 1−j bracket i=1(1 − xt ) /(1 − qt ) has roots at x = t for j = 1, . . . , n. 1−j mj Therefore, if we set x = t q (for some mj < νj) in (50) we get

n mj 1−j+n−i µi X −n(ν0) 2n(µ)−(n−1)|µ| Y (1 − q t ) 0 = q t s1(ν, µ) (65) (1 − qtn−i)µi µ(ν i=1 where the summation is over all partitions µ ( ν, that is all partitions µ ⊆ ν such that µj ≤ mj. This inequality follows from the vanishing property of the w functions (22). In the particular case, setting x = q in (48) gives

X −n(ν0) 2n(µ)−(n−1)|µ| 0 = q t s1(λ, µ) (66) µ⊆ν

14 Pn which is an analogue of k=0 s1(n, k) = 0 in the classical case. 1−j In another special case, setting x = t (i.e., mj = 0) in (48) and (50) would amount to vanishing of all brackets hxiµ except the ones in whose index the j-th part (thus all parts j + 1, . . . , n after j) are 0. That is, the brackets will be nonzero only for partitions such as µ = (µ1, . . . , µj−1, 0,..., 0), and others will vanish. This is particularly interesting, for the substitution x = 1−j Qn n−i µi n−i µi t the limit brackets i=1(1 − xt ) /(1 − qt ) also vanish except for partitions µ = (µ1, . . . , µn−j, 0,..., 0). In general, setting x = t1−j in (48) and (50) respectively gives n 1−j+n−i µi X −n(ν0) 2n(µ)−(n−1)|µ| Y (1 − t ) 0 = q t s1(ν, µ) (67) (1 − qtn−i)µi µ(ν i=1 where the sum is over all partitions µ ( ν such that µ = (µ1, . . . , µj−1, 0,..., 0). Likewise, X n(µ0) −2n(ν)+(n−1)|ν| 1−j 0 = q t s2(ν, µ) ht iµ (68) µ(ν where the sum is over all partitions µ ( ν such that µ = (µ1, . . . , µn−j, 0,..., 0). The particular cases when j = 1 in (67) and j = n in (68) show that the multiple Stirling numbers vanish when µ = 0¯ = (0,..., 0) ∈ Cn as in the classical case. That is, s1(λ, 0)¯ = s2(λ, 0)¯ = 0 for any n-part partition λ with λn 6= 0.

n (3) Recall that, s1(n, m) = −s2(n, m) = − 2 when n−m = 1 for the classical Stirling numbers. Similarly, if the index partitions satisfy |λ| − |λ˜| = 1, we have that ˜ ˜ s1(λ, λ) = −s2(λ, λ) exactly as in the one dimensional case. The proof follows easily from the inversion (60) relation, and the observa- tion that there are only two partitions between λ and λ˜ under the inclusion ordering, namely the two partitions themselves. That is, X ˜ 0 = δλλ˜ = s1(λ, µ) s2(µ, λ) (69) λ˜⊆µ⊆λ ˜ ˜ ˜ ˜ which implies that s1(λ, λ) s2(λ, λ) = −s1(λ, λ) s2(λ, λ). That the diagonal entries of both type of multiple qt-Stirling numbers are 1 by (61) is now enough to conclude.

15 5. Conclusion We have derived several interesting summation identities for the multiple qt-Stirling numbers in the present paper. We will write down additional properties such as the recurrence relations they satisfy, their explicit evalua- tions in certain special cases, their combinatorial interpretation, generating functions, and connections to other families of multiple special numbers in an upcoming article. The Stirling numbers have interesting connections to various branches in mathematics such as the one expressed in the classical Dobinski’s formula. Such relations will also be formulated in that paper.

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