formula for sum of infinite ap series

... values of Tables 1 and 2 into Theorems 1.1 and 1.2 yields the summa- tion formulas for sums of m-th power for m =1, 2, ,11. In particular, for 1 ≤ m ≤ 5 we recover the formulas (1.2)–(1.6) of Warnaar ... existence of a general formula for S m,n (q), and left it as an open problem. It is the purpose of this paper to provide such a general formula, which turns out to be a q-analogue of the Faulhaber formula ... G m,s (q), and H m,s (q) for small m in Section 4 and obtain summation formulas of (1.1) for m ≤ 11. Section 5 contains some further extensions of these sum- mation formulas. 2 Some Preliminary...

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... theorem of Krasnoselskii for a sum of contraction and compact mappings in Banach spaces. The cla ss of asymptotically nonexpansive m appi ngs includes properly the class of nonexpansive mappings ... prove some new g eneralized forms of the Krasnoselskii theorem on fixed points for the sum A + B of a weakly-strongly continuous mapping and an asymptotically nonexpansive mapping in Banach spaces. ... point theorems for the sum of continuous and asymptotically nonexpansive mappings in Banach spaces. Journal of Inequalities and Applications 2011 2011:28. Arunchai and Plubtieng Journal of Inequalities...

Ngày tải lên: 21/06/2014, 01:20

... scheme by the viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. ... of the set of solutions of an equilibrium problem and the set of common fixed points of infinite nonexpan- sive mappings in a Hilbert space. We prove a strong-convergence theorem under mild assumptions ... Takahashi [4], we introduce an iterative scheme for finding a common element of the set of solutions of EP(h) and the set of fixed points of infinite nonexpansive mappings in a Hilbert space. We obtain a...

Ngày tải lên: 22/06/2014, 06:20

... scheme by the viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. ... of the set of solutions of an equilibrium problem and the set of common fixed points of infinite nonexpan- sive mappings in a Hilbert space. We prove a strong-convergence theorem under mild assumptions ... Takahashi [4], we introduce an iterative scheme for finding a common element of the set of solutions of EP(h) and the set of fixed points of infinite nonexpansive mappings in a Hilbert space. We obtain a...

Ngày tải lên: 22/06/2014, 19:20

... f  n on I. Suppose that the infinite series  ∞ n=1 f n converges for at least one point of I and that the s eries of derivatives  ∞ n=1 f  n converges uniformly on I. Then there exists a real-valued ... M ∗ (λ 1 −K 0 M 2 ) −1 . Proof. Setting λ i = 0foralli>nin Corollary 3.6, the result follows.  The following theorem proves the existence of a solution for an iterative functional series equation. Given ... K  1 =  ∞ i=1 λ i L  i M 2(i−1) 1 . Proof. For a function F ∈ Ᏻ 1 δ (I,λ 1 ηM,M ∗ ), the mapping ˜ F defined by ˜ F(x) = (b −a)F(x) −bF(a)+aF(b) F(b) −F(a) ∀x ∈I (3.47) 228 Equations involving series of iterates Theorem...

Ngày tải lên: 23/06/2014, 00:20

... property of the Jeu de Taquin operation. A proof is given in the proof of Theorem 4.9.4 of [ S]. Lemma 3.1. Given a semistandard tableau Q of shape η, the number of semistan- dard tableaux T of shape ... 2, 1997 Abstract This paper gives a new formula for the plethysm of power -sum symmetric functions and complete symmet- ric functions. The form of the main result is that for à b and a with ... The choices for the initial position of F in each iteration might effect the outcome of the algorithm. But, it has been shown (see Section 3.9 of [S] for a proof) that for all choices of the F’s,...

Ngày tải lên: 07/08/2014, 06:20

... first formula. For the second formula, we note that L =1− K −1 , and a formula for the coefficients of K −1 follows from (8.4) and (5.10). Applying Lemma 2.5 to Theorem 8.3 gives hook length formulas ... of a vertex v in a forest is the number of descendants of v (including v). We will consider various kinds of trees and forests in this paper, and for each of them we will have a hook length formula ... class of ordered forests on n vertices corresponds to n! labeled ordered forests. An unlabeled forest is formally defined as an isomorphism class of forests, but we think of an unlabeled forest...

Ngày tải lên: 07/08/2014, 08:22

... we see that the free sum induces a product of the numerators of the Ehrhart series of the summands. It is the electronic journal of combinatorics 13 (2006), #N15 3 Thus, for all i,  a i 0 Q  ·  p q  =  a i 0 Q  ·   α m p m  β n q n  ≥ ... and α m , β n ≥ 0. the electronic journal of combinatorics 13 (2006), #N15 4 An Ehrhart Series Formula For Reflexive Polytopes Benjamin Braun Department of Mathematics Washington University, St. ... the d-dimensional cube, being the free sum of d copies of the interval [−1, 1] and the product of d copies of [−1, 1], respectively. the electronic journal of combinatorics 13 (2006), #N15 1 ...

Ngày tải lên: 07/08/2014, 13:21

... interpretation of the closed formula (10) for the number of con- vex permutominoes. We remark that the expression of f n resembles the expression (1) for the number c n of convex polyominoes of semi-perimeter ... operator ϑ performs on P an application of operations (β) and (γ) for any cell in c n . So, let k be the degree of P , the application of ϑ to P produces 2k different convex permutominoes of size n ... 05A15 Abstract In this paper we determine a closed formula for the number of convex permu- tominoes of size n. We reach this goal by providing a recursive generation of all convex permutominoes of size n+1...

Ngày tải lên: 07/08/2014, 15:22

... combinatorial sum involving the generalized Catalan numbers C k,γ (n) = γ kn+γ  kn+γ n  . From the combinatorial sum, we refine the formula for k-ary trees and obtain an implicit formula for the generating ... any obvious combinatorial proof for their special case, even for (1). But, shortly later, we found that a special case of (2) for α = β also appeared in an early paper of Gould [3] though he did ... noncombinatorial proof of (2) by a modified Riordan array theorem derived from theorems in Sprugnoli et al. [5]. Also we refine the formula for C β,γ (n) with aids of a Gould classes of inverse relation...

Ngày tải lên: 07/08/2014, 15:23

... Estimates for the number of sums and products and for exponential sums in fields of prime order, J. London Math. Soc. (2) 73 (2006), 380–398. [5] J. Bourgain, N. Katz and T. Tao, A sum- product ... expect that appropriate adaptation of techniques of [3] and [11] may lead to quantitative improvement of the exponent 1/25. 2 Lemmas Below in statements of lemmas all the subsets are assumed to ... They hold for subsets of any abelian group, but here we state them only for subsets of F p . Lemma 1. For any subsets X, Y, Z of F p we have |X − Z| ≤ |X − Y ||Y − Z| |Y | . Lemma 2. For any subsets...

Ngày tải lên: 07/08/2014, 15:23

... property of a t-connected sum of graphs. Corollary 3.3. Let t be a positive integer and let G be a t-connected sum of graphs G 1 , G 2 , . . . , G n . I f H is also a t-connected sum of G 1 , ... G n , then b k (G) = b k (H) for all k. Using Proposition 2.3, we get a formula for the special graded Betti number of a t-connected sum of two simplicial complexes for t  3. Corollary 3.4. Let ... combinatorial proof of a formula for Betti numbers of a stacked polytope Suyoung Choi ∗ Department of Mathematical Sciences KAIST, Republic of Korea choisy@kaist.ac.kr (Current) Department of Mathematics Osaka...

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