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Search: MSC category 40 ( Sequences, series, summability )

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1. CJM 2010 (vol 63 pp. 241)

Essouabri, Driss; Matsumoto, Kohji; Tsumura, Hirofumi
Multiple Zeta-Functions Associated with Linear Recurrence Sequences and the Vectorial Sum Formula
We prove the holomorphic continuation of certain multi-variable multiple zeta-functions whose coefficients satisfy a suitable recurrence condition. In fact, we introduce more general vectorial zeta-functions and prove their holomorphic continuation. Moreover, we show a vectorial sum formula among those vectorial zeta-functions from which some generalizations of the classical sum formula can be deduced.

Keywords:Zeta-functions, holomorphic continuation, recurrence sequences, Fibonacci numbers, sum formulas
Categories:11M41, 40B05, 11B39

2. CJM 2008 (vol 60 pp. 520)

Chen, Chang-Pao; Huang, Hao-Wei; Shen, Chun-Yen
Matrices Whose Norms Are Determined by Their Actions on Decreasing Sequences
Let $A=(a_{j,k})_{j,k \ge 1}$ be a non-negative matrix. In this paper, we characterize those $A$ for which $\|A\|_{E, F}$ are determined by their actions on decreasing sequences, where $E$ and $F$ are suitable normed Riesz spaces of sequences. In particular, our results can apply to the following spaces: $\ell_p$, $d(w,p)$, and $\ell_p(w)$. The results established here generalize ones given by Bennett; Chen, Luor, and Ou; Jameson; and Jameson and Lashkaripour.

Keywords:norms of matrices, normed Riesz spaces, weighted mean matrices, Nörlund mean matrices, summability matrices, matrices with row decreasing
Categories:15A60, 40G05, 47A30, 47B37, 46B42

3. CJM 2007 (vol 59 pp. 85)

Foster, J. H.; Serbinowska, Monika
On the Convergence of a Class of Nearly Alternating Series
Let $C$ be the class of convex sequences of real numbers. The quadratic irrational numbers can be partitioned into two types as follows. If $\alpha$ is of the first type and $(c_k) \in C$, then $\sum (-1)^{\lfloor k\alpha \rfloor} c_k$ converges if and only if $c_k \log k \rightarrow 0$. If $\alpha$ is of the second type and $(c_k) \in C$, then $\sum (-1)^{\lfloor k\alpha \rfloor} c_k$ converges if and only if $\sum c_k/k$ converges. An example of a quadratic irrational of the first type is $\sqrt{2}$, and an example of the second type is $\sqrt{3}$. The analysis of this problem relies heavily on the representation of $ \alpha$ as a simple continued fraction and on properties of the sequences of partial sums $S(n)=\sum_{k=1}^n (-1)^{\lfloor k\alpha \rfloor}$ and double partial sums $T(n)=\sum_{k=1}^n S(k)$.

Keywords:Series, convergence, almost alternating, convex, continued fractions
Categories:40A05, 11A55, 11B83

4. CJM 2006 (vol 58 pp. 548)

Galanopoulos, P.; Papadimitrakis, M.
Hausdorff and Quasi-Hausdorff Matrices on Spaces of Analytic Functions
We consider Hausdorff and quasi-Hausdorff matrices as operators on classical spaces of analytic functions such as the Hardy and the Bergman spaces, the Dirichlet space, the Bloch spaces and $\BMOA$. When the generating sequence of the matrix is the moment sequence of a measure $\mu$, we find the conditions on $\mu$ which are equivalent to the boundedness of the matrix on the various spaces.

Categories:47B38, 46E15, 40G05, 42A20

5. CJM 2005 (vol 57 pp. 941)

Berg, Christian; Durán, Antonio J.
Some Transformations of Hausdorff Moment Sequences and Harmonic Numbers
We introduce some non-linear transformations from the set of Hausdorff moment sequences into itself; among them is the one defined by the formula: $T((a_n)_n)=1/(a_0+\dots +a_n)$. We give some examples of Hausdorff moment sequences arising from the transformations and provide the corresponding measures: one of these sequences is the reciprocal of the harmonic numbers $(1+1/2+\dots +1/(n+1))^{-1}$.

Categories:44A60, 40B05

6. CJM 2001 (vol 53 pp. 866)

Yang, Yifan
Inverse Problems for Partition Functions
Let $p_w(n)$ be the weighted partition function defined by the generating function $\sum^\infty_{n=0}p_w(n)x^n=\prod^\infty_{m=1} (1-x^m)^{-w(m)}$, where $w(m)$ is a non-negative arithmetic function. Let $P_w(u)=\sum_{n\le u}p_w(n)$ and $N_w(u)=\sum_{n\le u}w(n)$ be the summatory functions for $p_w(n)$ and $w(n)$, respectively. Generalizing results of G.~A.~Freiman and E.~E.~Kohlbecker, we show that, for a large class of functions $\Phi(u)$ and $\lambda(u)$, an estimate for $P_w(u)$ of the form $\log P_w(u)=\Phi(u)\bigl\{1+O(1/\lambda(u)\bigr)\bigr\}$ $(u\to\infty)$ implies an estimate for $N_w(u)$ of the form $N_w(u)=\Phi^\ast(u)\bigl\{1+O\bigl(1/\log\lambda(u)\bigr)\bigr\}$ $(u\to\infty)$ with a suitable function $\Phi^\ast(u)$ defined in terms of $\Phi(u)$. We apply this result and related results to obtain characterizations of the Riemann Hypothesis and the Generalized Riemann Hypothesis in terms of the asymptotic behavior of certain weighted partition functions.

Categories:11P82, 11M26, 40E05

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