Multiple Analogues of Binomial Coefficients and Families of Related

Multiple analogues of binomial coeﬃcients and families of related special numbers

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

Abstract Multiple qt-binomial coefficients and multiple analogues of several celebrated families of related special numbers are constructed in this paper. These higher dimensional generalizations include the first and the second kind of qt-Stirling numbers, qt-Bell numbers, qt-Bernoulli numbers, qt-Catalan numbers and the qt-Fibonacci numbers. Certain significant applications are also studied including two discrete probability measures on the set of all integer partitions. Keywords: multiple binomial coefficients, multiple special numbers, well poised Macdonald functions, well poised Jackson coefficients, discrete probability measures 2000 MSC: 05A10, 33D67, 11B65

1. Introduction Many distinct families of special numbers are investigated and their properties are explored in number theory. One dimensional generalizations of the sequences of these numbers have also been studied extensively in most cases. An important class of generalizations is their one parameter q-extensions. In this paper, we give multidimensional basic qt-generalizations for several collections of special numbers including the binomial coeﬃcients and the classes of numbers associated with the names of Stirling, Bernoulli, Catalan, Bell and Fibonacci. We also point out how to construct multiple ordinary

Email address: hasan [email protected] (Hasan Coskun) URL: http://faculty.tamu-commerce.edu/hcoskun (Hasan Coskun)

Preprint submitted to Discrete Mathematics α-generalizations for the same number sequences. The definitions of these numbers and the properties they satisfy show great variety which makes this research area very interesting. Among many mathematicians who contributed to this line of research, L. Carlitz appears to be the first to study the q-extensions for several families of special numbers given in this paper. Many have made significant contributions since then investigating properties of q-generalizations, their applications and connections with other types of numbers. We will give references to some of these important work in Section 4 below. The present paper takes a step in generalizing the one dimensional q- special numbers to multiple qt-special analogues. These generalizations are given in terms of the qt-binomial coefficient defined in Section 3 in the form

|µ| 2n(µ)+(1−n)|µ| j−i λ q t Y (qt )µi−µj := W s↑(qλtδ(n); q, t) µ (qtn−1) (qtj−i−1) µ qt µ 1≤i

s↑ where λ and µ are n-part partitions and q, t ∈ C. The Wµ function that enters into the definition is a limiting case of the well–poised BCn Macdon- ald function Wλ. We first give a brief review of this remarkable family of functions and its one parameter generalization ωλ called well–poised BCn Jackson coefficients. The symmetric rational functions Wλ and ωλ are first introduced in the author’s Ph.D. thesis [15] in the basic (trigonometric) case, and later in [16] in the more general elliptic form.

2. Background

We start with the formal deﬁnition of the q-Pochhammer symbol (a; q)α, for q, α ∈ C, given in the form

(a; q)∞ (a; q)α := α (1) (aq ; q)∞

Q∞ i where the infinite product (a; q)∞ is defined as (a; q)∞ := i=0(1−aq ). Note that when α = m is a positive integer, the definition reduces to (a; q)m = Qm−1 k k=0 (1 − aq ). An elliptic analogue is defined [18, 40] by

m−1 Y m (a; q, p)m := θ(aq ) (2) k=0

2 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. Note that when p = 0, (a; q, p)m reduces to the standard (trigonometric) q-Pochhammer symbol. For any partition λ = (λ1, . . . , λn) and t ∈ C, deﬁne [43]

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

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

(a; q, p)λ1 = (a)λ1 . The following notation will also be used.

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

Now 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 hor- izontal strip; i.e. λ1 ≥ µ1 ≥ λ2 ≥ µ2 ≥ . . . λn ≥ µn ≥ λn+1 = µn+1 = 0. Following [17], we deﬁne

Hλ/µ(q, p, t, b) µi−µj−1 j−i λi+λj 3−j−i Y (q t )µj−1−λj (q t b)µj−1−λj := (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 (6) (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 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) (7) θ(bt1−2i) (bqt−2i) i=1 µi+λi+1 where q, p, t, x, a, b ∈ C. 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

3 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). (8) ν≺λ We will also need the elliptic Jackson coefficients below. Let λ and µ be again partitions of at most n-parts such that λ/µ is a skew partition. Then the Jackson coefficients ωλ/µ are defined by

−1 −1 (x , ax)λ (qbr x, qb/axr)µ ωλ/µ(x; r, q, p, t; a, b) := −1 (qbx, qb/ax)λ (x , ax)µ n −1 1−n −1 2−2i 2µi (r, br t )µ Y θ(br t q ) µi · qt2i−2 (qbr−2, qtn−1) θ(br−1t2−2i) µ i=1 j−i −1 3−i−j Y (qt )µi−µj (br t )µi+µj · (qtj−i−1) (br−1t2−i−j) 1≤i

X −k 2k k ωλ/τ (y, z; r; a, b) := ωλ/µ(r y; r; ar , br ) ωµ/τ (z; r; a, b) (10) µ

n−k k where y = (x1, . . . , xn−k) ∈ C and z = (xn−k+1, . . . , xn) ∈ C . A key result used in the development of the multiple special numbers, the cocycle identity for ωλ/µ, is written in [17] in the form

−1 2 ων/µ((uv) ; uv, q, p, t; a(uv) , buv) X −1 2 −1 2 = ων/λ(v ; v, q, p, t; a(vu) , bvu) ωλ/µ(u ; u, q, p, t; au , bu) (11) µ⊆λ⊆ν where the summation index λ runs over partitions.

4 Using the recurrence relation (10) the definition of ωλ/µ(x; r; a, b) can be extended from the single variable x ∈ C case to the multivariable function n ωλ/µ(z; r; a, b) with arbitrary number of variables z = (x1, . . . , xn) ∈ C . That ωλ/µ(z; r; a, b) is symmetric is also proved in [17] using a remarkable elliptic BCn 10ϕ9 transformation identity. Let the Z-space V denote, as in [17], the space of infinite lower–triangular matrices whose entries are rational functions over the field F = C(q, p, t, r, a, b), and are indexed by partitions with respect to the partial inclusion ordering ⊆ defined by µ ⊆ λ ⇔ µi ≤ λi, ∀i ≥ 1. (12) The condition that a matrix u ∈ V is lower triangular with respect to the inclusion ordering can be stated in the form

uλµ = 0, when µ 6⊆ λ. (13)

The multiplication operation in V is deﬁned by the relation X (uv)λµ := uλνvνµ (14) µ⊆ν⊆λ for u, v ∈ V .

