Partial Sums of Two Quartic q-Series
Wenchang Chu, Chenying Wang
利用部分求和的修正Abel引理研究了两个四次基本超几何级数的部分和.由此建立了几个求和和变换公式。
The partial sums of two quartic basic hypergeometric series are investigated by means of the modified Abel lemma on summation by parts. Several summation and transformation formulae are consequently established.
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Partial sums of hypergeometric series of unit argument
研究单位宗量广义超几何级数部分和的渐近性态。
We prove an exponential integral estimate for the quadratic partial sums of multiple Fourier series on large sets that implies some new properties of Fourier series.
In this paper we investigate some convergence and divergence of some specific subsequences of partial sums with respect to Walsh system on the martingale Hardy spaces. By using these results we obtain relationship of the ratio of convergence of the partial sums of the Walsh series with the modulus of continuity of martingale. These conditions are in a sense necessary and sufficient.
Sextic polynomial oscillator is probably the best known quantum system which is partially exactly {\it alias} quasi-exactly solvable (QES), i.e., which possesses closed-form, elementary-function bound states $ψ(x)$ at certain couplings and energies. In contrast, the apparently simpler and phenomenologically more important quartic polynomial oscillator is {\em not\,} QES. A resolution of the paradox is proposed: The one-dimensional Schrödinger equation is shown QES after the analyticity-violating symmetrization $V(x)=A|x|+B x^2+C|x|^3+x^4$ of the quartic polynomial potential.
In this paper we derive characterizations of boundedness of the subsequences of partial sums with respect to Vilenkin system on the martingale Hardy spaces when $ 0<p<1 $. Moreover, we find necessary and sufficient conditions for the modulus of continuity of $f\in H_{p}$ martingales, which provide convergence of subsequences of partial sums on the martingale Hardy spaces. It is also proved that these results are the best possible in a special sense. As applications, both some well-known and new results are pointed out.
Cesaro Means along Polynomial Subsequences of Fourier Partial Sums at Lebesgue Points
该研究对Zalcwasser1936年提出的两个问题给出肯定答案，证明了Fourier部分和沿特定多项式子序列的(C,α)平均在L¹(𝕋)函数的所有Lebesgue点处逐点收敛到原函数。
In this paper we investigate some strong convergence theorems for partial sums with respect to Vilenkin system.
In 1936 Zygmunt Zalcwasser asked with respect to the trigonometric system that how "rare" can a sequence of strictly monotone increasing integers $(n_j)$ be such that the almost everywhere relation $\frac{1}{N}\sum_{j=1}^N S_{n_j}f \to f$ is fulfilled for each integrable function $f$. In this paper, we give an answer to this question. It follows from the main result that this a.e. relation holds for every integrable function $f$ and lacunary sequence $(n_j)$ of natural numbers.
Almost everywhere divergence of Cesaro means of subsequences of Walsh--Paley Fourier partial sums
证明存在严格递增正整数序列和L^1函数，使得Walsh-Fourier部分和子序列的Cesàro均值几乎处处发散，揭示了二进结构下仍存在发散现象。
Let $f(z) = \sum_{k=0}^\infty d_k z^k$, $d_k\in\mathbb{C}\backslash\{ 0 \}$, $d_0=1$, be a power series with a non-zero radius of convergence $ρ$: $0 <ρ\leq +\infty$. Denote by $f_n(z)$ the n-th partial sum of $f$, and $R_{2n}(z) = \frac{ f_{2n}(z) }{ z^n }$, $R_{2n+1}(z) = \frac{ f_{2n+1}(z) }{ z^{n+1} }$, $n=0,1,2,...$. By the result of Hendriksen and Van Rossum there exists a linear functional $\mathbf{L}$ on Laurent polynomials, such that $\mathbf{L}(R_n R_m) = 0$, when $n\not= m$, while $\mathbf{L}(R_n^2)\not= 0$. We present an explicit integral representation for $\mathbf{L}$ in the abov…
Using sums of squares to prove that certain entire functions have only real zeros
给出了如何利用实值函数的平方和证明贝塞尔函数$J_α(z)$当$α\ge -1,$汇合超几何函数${}_0F_1(c\/; z)$当$c>0$或$0>c>-1$时，Laguerre多项式$L_n^α(z)$当$α\ge -2,$和Jacobi多项式$P_n^{(α,β)}(z)$当$α\ge -1$和$ β\ge -1.$时的零点的真实性。此外，还得到了$|F(z)|^2,$的新不等式，其中$F(z)$是这些函数之一，导出的恒等式导致$\partial |F(z)|^2/\partial y$和$\partial ^2 |F(z)|^2/\partial y^2,$的不等式，这也给出了零点真实性的新证明。
In this paper we investigate some strong convergence theorems for partial sums with respect to Vilenkin system.
A note on the Fourier coefficients and partial sums of Vilenkin-Fourier series
本文的主要目的是研究Vilenkin-傅里叶级数的部分和的Paley型和Hardy-Littlewood型不等式及强收敛定理。
In the present investigation our main aim is to give lower bounds for the ratio of some normalized $q$-Bessel functions and their sequences of partial sums. Especially, we consider Jackson's second and third $q$-Bessel functions and we apply one normalization for each of them.
In this paper we prove and discuss some new $\left( H_p,L_{p}\right)$ type inequalities for partial Sums and Fejér means with respect to Walsh system. It is also proved that these results are the best possible in a special sense. As applications, both some well-known and new results are pointed out.
A class of increasing sequences of natural numbers $(n_k)$ is found for which there exists a function $f\in L[0,1)$ such that the subsequence of partial Walsh-Fourier sums $(S_{n_k}(f))$ diverge everywhere. A condition for the growth order of a function $φ:[0,\infty)\rightarrow[0,\infty)$ is given fulfilment of which implies an existence of above type function $f$ in the class $φ(L)[0,1)$.
On the partial sums of Vilenkin-Fourier series
本文主要研究Vilenkin-傅里叶级数部分和的加权极大算子。当$H_{p},$
Partial sums of the M{ö}bius function
假设黎曼假设，我们建立了一个上界的总和M{\”o}bius函数的$x$。我们的方法是基于估计的频率与一个给定的长度的间隔可以包含一个不寻常的数目的零点的黎曼ζ函数的纵坐标。
Discrete matrix models for partial sums of conformal blocks associated to Painlevé transcendents
Gamayun，Iorgov和Lisovyy最近提出的一个猜想给出了Jimbo-Miwa-Ueno isomonodromic $τ$ -函数对于某些Painlevé超越的渐近展开式。这个展开式中的系数是以二维共形场论的共形块的形式给出的，它可以写成在成对的分区上的无穷和。本文提出了一种格上的离散矩阵模型，利用该模型的配分函数，可以得到Painlevé共形块的长度受限部分和的重积分表示.这导致表达式的部分和涉及Hänkel行列式相关联的离散措施的矩阵模型，或等价地，Wronskians的相应的时刻生成函数，这是广义超几何类型。
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