二次曲面的求和公式

一类仿射Ψ丛的Plancherel公式与泊松求和公式

Plancherel and Poisson summation formulae for a family of affine $Ψ$-bundles

本文证明一类仿射Ψ丛的Plancherel公式，构造对应傅里叶变换与渐近Schwartz空间，在适当假设下证明泊松求和公式，其研究动机源于三重乘积L-函数的应用。

Watson-type 3F2-series and summation formulae involving generalized harmonic numbers

In terms of the derivative operator and Watson-type $_3F_2$-series identities, three families of summation formulae involving generalized harmonic numbers are established.

Whipple-type $_3F_2$-series and summation formulae involving generalized harmonic numbers

By means of the derivative operator and Whipple-type $_3F_2$-series identities, two families of summation formulae involving generalized harmonic numbers are established.

多项式、超函数与广义Lipschitz求和公式

Polylogarithms, hyperfunctions and generalized Lipschitz summation formulae

提出了经典Lipschitz求和公式的一个推广。它涉及新的多对数有理函数通过傅立叶展开的某些序列的伯努利型多项式。并构造了相关的一维超函数族。

Euler和Poisson求和公式的类比

Analogues of Euler and Poisson summation formulae

本文统一地得到了$\sum_{a < n \leq b} χ(n) f(n)$，$\sum_{a < n \leq b} d(n) f(n)$，$\sum_{a < n \leq b} d(n) χ(n) f(n)$的Euler-Maclaurin和Poisson类比，其中$(χ(n))$是周期复序列，$d(n)$是除数函数，$f(x)$是$[a,b]$上的充分光滑函数.给出了广义Abel求和公式、广义Euler求和公式和多元Euler求和公式。

A dimensionally continued Poisson summation formula

We generalize the standard Poisson summation formula for lattices so that it operates on the level of theta series, allowing us to introduce noninteger dimension parameters (using the dimensionally continued Fourier transform). When combined with one of the proofs of the Jacobi imaginary transformation of theta functions that does not use the Poisson summation formula, our proof of this generalized Poisson summation formula also provides a new proof of the standard Poisson summation formula for dimensions greater than 2 (with appropriate hypotheses on the function being summed). In general, ou…

Voronoï summation for half-integral weight automorphic forms

A general Voronoï summation formula for the (metaplectic) double cover of $\text{GL}_2$ is derived via the representation theoretic framework à la Ichino--Templier. The identity is also formulated classically and used to establish Voronoï summation formulae for half-integral weight modular forms and Maaß forms.

On the asymptotic formulae for some multiplicative functions in short intervals

In this paper we prove the mean values of some multiplicative functions connected with the divisor function on the short interval of summation.

A New Summation Formula for WP-Bailey Pairs

Let $(α_n(a,k),β_n(a,k))$ be a WP-Bailey pair. Assuming the limits exist, let \[ (α_n^*(a),β_n^*(a))_{n\geq 1} = \lim_{k \to 1}\left(α_n(a,k),\frac{β_n(a,k)}{1-k}\right)_{n\geq 1} \] be the \emph{derived} WP-Bailey pair. By considering a particular limiting case of a transformation due to George Andrews, we derive new basic hypergeometric summation and transformation formulae involving derived WP-Bailey pairs. We then use these formulae to derive new identities for various theta series/products which are expressible in terms of certain types of Lambert series.

Dirichlet twists of $\operatorname{GL}_n$-automorphic $L$-functions and hyper-Kloosterman Dirichlet series

We calculate mean values of $\operatorname{GL}_n$-automorphic $L$-functions twisted by primitive even Dirichlet characters of prime-power conductor, at arbitrary points within the critical strip, by derivation of special Voronoi summation formulae. Our calculation is novel in that the twisted sum can be expressed in terms of the average itself, and also that it sees the derivation of various new summation formulae in the setting of prime-power modulus. One consequence, as we explain, is to show the analytic continuation and additive summation formulae for hyper-Kloosterman Dirichlet series ass…

Some implications of Chu's $_{10}ψ_{10}$ extension of Bailey's $_{6}ψ_{6}$ summation formula

Lucy Slater used Bailey's $_6ψ_6$ summation formula to derive the Bailey pairs she used to construct her famous list of 130 identities of the Rogers-Ramanujan type. In the present paper we apply the same techniques to Chu's $_{10}ψ_{10}$ generalization of Bailey's formula to produce quite general Bailey pairs. Slater's Bailey pairs are then recovered as special limiting cases of these more general pairs. In re-examining Slater's work, we find that her Bailey pairs are, for the most part, special cases of more general Bailey pairs containing one or more free parameters. Further, we also find ne…

$\mathrm{GL}_1$ 上的某些 Fourier 算子及其相关的 Poisson 求和公式

Certain Fourier Operators and their Associated Poisson Summation Formulae on $\mathrm{GL}_1$

本文推广Tate的工作，利用$\GL_1$上的调和分析研究Langlands自守$L$-函数，引入了$(\sig,ρ)$-Schwartz函数与Fourier算子，证明了$\GL_n$标准表示下的Poisson求和公式猜想，并给出了标准$L$-函数临界零点的谱解释。

SL(2, Z) 部分元素求和的热带公式

Tropical formulae for summation over a part of SL(2, Z)

本文研究SL(2,Z)相关格点平行四边形的三角不等式缺陷幂次和的收敛性，推导精确求和公式，提出基于热带焦曲线的通用方法，为凸域不变量研究提供新工具。

$q$-Analogues of $π$-Related Formulae from Jackson's $_8ϕ_7$-Series via Inversion Approach

By making use of the multiplicate form of the extended Carlitz inverse series relations, we establish two general `dual' theorems of Jackson's summation formula for well--poised $_8ϕ_7$-series. Their duplicate forms under the partition pattern $n=\lfloor{\frac{n}2}\rfloor+\lfloor{\frac{n+1}2}\rfloor$ are explored and yield numerous $q$-series identities whose limiting cases as $q\to1$ result in classical $π$-related Ramanujan--like series of convergence rate ``$\frac1{16}$" including one for $1/π^2$ discovered by Guillera (2003). The triplicate dual formulae under the partition pattern $n=\lfl…

Γ函数的对数导数的Ramanujan公式

Ramanujan's formula for the logarithmic derivative of the gamma function

我们证明了一个显着的拉马努金公式的对数导数的Γ函数，收敛速度比经典的扩展，这是没有证明拉马努金的笔记本电脑。该公式有一些非常有趣的后果，我们得出，包括一个优雅的双曲求和，拉马努金公式的黎曼ζ函数评价在奇数正整数，和新的公式欧拉常数，伽玛。

Moments of Central $L$-values for Maass Forms over Imaginary Quadratic Fields

In this paper, over imaginary quadratic fields, we consider the family of $L$-functions $L (s, f)$ for an orthonormal basis of spherical Hecke--Maass forms $f$ with Archimedean parameter $t_f$. We establish asymptotic formulae for the twisted first and second moments of the central values $L\big(\frac 1 2, f\big)$, which can be applied to prove that at least $33 \%$ of $L\big(\frac 1 2, f\big)$ with $t_f \leqslant T$ are non-vanishing as $T \rightarrow \infty$. Our main tools are the spherical Kuznetsov trace formula and the Voronoï summation formula over imaginary quadratic fields.