2.1. Limiting Cases The limiting cases of the basic (the p = 0 case of the elliptic) W functions Wλ/µ(x; q, t, a, b) = Wλ/µ(x; q, 0, t, a, b) will be used in computations in what follows. To simplify the exposition, some more notation will be helpful. We set ab Wλ/µ(x; q, t, s) := lim Wλ/µ(x; q, t, a, as) (15) a→0 and s↑ |λ|−|µ| ab W (x; q, t) := lim s Wλ/µ(x; q, t, s) (16) λ/µ s→∞ and ﬁnally, s↓ ab W (x; q, t) := lim Wλ/µ(x; q, t, s) (17) λ/µ s→0 The existence of these limits can be seen from (p = 0 case of) the deﬁni- tion (7), the recursion formula (8) and the limit rule

|µ| |µ| |µ| −n(µ) n(µ0) lim a (x/a)µ = (−1) x t q (18) a→0

5 Pn Pn 0 Pn µi where |µ| = i=1 µi and n(µ) = i=1(i − 1)µi, and n(µ ) = i=1 2 . These functions are closely related to the Macdonald polynomials [33, 37] and BCn abelian functions [38]. We now make these deﬁnitions more precise. Let Hλ/µ(q, t, b) denote Hλ/µ(q, 0, t, b), and deﬁne

Hλ/µ(q, t) := lim Hλ/µ(q, t, b) b→0 µi−µj−1 j−i λi−µj−1+1 j−i−1 Y (q t )µj−1−λj (q t )µj−1−λj = (19) (qµi−µj−1+1tj−i−1) (qλi−µj−1 tj−i) 1≤i

By setting b = as in the deﬁnition of Wλ/µ function, and sending a → 0 we ab deﬁne the family of symmetric rational functions Wλ/µ in the form

ab Wλ/µ(x; q, t, s) := lim Wλ/µ(x; q, t, a, as) a→0 −1 −n(λ)+|µ|+n(µ) (x )λ(qs/(xt))µ = t Hλ/µ(q, t) −1 (20) (x )µ(qs/x)λ ab for x ∈ C. Using (8) we get the following recurrence formula for Wλ/µ function ab X ab −` −` ab Wλ/µ(y, z; q, t, s) = Wλ/ν(yt ; q, t, st ) Wν/µ(z; q, t, s) (21) ν≺λ where y ∈ C and z ∈ C` as before. s↑ s↓ Similarly, the Wλ/µ and Wλ/µ are deﬁned as follows.

s↑ |λ|−|µ| ab W (x; q, t) := lim s Wλ/µ(x; q, t, s) λ/µ s→∞ −1 −|λ|+|µ| −n(λ0)+n(µ0) (x )λ = (−q/x) q Hλ/µ(q, t) −1 (22) (x )µ s↑ The recurrence formula for Wλ/µ function turns out to be

s↑ X `(|λ|−|ν|) s↑ −` s↑ Wλ/µ(y, z; q, t) = t Wλ/ν(yt ; q, t) Wν/µ(z; q, t) (23) ν≺λ for y ∈ C and z ∈ C`. In the same way, we deﬁne s↓ ab W (x; q, t) := lim Wλ/µ(x; q, t, s) λ/µ s→0 −1 −n(λ)+|µ|+n(µ) (x )λ = t Hλ/µ(q, t) −1 (24) (x )µ

6 s↓ The recurrence formula for Wλ/µ function becomes

s↓ X s↓ −` s↓ Wλ/µ(y, z; q, t) = Wλ/ν(yt ; q, t) Wν/µ(z; q, t) (25) ν≺λ where again y ∈ C and z ∈ C`.

3. qt-Binomial Coefficients A common property for all types of special numbers we develop in this paper is that they are closely connected with binomial coefficients. Therefore, we start with the definition of qt-binomial coefficients which will be proved using a multiple analogue of the terminating version of qt-binomial theorem. We first derive a multiple terminating 2φ1 sum as a limit of Jackson’s 8φ7 summation formula.

Theorem 1. For an n-part partition λ, we have

−1 −1 j−i (sx )λ X |µ| 2n(µ) (x )µ Y (qt )µi−µj = q t n−1 j−i−1 (s)λ (qt )µ (qt )µ −µ µ⊆λ 1≤i

−1 2 X −1 2 ωλ(s x; s; as , bs) = ωλ/µ(s ; s; as , bs) ωµ(x; s; a, b) (27) µ⊆λ can be written explicitly in the form

−1 1−n (sx , asx)λ X (s, as)λ (bt , qb/as)µ = n−1 (qbx, qb/ax)λ (qb, qb/a)λ (qt , as)µ µ⊆λ n 2−2i 2µi j−i 3−i−j Y (1 − bt q ) Y (qt )µi−µj (bt )µi+µj · (qt2i−2)µi (1 − bt2−2i) (qtj−i−1) (bt2−i−j) i=1 1≤i

7 The terminating qt-binomial theorem follows as a corollary. Corollary 2. For an n-part partition λ, we have

X |µ| |µ|+n(µ0) n(µ)+(1−n)|µ| 1 (x)λ = (−1) q t n−1 (qt )µ µ⊆λ j−i Y (qt )µi−µj · W s↑(qλtδ(n); q, t) · x|µ| (29) (qtj−i−1) µ 1≤i

This latter identity is a multiple analogue of Cauchy’s q-binomial theorem [24]. Using (29) we give the deﬁnition of a multiple analogue of the binomial coeﬃcient as promised.

Definition 1. Let λ and µ be n-part partitions. Then the qt-binomial coefficient is defined by

|µ| 2n(µ)+(1−n)|µ| j−i λ q t Y (qt )µi−µj := W s↑(qλtδ(n); q, t) (30) µ (qtn−1) (qtj−i−1) µ qt µ 1≤i

8 which are also known as the Gaussian polynomials that are studied extensively in the literature including but not limited to the works in [4, 5, 24, 6, 26, 34, 25, 14, 39]. Some of the main properties of the binomial coefficients readily generalize to the multiple case. For example, the identities n n X n X n 2n = and 0 = (−1)k (33) k k k=0 k=0 have the following multiple analogues. Theorem 3. For an n-part partition λ X 0 λ (−1) = qn(µ )t−n(µ) (34) λ µ µ⊆λ qt and X 0 λ 0 = (−1)|µ|qn(µ )t−n(µ) (35) µ µ⊆λ qt Proof. These identities follow immediately from (31) by setting z = −1 and z = 0, respectively. Before investigating multiple analogues of other binomial identities, we first note that the definition (1) makes sense even for generalized partitions µ = (µ1, µ2, . . . , µn) such that µ1 ≥ µ2 ≥ · · · ≥ µn with possibly negative parts µi starting with some i ∈ [n] := {1, 2, . . . , n}.

