Báo cáo toán học: "On Kadell’s two Conjectures for the q-Dyson Product" pot

... the main theorem. Proof of Theorem 4.1. If m = 0, then the theorem reduces to the q-Dyson Theorem. So we assume that m  1. Applying Theorem 2 .3 to the constant term in the left-hand side of (4.6) ... (2.7) The idea to prove this theorem is by iterating Lemma 2.2 to transform the random i 1 in (2.6 ) to zero and then a pplying Theorem 2.1. But in the proof there are many te...

Ngày tải lên: 08/08/2014, 14:23

... − x 1 x r )   x 1 =x −1 r for r ≥ 2. Now clearly we have CT x 1 Q(x) = A 0 + A 2 + ···+ A n . This proves the first part of the lemma. For the second part, we need to rewrite A r in the right f orm. For r = 0 ... two prized constant term identities. For one of the identities, Zeilberger asked for a simple proof that may give rise to a simple proof of Andrews theorem fo...

Ngày tải lên: 08/08/2014, 14:23

... elements of the zero block, so the number of the τ i in (2) is now 2n. For the partition of Figure 4, if the blocks {1, 4, 6}, {2}, labeled with 1 and 2, respectively, have the sign + then the associated ... the arrangement which consists of the hyperplanes in d of the form β i (x)=β j (x)fori=j,whereβ i are the affine forms x i ,x i +1, ,x i +λ i −1 for 1 ≤ i ≤ d, −x...

Ngày tải lên: 07/08/2014, 06:22

... combinatorial formula for the character polynomials which has some kinship with the Murnaghan-Nakayama rule, and is quite suitable for hand calculations. Theorem I.1 For a given µ ⊢ k, let bd(i, µ) be the ... the convention h a = 0 when a < 0, the inductive hypothesis and equation 5.14 give for all m: k h m−k = p ⊥ k h m + (k − 1)h m−k that is p ⊥ k h m = h m−k and 5.14 a...

Ngày tải lên: 08/08/2014, 01:20

... E(F n |F 1 σ )) for any σ  n, then for each x ∗ ∈ X ∗ , the process {s(x ∗ ,F t } t∈J is also a uniformly integrable real 1-mil. It follows from Lemma 3.1 that for each x ∗ ∈ D ∗ 1 there exists ... Furthermore, restricted to the one-parameter discrete case, the notion of 1-mils coincides with the original notion of mils introduced by Talagrand [16]. The following lemma will b...

Ngày tải lên: 06/08/2014, 05:20

... fact. Furthermore, we derive formulas for the generating functions for the numbers u(4n), u(4n + 1), u(4n + 2) and u(4n + 3), which are analogous to the generating function formulas for the partition ... Meanwhile, the hook lengths of other squares remain the same. Furthermore, if λ i and λ ′ j have the same parity, then the number of squares in T (λ, v) is even. This im...

Ngày tải lên: 08/08/2014, 11:20

... and Guo [1] gave another proof for Yang’s formula. The objective of this paper is to give combinatorial proofs of Yang’s formula for k-ary trees and the other formula of Han for binary trees. Recall ... checked that the maps ϕ and φ are inverses of each other. This completes the proof. In particular, for k = 2, the proof of Theorem 2.1 reduces to a combinatorial proof of Han...

Ngày tải lên: 08/08/2014, 14:23

... for the first time for k = 66, which corresponds to c 4 = 2r 4 − 1. But neither r 4 = 66 nor c 3 = 99 are prime. 2 Relation between the conjectures We start by giving the alternative formula for ... We consider the former statement more natural (although less general) and, as we shall see, there are some differences between the two situations. We refer the reader to [1] f...

Ngày tải lên: 08/08/2014, 14:23

Ngày tải lên: 05/08/2014, 15:20

Ngày tải lên: 05/08/2014, 15:20