Poisson summation formulas involving the sum-of-squares function

加权勒贝格空间中的泊松求和公式

Poisson summation formula in weighted Lebesgue spaces

本文刻画了加权勒贝格空间中使泊松求和公式成立的参数$(α,β,p,q)$，完成了Kahane与Lemarié-Rieusset的相关研究。

Some applications of the Dirichlet integrals to the summation of series and the evaluation of integrals involving the Riemann zeta function

Using the Dirichlet integrals, which are employed in the theory of Fourier series, this paper develops a useful method for the summation of series and the evaluation of integrals.

测度、模形式与 Poisson 型求和公式

Measures, modular forms, and summation formulas of Poisson type

本文建立 $k$-球面 Fourier 本征测度与 metaplectic 群下模型 Fourier 级数的对应，并据此构造测度、导出 Poisson 型求和公式及高维推广。

Around Poisson--Mehler summation formula

We study polynomials in $x$ and $y$ of degree $n+m:\allowbreak \{Q_{m,n}(x,y|t,q)\}_{n,m\geq 0}$ that appeared recently in the following identity: $γ_{m,n}(x,y|t,q) \allowbreak =\allowbreak γ_{0,0}(x,y|t,q) \allowbreak Q_{m,n}(x,y|t,q) $ where $γ_{m,n}(x,y|t,q) \allowbreak =\allowbreak \sum_{i\geq 0}\frac{t^{i}}{[i]_{q}}H_{i+n}(x|q) H_{m+i}(y|q)$, $\allowbreak $ $\{H_{n}(x|q)}_{n\geq -1}$ are the so-called $q-$% Hermite polynomials (qH). In particular we show that the spaces $span\{Q_{i,n-i}(x,y|t,q) :i=0,...,n\}_{n\geq 0}$ are orthogonal with respect to a certain measure (two-dimensional $(t,…

Higher derivatives of the inverse tangent function and a summation formula involving binomial coefficients

In 2017, O. Deiser and C. Lasser obtained an explicit formula for the $n$-th derivative of the inverse tangent function. We calculate this derivative by a different method based on Faà di Bruno's formula. Comparing the two results leads to the following identity for binomial coefficients: $$\sum_{i=m}^{[n/2]}\frac{(-1)^i}{4^i}\binom{i}{m}\binom{n-i}{i}=\frac{(-1)^m}{2^n}\binom{n+1}{2m+1},$$ where $n,m\in \mathbb{N}_0$ and $m\leq [n/2]$. As was pointed out to the author by C. Krattenthaler, this formula is a special case of Gauß's formula for the hypergeometric function $_2F_1$.

Two q-summation formulas and q-analogues of series expansions for certain constants

From two q-summation formulas we deduce certain series expansion formulas involving the q-gamma function. With these formulas we can give q-analogues of series expansions for certain constants.

Distributions and Analytic Continuation of Dirichlet Series

This paper is second in a series of three papers; the first of which is "Summation Formulas, from Poisson and Voronoi to the Present" (math.NT/0304187), and the third of which is "Automorphic Distributions, L-functions, and Voronoi Summation for GL(3)". The first paper is primarily an expository paper, while the third proves a Voronoi-style summation formula for the coefficients of a cusp form on GL(3,Z)\GL(3,R). This present paper contains the distributional machinery used in the third paper for rigorously deriving the summation formula, and also for the proof of the GL(3)xGL(1) converse theo…

傅里叶求和公式与晶体测度的分类

A classification of Fourier summation formulas and crystalline measures

本文对傅里叶求和公式和晶体测度进行了分类，并引入了eta-quotients方法推广了Guinand的例子。

Summation of certain trigonometric series with logarithmic coefficients

The sums of three trigonometric series with logarithmic coefficients are derived by extending an approach first utilized by Lerch. By applying Frullani's theorem to two of these series, two non-trivial integrals involving hyperbolic functions are evaluated in terms of the digamma function.