Theorem 4. The Wλ function is well-defined for the generalized partitions λ. In fact, the evaluation of Wλ function that enters into the definition of qt- binomial coefficient can be computed for n-tuples of complex numbers λ ∈ Cn n ab s↑ s↓ or µ ∈ C . The limiting Wλ , Wλ and Wλ functions are also defined in the general case.

Proof. Let λ be an n-part partition with λn 6= 0 and 0 ≤ k ≤ λn for some n integer k, and let z = (z1, . . . , zn) ∈ C . We have shown [16] that j−2i Y (qbt )2k W (z; q, t, a, b) = λ (qbtj−1−2i) 1≤i

9 n n where λ − k denotes λ − k = (λ1 − k, . . . , λn − k). Among other applications, this identity extends the deﬁnition of Wλ function to general partitions. For a generalized partition λ, the index λ − kn on the right hand side will be a standard partition. The duality formula for Wλ functions from [16] may be used to compute the special evaluation of Wλ that occur in many application in this paper even for λ ∈ Cn. The formula can be stated as

n−1 −1 ν δ 2 (qbt )λ(qb/a)λ Wλ k q t ; q, t, k a, kb · n−1 (k)λ(kat )λ j−i 0 2n−i−j−1 Y (t )λi−λj (qa t )λi+λj · (tj−i+1) (qa0t2n−i−j) 1≤i

q−λ1ν1 (k) (q/k) (kat2(n−1)qν1 ) λ1 ν1 λ1 (38) 1−λ1 n−1 1+ν1 1−n 1−ν1 (q /k)ν1 (bt q , (bt /a)q )λ1 corresponding to the dominant weight λ1 from the W function on the left hand side, and one term

((atn−1/b)q−λ1 , akt2(n−1)qλ1 ) ν1 (39) n−1 1+λ1 1−λ1 (bt q , q /k))ν1

10 for ν1 from the W on the right hand side to get

q−λ1ν1 (k) (q/k) (kat2(n−1)qν1 ) λ1 ν1 λ1 W 0 k−1qνtδ; q, t, k2a, kb 1−λ1 n−1 1+ν1 1−n 1−ν1 λ (q /k)ν1 (bt q , (bt /a)q )λ1 n−1 j−i n−i−j (qbt )λ(qb/a)λ Y (t )λi−λj (qkbt )λi+λj · −1 n−1 j−i+1 n−i−j+1 (k) (kt ) 0 (kat ) (t ) (qkbt ) λ1 λ λ 1≤i

ν δ Wλ q t ; q, t, a, b

λ1ν1 −ν1 −1 n−1 j−i+1 n−i−j+1 q (q )λ (t )λ0 (at )λ Y (t )λi−λj (qbt )λi+λj = 1 (q) (qbtn−1) (qb/a) (tj−i) (qbtn−i−j) ν1 λ λ 1≤i

This formula allows us to extend the evaluation of the Wλ function on the left hand side from an n-part partition λ to any λ ∈ Cn, because λ appears as a variable on the right hand side. The front factors that involve λ can be extended to the complex case by the deﬁnition of q-Pochhammer symbol (1). ab s↑ s↓ Similar evaluations for the limiting cases Wλ , Wλ and Wλ can easily be computed from this result. We now turn to the study of main properties of qt-binomial coeﬃcients. Theorem 5. For an n-part partition λ and generalized n-part partition µ, we have λ λ = n = 1. (42) λ qt 0 qt

11 where 0n is the n-part partitions whose parts are all zero. In addition, λ = 0 (43) µ qt when µ 6⊆ λ or 0n 6⊆ µ.

Proof. Recall that the normalizing coeﬃcients for Wλ function is written [16] in the form

λ δ(n) Wλ(q t ; q, t, a, b) n (qbtn−k, qtn−k) (at2n−2k) Y λk 2λk (n+1−2k)λk = n−k n−k n+1−2k t ((a/b)t , at )λ (qbt )2λ k=1 k k j−i−1 2n−i−j Y (qt )λi−λj (at )λi+λj · (a/(qb))|λ| · (44) (qtj−i) (at1+2n−i−j) 1≤i

Setting b = as and sending a → 0 and s → ∞ (after multiplying by s|λ|) gives

j−i−1 Y (qt )λi−λj W s↑(qλtδ(n); q, t) = (qtn−1) t(n−1)|λ|−2n(λ)q−|λ| (45) λ λ (qtj−i) 1≤i

λ Substituting this evaluation into λ qt yields 1 as desired. It is easy to see from the deﬁnition of Wλ function that Wµ(x; q, t, a, b) = 1 n s λ when µ = 0 . Therefore W0n (x; q, t) = 1 and 0n qt = 1. Finally, the fundamental vanishing property of W functions states [16] that λ δ Wµ(q t ; q, t, a, b) = 0 (46) when µ 6⊆ λ. Hence, W s(qλtδ(n); q, t) = 0 and therefore λ = 0 in that µ µ qt λ n−1 case. Note also that if µn < 0 then = 0 due to the (qt )µ factor in the µ qt denominator.

We introduce some notation before writing a multiple analogue of the important recurrence relation n + 1 n n = + , 1 ≤ k ≤ n k k k − 1

12 for classical binomial coeﬃcients. Let ei = (0,..., 1,..., 0) denote the n- dimensional vector whose i-th coordinate is 1 and all others are 0. Then λi i stands for λ + ei whenever λ is a partition. Finally, we write λ ` k to denote that λ is a partition of the positive integer k.

Theorem 6. For an n-part partition λ and k ≤ |λ|, we have

i X 0 λ qn(µ )t−n(µ) µ µ`k qt X 0 λ X 0 λ = qn(τ )t−n(τ) + qn(ν )+λi t−n(ν)+1−i (47) τ ν τ`k qt ν`(k−1) qt where µ ⊆ λi and τ, ν ⊆ λ.

Proof. The proof is a simple application of the qt-binomial theorem (31). 1−i λi Since (x)λi = (1 − xt q )(x)λ, we get

i X 0 λ (−1)|µ|qn(µ )t−n(µ) x|µ| µ µ⊆λi qt X 0 λ = (1 − xt1−iqλi ) (−1)|τ|qn(τ )t−n(τ) x|τ| (48) τ τ⊆λ qt by a double application of (31). This may be written as

i |λ | i X X 0 λ (−1)|µ|qn(µ )t−n(µ) x|µ| µ k=0 µ`k qt |λ| X X 0 λ = (1 − xt1−iqλi ) (−1)|τ|qn(τ )t−n(τ) x|τ| (49) τ `=0 τ`` qt

Note that the coeﬃcient of xk on the left hand side is

i X 0 λ (−1)|µ|qn(µ )t−n(µ) (50) µ µ`k qt

13 The coeﬃcient of xk that comes from the two pieces on the right hand side becomes X 0 λ X 0 λ (−1)|τ|qn(τ )t−n(τ) + (−1)|ν|+1qn(ν )+λi t−n(ν)+1−i (51) τ ν τ`k qt ν`(k−1) qt

Note also that µ ⊆ λi and τ, ν ⊆ λ, for otherwise the binomial coeﬃcients vanish by Theorem (5) above. Canceling out common factors gives the result.

