Báo cáo toán học: "Two finite forms of Watson’s quintuple product identity and matrix inversion" pptx

... [4] and Guo-Zeng [6] found independently that Watson’s quintuple product identity follows surprisingly from two basic algebraic identities, called finite forms of Watson’s quintuple product identity. ... proof of theorem. By the same technique, we derive the following equivalent form of Eq. (1.3). the electronic journal of combinatorics 13 (2006), #R52 5 Two fin...

Ngày tải lên: 07/08/2014, 13:21

... this part of the proof is similar to Kongsiriwong and Liu’s proof of the Jacobi triple product identity [5] and Chan’s proof of the quintuple product identity [2]. By putting a = ω and b = −ω ... proofs of Winquist’s identity. In the first pro of, we u se basic properties of cube roots of unity and the Jacobi triple product identity. The latter does not use t...

Ngày tải lên: 08/08/2014, 12:22

... FU(<α i > ∞ i=1 )={  i∈A α i : A ∈F}.(FU stands for finite unions.” One may consider the set of finite unions of a finite sequence as well, of course.) If G⊂F,letG k < be the set of k-tuples (α 1 , ,α k )inG k for ... of a finitary case of Furstenberg and Katznelson’s multiparameter elaboration of a theorem, due to Carlson, about variable words. The second is...

Ngày tải lên: 07/08/2014, 06:20

... canonical form of a loopfree game G is the simplest game infinites- imally close to G. Reduced canonical forms were introduced by Calistrate, and Grossman and Siegel provided an alternate proof of their ... L 0 (H) and R 0 (G) ≥ R 0 (H). the electronic journal of combinatorics 13 (2006), #R57 6 Proof of Lemma 2.9. Let α −β be a follower of γ −δ.Wesaythatα−β is safe if Left ca...

Ngày tải lên: 07/08/2014, 13:21

... example is Garsia and Milne’s proof of the Rogers-Ramanujan identities [16], making use of the involution principle. Bressoud and Zeilberger gave an alternative, much shorter proof of these identities ... [5]. Zeilberger gave a q-Foata proof of the q-Pfaff-Saalsch¨utz identity [20], inspired by Foata’s bijective proof of the Pfaff-Saalsch¨utz identity [6]. In view of the fac...

Ngày tải lên: 07/08/2014, 06:22

... electronic journal of combinatorics 9 (2002), #N9 2 On a theorem of Erd˝os, Rubin, and Taylor on choosability of complete bipartite graphs Alexandr Kostochka ∗ University of Illinois at Urbana–Champaign, ... with g(v)=i and let f g (w)=v. Then vertices in different W i cannot get the same color, and f is a coloring from the lists of vertices in G.ThisprovesthatN(k, r) ≥ p(r, k)...

Ngày tải lên: 07/08/2014, 06:23

... proof reveals interesting properties of pairs of permutations. We will first give a brief history of this identity, indicating methods of proof. The permanent of a square matrix is the sum of ... combinatorial model of Borchardt’s identity, and in Section 3 we prove the identity by means of sign- reversing involutions. 2 Combinatorial Model of Borchardt’s Identity B...

Ngày tải lên: 07/08/2014, 08:20

... left-hand (resp. right-hand) side of (2). The aim of this paper is to find a combinatorial proof of (2). In Section 2 we construct the sets F bi n of labeled bicolored forests on [n]andD n of certain ... same talk, he also asked for a combinatorial proof of identity (1). Multiplying both sides of (1) by 2 n and expanding the product in the right-hand side yields 2 n (n +1)...

Ngày tải lên: 07/08/2014, 08:22

... variant of Vandermonde’s determinant. the electronic journal of combinatorics 11(2) (2005), #R19 10 From the computational point of view, with the help of Maple or other softwares, it is, of course, ... Garrett and K. Hummel, A combinatorial proof of the sum of q-cubes, Electron. J. Combin. 11 (2004), #R9. [3] G. Gasper and M. Rahman, Basic Hypergeometric Series, Encyclopedi...

Ngày tải lên: 07/08/2014, 08:22

... vertices v of T ,andh(v)isthehook length of v, which is the number of descendants of v, including v. Direct combinatorial proofs of Postnikov’s identity were given by Seo [15], Du and Liu [5], and Chen ... vertices v of T ,andh(v ) is the number of descendants of v (including v). Our results give analogues of Postnikov’s formula for other types of trees, and we als...

Ngày tải lên: 07/08/2014, 08:22