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Multiple Stirling Number Identities

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Multiple Stirling Number Identities 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 sec- ond kind was recently constructed by the author. Certain summation iden- tities, 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 bino- mial, 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 jµj 2n(µ)+(1−n)jµj 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<j≤n µi−µj n where µ is a partition of at most n parts, z 2 C and q; t 2 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 2 Cn. 2. Background The basic q-Pochhammer symbol (a; q)α may be defined formally for com- plex parameters q; α 2 C as (a; q)1 (a)α = (a; q)α := α (1) (aq ; q)1 Q1 i where the infinite product (a; q)1 is defined by (a; q)1 := 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 2 C, m is a positive integer, and the normalized elliptic function θ(x) is given by θ(x) = θ(x; p) := (x; p)1(p=x; p)1 (3) for x; p 2 C with jpj < 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 2 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 ma- trices 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; 8i ≥ 1: (6) The condition that a matrix u 2 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 2 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 fami- lies 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 gen- eral 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<j≤n µj−1−λj µj−1−λj 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 2 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` 2 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 jµj jµj jµj −n(µ) n(µ0) lim a (x=a)µ = (−1) x t q (12) a!0 Pn Pn 0 Pn µi where jµj = 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 2 C define jλ|−|µj wλ/µ(x; q; t) := lim s lim Wλ/µ(x; q; t; a; as) s!1 a!0 −1 −|λj+jµj −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 `(jλ|−|νj) −` wλ/µ(y; z; q; t) = t wλ/ν(yt ; q; t) wν/µ(z; q; t) (14) ν≺λ 4 Similarly, for x 2 C define the dual function w^λ/µ(x; q; t) := lim lim Wλ/µ(x; q; t; a; as) s!0 a!0 −1 −n(λ)+jµj+n(µ) (x )λ = t Hλ/µ(q; t) −1 (15) (x )µ The recurrence formula for the dualw ^λ/µ(x; q; t) may be written as X −` w^λ/µ(y; z; q; t) = w^λ/ν(yt ; q; t)w ^λ/µ(z; q; t) (16) ν≺λ for y 2 C and z 2 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) 2 C . (1) The wµ and its dual w^µ are flipped versions of one another. That is, −|µj −2n(µ)−(n−1)jµj w^µ(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 jµj −1 jµj 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 = maxfλ1 −λ2; λ2 −λ3; : : : ; λn−1 −λng, and m is the maximum of m = maxfλ1 −µ1; µ1 −λ2; λ2 −µ2; : : : ; µn−1 −λng.
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