A multiple analogue of the symmetry property

n n = , 0 ≤ k ≤ n k n − k for classical binomial coeﬃcients now follows.

Corollary 7. For an n-part partition λ and k ≤ |λ|, we have

X λ X λ = (52) µ τ µ`k qt τ`(|λ|−k) qt

Proof. This result follows from the qt-binomial theorem (31) through a similar argument used in the proof of the previous theorem.

We will now write an analogue of the identity

∞ zk X n = zn, |z| < 1 (53) (1 − z)k+1 k n=k

Let λ be an n-part partition and consider the C(q, t)-space Pλ of all polynomials of degree less then or equal to |λ| in a single complex variable x. Note k that both sets β1 = {x : k = 1,..., |λ|} and β2 = {(x)µ : µ ⊆ λ} form bases for Pλ. In fact, qt-binomial theorem stipulates just this, providing a change of basis formula between the two bases. Similarly, we view the terminating 2φ1 sum (26) as a matrix representation of the shift operator acting on β2, and prove an simpler version of the cocycle identity [16] for ωλ functions as follows.

14 Theorem 8. For n-part partitions ν and µ, we have

ab ν δ(n) −1 n−1 (sr)ν Wµ (q t ; q, t, (sr) t ) j−i X |λ| 2n(λ) (s)ν Y (qt )λi−λj ab ν δ(n) −1 n−1 = q t n−1 j−i−1 Wλ (q t ; q, t, s t ) (qt )λ (qt )λ −λ µ⊆λ⊆ν 1≤i

Proof. Write the identity (26) in the form

j−i X |µ| 2n(µ) (s)λ Y (qt )µi−µj (sx)λ = q t n−1 j−i−1 (qt )µ (qt )µ −µ µ⊆λ 1≤i

Starting with the basis β2, apply the shift operator by a shift factor of r followed by a shift by s. This double shift can be achieved by a composite shift by a factor of sr. Writing this argument explicitly using the identity above and simplifying gives the desired result.

Using this result, we now write a multiple analogue of the binomial iden- Qn 1−i tity (53) given above. Here (a)∞n denotes i=1(at )∞. Theorem 9. For n-part partitions ν and µ, we have

|µ| j−i+1 j−i+1 z Y (t )µi−µj X |λ| Y (t )λi−λj λ j−i = z j−i · (56) (qz)∞n (t )µ −µ (t )λ −λ µ 1≤i

Proof. First send r → 0 in the (54) to get

s ν δ(n) Wµ(q t ; q, t) j−i X |λ| 2n(λ) (s)ν Y (qt )λi−λj ab ν δ(n) −1 n−1 = q t n−1 j−i−1 Wλ (q t ; q, t, s t ) (qt )λ (qt )λ −λ µ⊆λ⊆ν 1≤i

15 Next we send ν → ∞. Note that setting ν = kn, where kn = (k, k, . . . , k) denotes the n-part partition whose parts all equal k, simpliﬁes the W function evaluations in the identity. To make this precise, we recall the Weyl denominator formula [16] which states that

δ(n) Wµ(xt ; q, t, a, b) −1 n−1 j−i+1 n−i−j+1 (x , axt )µ Y (t )µi−µj (qbt )µi+µj = (58) (qbxtn−1, qb/(ax)) (tj−i) (qbtn−i−j) µ 1≤i

−1 j−i+1 (x )µ Y (t )µi−µj W ab(xtδ(n); q, t, s−1tn−1) = (59) µ (qs−1tn−1/x) (tj−i) µ 1≤i

Multiplying both sides by (s−1tn−1)|µ|, and sending s → 0 further implies that j−i+1 0 Y (t )µi−µj W s↑(xtδ(n); q, t) = (−1)|µ|x|µ|tn(µ)q−|µ|−n(µ )(x−1) (60) µ µ (tj−i) 1≤i

On setting x = qk, sending k → ∞ in the last two evaluations and substituting them into the identity (57), we get

|µ| −|µ| j−i+1 2n(λ)−(n−1)|λ| |λ| s q Y (t )µi−µj X t s j−i = n−1 (s)∞n (t )µ −µ (qt )λ 1≤iseries converges follows from a multiple analogue of the dominated convergence theorem introduced in [17]. Consider the multiple series X hλ(k) (62) + λ∈Lk + n where Lk is the sub–alcove {λ ∈ Z : k ≥ λ1 ≥ . . . λn ≥ 0}. The theorem + states that if the pointwise limit hλ := limk→∞ hλ(k) exists for all λ ∈ Lk ,

16 h h and we can find mλ for each λ such that |hλ(k)| ≤ mλ for all k ≥ λ1, and P h that the series λ∈L+ mλ is convergent, then the original series converges. That the pointwise limit exists on both sides in (57) is already verified above. Note that the index µ of the W function inside the summand is a fixed partition. Note also that, except the powers of q, t and s, all other α α factors in the summand of (57) can be put into the form (uq )∞/(vq )∞ using the definition of q-Pochhammer symbol (1). Standard theorems on infinite products and sequences imply that such factors are bounded when α is a non–negative integer, u, v ∈ C, and that v is such that the denominator never vanishes. This is because

α (uq )∞ lim α = 1 (63) α→∞ (vq )∞ when |q| < 1. Therefore it follows that for a constant Ch that may depend only on q, t and s, and is independent of k and λ, we have found

n Y h 2n(λ)−(n−1)|λ| |λ| (2i−n−1)λi λi mλ = Ch t s = Ch t s (64) i=1

P h Finally, we need to show that λ∈L+ mλ is convergent which may be veriﬁed using the multiple series ratio test. Let ei be deﬁned as above. Then, we see that mh λ+ei (2i−n−1) h = t qz (65) mλ for each i ∈ [n] where possible (i.e., when λ + ei is a partition). Therefore, P h λ∈L+ mλ converges when

(2i−n−1) max{ t qz } < 1. i∈[n]

The original series also converges under the same condition.