Summation formulae for noncommutative hypergeometric series

We establish several summation formulae for hypergeometric and basic hypergeometric series involving noncommutative parameters and argument. These results were inspired by a recent paper of J. A. Tirao [Proc. Nat. Acad. Sci. 100 (14) (2003), 8138-8141].

Further summation formulas for the Kampé de Fériet function

The aim of this research is to provide thirty-two interesting summation formulas for the Kampé de Fériet function in general forms, which are given in sixteen theorems. The results are established with the help of the identities in Liu and Wang \cite{Li-Wa} and generalizations of Kummer's summation theorem, Gauss' second summation theorem and Bailey's summation theorem obtained earlier by Rakha and Rathie \cite{Ra-Ra}. Some special cases and relevant connections of the results presented here with those involving certain known identities are also indicated.

二次曲面的求和公式

Summation formulae for quadrics

本文证明偶数变量二次曲面零点集上无支集限制的 Poisson 求和公式，边界项由常数或相关较小二次曲面求和表示，并借助 Arthur 截断 theta 提升建立其与正交群表示及二次型解数估计的联系。

超几何恒等式的应用：生成广义超几何缩减公式

On the application of a hypergeometric identity to generate generalized hypergeometric reduction formulas

本文通过新的超几何恒等式推导出新的求和公式，并验证了相关函数的递推关系

A bilateral series involving basic hypergeometric functions

We prove a summation formula for a bilateral series whose terms are products of two basic hypergeometric functions. In special cases, series of this type arise as matrix elements of quantum group representations.

The Abel-Zeilberger Algorithm

We use both Abel's lemma on summation by parts and Zeilberger's algorithm to find recurrence relations for definite summations. The role of Abel's lemma can be extended to the case of linear difference operators with polynomial coefficients. This approach can be used to verify and discover identities involving harmonic numbers and derangement numbers. As examples, we use the Abel-Zeilberger algorithm to prove the Paule-Schneider identities, the Apery-Schmidt-Strehl identity, Calkin's identity and some identities involving Fibonacci numbers.

在六边形区域上通过Taylor-Abel-Poisson平均进行逼近

Approximation on hexagonal domains by Taylor-Abel-Poisson means

本文研究了积分度量下关于六边形区域周期的多元函数Fourier级数的Taylor-Abel-Poisson线性求和法的逼近性质，证明了直接和逆定理，并得到了Poisson核高阶径向导数$L_1$-范数的Bernstein型不等式。

涉及A^2类Appell多项式的Szász-Păltănea型算子的若干逼近性质

Some Approximation Properties by Szász-P{ă}lt{ă}nea type Operators involving the Appell Polynomials of class $A^2$

本文利用A^2类Appell多项式构造Szász-Păltănea型算子，验证Bohman-Korovkin定理并证明Lipschitz型空间、Voronvaskaja型渐近公式、连续模及加权连续模等逼近性质。

More remarkable sinc integrals and sums

We use Poisson summation formula to calculate integrals of producs of sinc functions (cf. [4]) and related integrals as in [5] and [3]. We also generalize the one in [5] and introduce other remarkable integrals. Finally we give a sum version of Siegel-type lower bound. (cf. [2], Theorem 3)

On the asymptotics of a cotangent sum related to the Estermann zeta function

The sum $c_0(1/k)=-\sum_{m=1}^{k-1}(m/k)\cot(mπ/k)$ is related to the Estermann zeta function. A recent paper computes the first two terms of the large-$k$ asymptotic expansion of $c_0(1/k)$. Using the Poisson summation formula for finite sums, we find three additional terms.

一系列涉及上不完全Gamma函数的定积分

A series of definite integrals involving upper incomplete Gamma functions

本文利用概率论推导出涉及上不完全Gamma函数的一系列定积分之和的表达式，其中正态方差混合分布起关键作用，并给出了第一类Bessel函数与Struve函数导数的新求和公式。