Next, we write a multiple analogue of the identity

n n X n k 2n−m = (66) m k m k=m for qt-binomial coeﬃcients.

17 Theorem 10. For n-part partitions ν and µ, we have 0 (−1)ν ν X 0 ν λ t−n(µ)qn(µ ) = t−n(λ)qn(λ ) (67) (−1)µ µ λ µ qt µ⊆λ⊆ν qt qt

Proof. The cocycle identity [16] for ω functions

−1 2 X −1 2 −1 ωλ/µ((sr) , sr, as , bs) = ωλ/ν(s , s, as , bs) ων/µ(r ; r, a, b) (68) ν may be written explicitly in the form

−1 −1 (rs)λ(asr )λ (qb)λ(qb/a)λ (qb/as)µ (ar )µ −1 −1 (qbr )λ(qbr/a)λ (s)λ(as)λ (asr )µ (qb/a)µ 2−2n −1 1−n λ δ(n) Wµ(t, bst , br t ; q t ; q) 1−n −1 X (qb/as)ν (bt )ν (r)ν(ar )ν = (as) (qtn−1) (qbr−1) (qbr/a) ν ν ν ν ν n 2−2i 2νi j−i 3−i−j Y (1 − bt q ) νi Y (qt )νi−νj (bt )νi+νj qt2i−2 (1 − bt2−2i) (qtj−i−1) (bt2−i−j) i=1 1≤i

Setting b = azrs/q, and sending a → 0 and r → 0 gives

(zs)ν (z)µ |µ| s ν δ(n) s Wµ(q t ; q, t) (s)ν (zs)µ j−i X (z)λ |λ| 2n(λ) Y (qt )λi−λj = n−1 q t · j−i−1 (qt )λ (qt )λ −λ µ⊆λ⊆ν 1≤i

We will use a multiple analogue of Bailey’s 10φ9 transformation formula from [16] to transform the series on the right hand side. The 10φ9 trans-

18 formation can be written explicitly in the form

0 2 −1 0 2 2 (v)ν(a s v )ν (qbr)ν(qbr/a )ν (Av /s r)µ (qbrs/Av)µ 0 0 2 2 (qbrs/v)ν(qbrv/sa )ν (s)ν(a s)ν (qbrs /Av )µ (Av/sr)µ 2 2 −1 1−n 0 0 −1 X (qbrs /Av )λ ((brsv t )λ (rs/v)λ(Av/sr)λ (qbr/a s)λ (a sv )λ · n−1 2 2 2 0 2 −1 0 (Av/s)λ (qt )λ (qb)λ(qbr s /Av )λ (a s v )λ (qbr/a )λ λ µ⊆λ⊆ν n −1 2−2i 2λi j−i −1 3−i−j Y (1 − brsv t q ) λi Y (qt )λi−λj (brsv t )λi+λj · qt2i−2 (1 − brsv−1t2−2i) (qtj−i−1) (brsv−1t2−i−j) i=1 1≤i

19 Corollary 11. For n-part partitions ν and µ, we have

n ! νi 1−i 0 (−1)ν X (1 − q ) t t−n(µ)qn(µ ) (−1) 1 − q µ i=1 n ! λi 1−i X 0 X (1 − q ) t ν = t−n(λ)qn(λ ) (74) 1 − q λ e1⊆λ⊆ν i=1 qt

Proof. This follows immediately by setting µ = e1 in (67) and using the special evaluations (85) studied below.

One of the most important families of discrete probability distributions is the binomial distribution whose density is given by n f(i; n, p) := pi(1 − p)n−i (75) i where 0 < p < 1 and i = 0, 1, . . . , n. Here the density f(i; n, p) denotes the probability that the “event i” occurs for fixed parameters n and p. Changing the roles of i and n − i would give an equivalent definition. We now define an analogous probability measure on the set of all n-part partitions contained in λ under the partial inclusion ordering. Definition 2. Let λ be an n-part partition. For any partition µ ⊆ λ, the qt-binomial density function is defined by λ |λ|−|µ| g(µ; λ, z) := z (z)µ (76) µ qt where p ∈ R with 0 < p < 1. An alternative definition would be 0 λ (z) f(µ; λ, z) := t−2n(µ)q2n(µ ) λ z|µ| (77) µ qt (z)µ It needs to be verified that the density function is non–negative for any µ ⊆ λ, and that the total probability adds up to 1 when summed over all µ ⊆ λ. This is what we verify next.

Theorem 12. The multiple discrete density functions g(µ; λ, z) and f(µ; λ, z) are valid densities.

20 Proof. Set a = qb/r in (28) and send b → 0 and s → 0 to get (r)λ X (1/x)µ 0 λ = (−1)|µ| r|µ|t−n(µ)qn(µ ) (78) (r/x)λ (r/x)µ µ µ⊆λ qt

Setting z = 1/x now and sending r → ∞ gives

X λ 1 = z|λ|−|µ|(z) (79) µ µ µ⊆λ qt which shows that g(µ; λ, z) add up to 1. Similarly, shifting x to xr in (78) and sending r → 0 instead gives X (z)λ 0 λ 1 = z|µ|t−2n(µ)q2n(µ ) (80) (z)µ µ µ⊆λ qt verifying total probability for the alternative deﬁnition f(µ; λ, z). It is obvious from the deﬁnitions (4) and (1) that both densities are non– negative when 0 < q, t, z < 1 and z < tn−1, or alternatively when q, t > 1 and z < t1−n.

For a ﬁxed n-part partition ν, let Sν be the set of all partitions that are under ν with respect to the partial inclusion ordering. That is,

Sν := {λ : λ ⊆ ν}

It then follows from Theorem 12 that the distribution X ν F (λ; ν, z) := z|ν|−|µ|(z) (81) µ µ µ⊆λ qt defines a probability measure on Sν. An alternative distribution may be defined using the f density function defined above as well. The relation between this measure and the one given in [19] on the set of all partitions is to be investigated in another publication. We will, however, define a multiple analogue of another important family of density function for the Poisson distribution in this section. It will be defined as a limiting case of the binomial as in the classical case. First we give two multiple analogues of the exponential function ez.

21 Theorem 13. The following functions are multiple analogues of the exponential function ez.

0 X z|µ|qn(µ )tn(µ)+(1−n)|µ| n Eq(z) := (−z)∞ = n−1 (qt )µ µ∈Pn j−i j−i+1 Y (qt )µi−µj (t )µi−µj · (82) (qtj−i−1) (tj−i) 1≤i

1 X z|µ|t2n(µ)+(1−n)|µ| eq(z) := = n−1 (z)∞n (qt )µ µ∈Pn j−i j−i+1 Y (qt )µi−µj (t )µi−µj · (83) (qtj−i−1) (tj−i) 1≤i

Proof. Setting λ = kn in qt-binomial theorem (31), sending k → ∞ by using the identity (59) and applying the multiple analogue of the dominated convergence theorem employed in the proof of the previous theorem, and replacing x by −z gives the expression for Eq(z). Similarly, sending x → ∞ in (26), setting λ = kn as above and sending k → ∞ using the identity (60), and replacing s by z gives eq(z). Note in the latter case that the multiple dominated convergence theorem requires the condition max{ zt(2i−n−1) :i ∈ [n]} < 1 to be satisﬁed. We point out only two obvious properties of these functions. Namely, that eq(z) Eq(−z) = 1, and that

z z lim lim Eq(z(1 − q)) = e , and lim lim eq(z(1 − q)) = e . q→1 t→1 q→1 t→1 Both of these properties follow immediately from their deﬁnitions given in the previous theorem. Other properties Eq(z) and eq(z) satisfy will be investigated in a future publication. We now give a multiple analogue of the qt-Poisson distribution.

22 Theorem 14. The function

z|µ| q2n(µ0)t(1−n)|µ| f(µ; z) := Eq(−z) · n−1 (z)µ(qt )µ j−i j−i+1 Y (qt )µi−µj (t )µi−µj · (84) (qtj−i−1) (tj−i) 1≤i

Proof. Similar to the classical case, we set λ = kn and send k → ∞ in the qt-binomial density f(µ; λ, z) using the identity (60) to get the qt-Poisson density f(µ; z) function as desired. That the total probability P f(µ; z) add up to 1, and that f(µ; z) is µ∈Pn always non–negative follow from the construction.

Finally, we study certain special evaluations of the qt-binomials. It turns out that when either λ or µ is a rectangular partition kn = (k, k, . . . , k), then λ has a closed form product representation. µ qt Theorem 15. For n-part partitions λ and µ, the special cases λ kn λ n , and (85) k qt µ qt e1 qt have closed form product representations.

Proof. The λ = kn case of identity (36) becomes

j−2i n −1 Y (qbt )2k Y (zi )k(azi)k W n (z; q, t, a, b) = (86) k (qbtj−1−2i) (qbz ) (qb/(az )) 1≤i

n −1 ab Y (zi )k W n (z; q, t, s) = (87) k (qs/z ) i=1 i k Furthermore, multiplying both sides by snk and sending s → ∞ gives

n s↑ −nk Y 1−k Wkn (z; q, t) = q (q zi)k (88) i=1

23 after ﬂipping certain factors using

1−n (n) n (a)n = (q /a)nq 2 (−a) . (89) Therefore we get n 1−k+λi n−i λ Y (q t )k = (90) kn (qtn−i) qt i=1 k It is clear from the deﬁnition that

s↑ Wλ/λ(x; q, t) = 1 (91) and s↑ −1 W n (x; q, t) = q (1 − x) (92) e1/0 for x ∈ C. It then follows from the recurrence relation (23) that n n ! s↑ X −1 −1 n−i −1 X i−1 We1 (z; q, t) = −q xi(1 − xi t ) = q xi + t (93) i=1 i=1 n for z = (x1, . . . , xn) ∈ C . Multiplying by the front factors gives n n λ X (1 − qλi ) t1−i X = = [λ ] t1−i (94) e 1 − q i q 1 qt i=1 i=1 1 − qn where [n] := is the so–called q-number or q-bracket. q 1 − q The proof of the last result follows immediately from the identity (60). Setting λ = kn and manipulating factors gives

n n 1+k−µi i−1 k Y (q t )µ = t2n(µ)+(1−n)|µ| i µ (qtn−i) qt i=1 µi j−i j−i+1 Y (qt )µi−µj (t )µi−µj · (95) (qtj−i−1) (tj−i) 1≤i

In the light of last theorem, we give a definition for qt-number [z]qt ex- tending that of a q-number defined above. The definition will be used in the next section in the discussion of the qt-Stirling numbers. First, we write an extension of our qt-binomial coefficients.

24 n Definition 3. Let z = (x1, . . . , xn) ∈ C and µ be n-part partitions. Then the extended qt-binomial coefficient is defined by

|µ| 2n(µ)+(1−n)|µ| j−i z q t Y (qt )µi−µj := W s↑(qztδ(n); q, t) (96) µ (qtn−1) (qtj−i−1) µ qt µ 1≤i

With this extension, a deﬁnition for the qt-number [z]qt may be written as follows.

n Deﬁnition 4. Let z = (x1, . . . , xn) ∈ C . Then n z Y (1 − qxi tn−i) [z] = [z; q, t] := = (97) qt 1n (1 − qtn−i) qt i=1

We also deﬁne the partition shifted generalization [z; µ]qt = [z; µ, q, t] of the qt-bracket in the form of a normalized W s↑ function as follows.

n |µ|+n(µ0) Y 1 [z; µ]qt := q (1 − qtn−i)µi i=1 j−i Y (t )µi−µj · W s↑(qztδ(n); q, t) (98) (tj−i+1) µ 1≤i

|µ| −1 |µ| n(µ) −n(µ) −1 −1 x (x , q, t)µ = (−1) q t (x; q , t )µ (100) to simplify the W s↑ function. Similarly, using (88) we get

n 1−k xi n−i n n k Y (q q t )k [z; k ] := q (2) (101) qt (1 − qtn−i)k i=1

25 In particular for k = 1, we recover the deﬁnition (97) above. Note also 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)).

4. Multiple basic and ordinary special numbers In this section we give multiple basic and multiple ordinary analogues (or the qt- and α-analogues, respectively) of several celebrated families of special numbers including Stirling numbers, Bernoulli numbers, Bell numbers, Catalan numbers and Fibonacci numbers. The deﬁnition and some immediate properties of these numbers are studied in each case. Their fascinating deep properties such as the recurrence relations they satisfy, closed form evaluations in certain special cases and their combinatorial interpretations are carried out in other publications. The multiple special number sequences we introduce in this section are indexed by partitions. For each class of multiple basic qt-special number, the corresponding multiple ordinary α-special number analogue can be found by setting t = qα and sending q → 1.

4.1. qt-Stirling numbers The 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 (102) n n 1 k=0 The q-analogue of these numbers are deﬁned in [7] and their properties are studies in [36, 27, 31, 45] and others. We give the deﬁnition of the multiple qt-Stirling numbers generalizing the one dimensional q-analogues as follows.

Definition 5. For an n-part partition λ, the qt-Stirling numbers s1(λ, µ) of the first kind are defined by

n X Y n−i µi [x; λ]qt = s1(λ, µ) [ xt ]qt (103) µ⊆λ i=1 where x ∈ C. Consider the inﬁnite dimensional lower triangular matrix m whose entries are s1(λ, µ). That is, mλµ = s1(λ, µ). The qt-Stirling numbers

26 −1 s2(λ, µ) of the second kind are deﬁned to be the entries of the inverse m of −1 the matrix m. That is, s2(λ, µ) := mλµ .

We now write an explicit formula for the qt-Stirling numbers generalizing one dimensional q-analogues in the next theorem. The proof of the theorem makes clear, by presenting an inverse for the matrix m, that the deﬁnition indeed makes sense.

Theorem 16. 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)+(n−1)|µ| s1(ν, µ) := n Q n−i νi−µi i=1(1 − qt ) X · u(ν, λ) t(1−n)|λ| lim v(λ, µ, 1/q, 1/t) (104) q→1 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 · lim u(ν, λ, 1/q, 1/t) t(n−1)|λ| v(λ, µ, q, t) (105) q→1 µ⊆λ⊆ν where u(λ, µ) and v(λ, µ) are deﬁned by

|µ| 2n(µ) j−i q t Y (qt )µi−µj u(λ, µ, q, t) := W s↓(qλtδ(n); q, t) (106) (qtn−1) (qtj−i−1) µ µ 1≤i

(−1)|µ|q|µ|+n(µ0)tn(µ)+(1−n)|µ| v(λ, µ, q, t) := n−1 (qt )µ j−i Y (qt )µi−µj · W s↑(qλtδ(n); q, t) (107) (qtj−i−1) µ 1≤i

27 Proof. We write the W -Jackson sum [16] 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

−1 −1 X |µ| (x; q , t )λ = u(λ, µ) x (109) µ⊆λ where u(λ, µ) is deﬁned as above. Similarly, replace x by x/s in (108) and send s to 0 to get |λ| X −1 −1 x = v(λ, µ)(x; q , t )µ (110) µ⊆λ where v(λ, µ) is deﬁned as in the theorem. Then it is clear, by a change of basis argument, that X δνλ = u(ν, λ) v(λ, µ) (111) µ⊆λ⊆ν Note that the left hand side (110) does not depend on q or t. Also, n n x n−i µi n−1 Y n−i µi Y (1 − q t ) [xt ; µ, 1, t] = [xt ]qt = (112) (1 − qtn−i)µi i=1 i=1 for x ∈ C. By sending q → 1 in identity (110) and substituting it in (109) after replacing x by xtn−1, we get

−1 −1 X n−1 −1 (x; q , t )λ = (xt ; 1, t )µ µ⊆ν X · u(ν, λ) t(1−n)|λ| lim v(λ, µ, 1/q, 1/t) (113) q→1 µ⊆λ⊆ν

28 This gives the explicit expression for the qt-Stirling numbers of the ﬁrst kind after multiplying both sides by appropriate factors coming from the deﬁnition (99) of [x; λ, q, t]. Similarly, using the inversion relation (111) gives the explicit formula for the qt-Stirling numbers of the second kind, s2(ν, µ).

These identities reduces in one dimensional case (i.e., n = 1) to the formulas for the q-Stirling numbers given in [31]. An obvious property that follows from the inversion relation (111) in the last proof is that, considered as matrix entries, the qt-Stirling numbers of the ﬁrst and second kind form inverse matrices. That is, X X δνλ = s1(ν, λ) s2(λ, µ) = s2(ν, λ) s1(λ, µ) (114) µ⊆λ⊆ν µ⊆λ⊆ν

Leaving the investigation of other properties to a future publication, we point out another immediate property in the next theorem.

Theorem 17. For an n-part partition λ

s1(λ, λ) = s2(λ, λ) = 1. (115)

Proof. Setting b = astn−1 and sending a → 0, and then s → 0 in (44) gives

s↓ λ δ(n) Wλ (q t ; q, t) j−i−1 0 Y (qt )λi−λj = (−1)|λ| t−n(λ)q−|λ|−n(λ )(qtn−1) · (116) λ (qtj−i) 1≤i

Substitute this in u(λ, λ) to get

u(λ, λ) = (−1)|λ| tn(λ)q−n(λ0) (117)

Similarly, substituting (45) into v(λ, λ) gives

v(λ, λ) = (−1)|λ|qn(λ0) t−n(λ) (118)

In addition, the front factors

qn(ν0)t−2n(µ)+(n−1)|µ| n (119) Q n−i νi−µi i=1(1 − qt )

29 and its reciprocal reduce to q±n(ν0)t∓2n(λ)∓(n−1)|λ| when ν = µ. This is precisely what we want to complete the proof, because the factors q∓n(λ0) in u(λ, λ) and v(λ, λ) becomes 1 in the limit as q → 1. Finally, the inter- mediate term and the ﬂips q → 1/q and t → 1/t cancel with the factor t−2n(λ)+(n−1)|λ|. It should be pointed out again that setting t = qα and sending q → 1 gives the multiple ordinary α-Stirling numbers of the ﬁrst and second kind.

4.2. qt-Bernoulli numbers Another very important family of special numbers is the classical Bernoulli numbers βk that may be defined in terms of a recurrence relation. Namely, β0 = 1 and βk, for all integers k ≥ 1, satisfy n−1 X n 0 = β k k k=0 where n ≥ 2. Generalizing this approach in terms of multiple basic qt- binomial coefficients, we now give the definition of qt-Bernoulli numbers. 1 The vector e1 and the partition λ are defined as above. Definition 6. For an n-part partition λ, the qt-Bernoulli numbers are defined by the recurrence relation 1 X 0 λ 0 = t−n(µ)qn(µ ) β (120) µ µ µ⊆λ qt with the initial condition that β0n = β(0,0,...,0) = 1.

Solving the equation (120) for βλ gives 1−1 1 0 λ X 0 λ β = −tn(λ)q−n(λ ) t−n(µ)qn(µ ) β (121) λ λ µ µ qt µ⊂λ qt which can be used to compute βµ for all partitions µ. In the one dimensional (i.e., n = 1) case, the deﬁnition of qt-Bernoulli numbers reduce essentially to q-Bernoulli numbers deﬁned in [7] and studied in the works [32, 31, 11, 10] and others. In the limit as q → 1, the qt-Bernoulli numbers reduce to the classical Bernoulli numbers in the one dimensional case when n = 1. We conclude this section by the following conjecture generalizing the corresponding one dimensional result.

30 Conjecture 18. In the limit as t = qα and q → 1, the qt-Bernoulli numbers βλ vanish for all n-part partitions λ whose weight is |λ| = 2k + 1 for any integer k ≥ 1.

A proof of this conjecture among many other deep properties of these numbers will be investigated in a future publication.

4.3. qt-Bell numbers Another important class of special numbers intimately related to the binomial coeﬃcients is the Bell numbers deﬁned by

n X Bn := s2(n, k) k=0 This deﬁnition also generalize readily to the multiple case as follows.

Deﬁnition 7. Let λ be an n-part partition. The qt-Bell numbers are deﬁned by the relation X −n(µ) n(µ0) Bλ := t q s2(λ, µ) (122) µ⊆λ where s2(λ, µ) is the qt-Stirling numbers of the second type. The classical Bell numbers satisfy the recurrence

n X n B = B n+1 k k k=0

Among other properties, a multiple analogue of this result for Bλ in terms of λ will be investigated in another publication. µ qt For n = 1 this deﬁnition reduces to the q-Bell numbers which are deﬁned and studied in [35, 44] and others.

4.4. Multiple qt-Catalan numbers A qt-generalization of the Catalan numbers already deﬁned in [22] and developed in [23, 29, 30] and others. These numbers are one dimensional as they are indexed by positive integers (the weights of partitions involved) and they reduce to the q-Catalan numbers under the specialization of the t parameter. To avoid any possible confusion, we call the family of numbers

31 defined below ‘multiple qt-Catalan numbers’. The multiple qt-Catalan numbers appear to be related to the qt-Catalan numbers given in [22], however the precise relationship is still to be investigated. The multiple qt-Catalan numbers reduce, when n = 1, to the one-dimensional q-Catalan numbers that are defined in [9] and studied in [2, 3, 42, 20] and others. The classical Catalan numbers are defined by the relation

1 2n C := (123) n n + 1 n

We now give the following multiple qt-analogue of these numbers.

Deﬁnition 8. Let λ be an n-part partition. Then the multiple qt-Catalan number Cλ is deﬁned by 1 2λ Cλ := 1 (124) [λ ]qt λ qt 1 where 2λ = (2λ1,..., 2λn) and λ = λ + e1 = (λ1 + 1, λ2, . . . , λn) as before. In the special case when λ = kn is a rectangular partition, we get

n−1 n n−i n 1+k n−i (1 − qt ) Y (1 − qt ) Y (q t )k C n := (125) k (1 − qk+1tn−1) (1 − qktn−i) (qtn−i) i=2 i=1 k using the evaluation (90) from above. In the particular case when k = 1, this reduces to n Y (1 − q2tn−i) C n := (126) 1 (1 − qtn−i) i=2 which, as an empty product, becomes 1 when n = 1. Similarly, using (94) we see that 1 2 e1 Ce1 := 1 = 1 (127) [ e1]qt e1 qt Other properties of multiple qt-Catalan numbers will be investigated in a future publication.

32 4.5. qt-Fibonacci numbers

Recall that the classical Fibonacci numbers Fn have the representation

X k F = (128) n+1 m n=k+m

Taking a similar approach, we extend the deﬁnition of Fibonacci numbers to the multiple basic case as follows.

Definition 9. For an n-part partition λ, the qt-Fibonacci numbers are defined by the relation X 2n(µ0) 2(n−1)|µ|−2n(µ) ν F 1 := q t (129) λ µ ν+µ=λ qt where the sum runs over all n-part partitions ν ⊆ λ and µ ⊆ λ whose sum equals λ. Here, the sum ν + µ of two partitions ν and µ is defined to be the coordinate–wise summation of the two. That is,

ν + µ := (ν1 + µ1, ν2 + µ2, . . . , νn + µn) which itself is a partition.

In one dimensional case when n = 1, the qt-Fibonacci numbers are equivalent to the q-Fibonacci numbers deﬁned in [41] and studied in [8, 1, 21, 12, 13, 28] and others. In particular, in the limiting case q → 1 for n = 1, both deﬁnitions generate the classical Fibonacci sequence. Among a myriad of fascinating properties of the Fibonacci numbers, the fundamental recurrence relation

Fn = Fn−1 + Fn−2, which is used to deﬁne the Fibonacci sequence with the initial conditions F0 = 0 and F1 = 1, can be generalized to qt-Fibonacci numbers Fλ using Theorem 6. This fundamental result and other properties will be given in a separate publication. We close this section by pointing out that the relation between the so– called ﬁnite form of the classical Rogers–Ramanujan identities and the q- Fibonacci sequence holds true for our multiple qt-Fibonacci numbers. More

33 precisely, it is known [1, 12] that the q-Fibonacci sequence may be generated by bn−1c 2 X 2 n − k F (x) = xkqk −k (130) n+1 k k=0 q in the limit as q → 1 for x = 1. The ﬁnite form of Rogers–Ramanujan identities is essentially a limiting case of the Watson transformation. The multiple analogue of the Watson transformation given in [17] can be written in the form

(qb, qb/ρ2σ2)λ X |µ| 2n(µ) (σ2, ρ2, qb/ρ1σ1)µ q t n−1 (qb/σ2, qb/ρ2)λ (qt , qb/σ1, qb/ρ1)µ µ⊆λ j−i Y (qt )µi−µj · W ab(qλtδ(n); q, t, ρ σ tn−1/qb) (qtj−i−1) µ 2 2 1≤i

The limiting case of this identity as σ1, ρ1, σ2, ρ2 → ∞ becomes X 0 λ (qb) q|µ|+2n(µ )t−2n(µ)b|µ| λ µ µ⊆λ qt n 1−n 2−2i 2µi X |µ| 3|µ|+5n(µ0) −3n(µ) 2|µ| (bt )µ Y (1 − bt q ) = (−1) q t b n−1 2−2i (qt )µ (1 − bt ) µ⊆λ i=1 j−i 3−i−j Y (qt )µi−µj (bt )µi+µj · W a(qλtδ(n); q, t, bt1−n) (132) (qtj−i−1) (bt2−i−j) µ 1≤i

34 5. Conclusion The classical sequences of special numbers have fascinated mathematicians for centuries, and great many properties of these numbers have been studied. Several one or two parameter extensions of these numbers have been developed as well. However, these extensions were mostly one dimensional, and multiple analogues of these numbers have been missing. This paper takes an important step and presents multiple (basic and ordinary) analogues of several important classes of special numbers and develops some of their basic properties. Needless to say, the extremely rich properties of these numbers can not be fully covered in a single article. In addition, no attempt is made to include other families of related special numbers in this paper. These ideas will be pursued in future publications.

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