Dirichlet样条的矩及其在超几何函数中的应用

On a polyharmonic Dirichlet problem and boundary effects in surface spline approximation

For compact domains with smooth boundaries, we present an approximation scheme for surface spline approximation that delivers precise $L_p$ approximation orders on well known smoothness spaces. This scheme overcomes the boundary effects when centers are placed with greater density near to the boundary. It owes its success to an integral identity using a minimal number of boundary layer potentials, which, in turn is derived from the boundary layer potential solution to the Dirichlet problem for the $m$-fold Laplacian. Furthermore, his integral identity is shown to be the "native space extension…

Series acceleration formulas obtained from experimentally discovered hypergeometric recursions

In 2010, Kh. Hessami Pilehrood and T. Hessami Pilehrood introduced generating function identities used to obtain series accelerations for values of Dirichlet's $β$ function, via the Markov--Wilf--Zeilberger method. Inspired by these past results, together with related results introduced by Chu et al., we introduce a variety of hypergeometric recurrences. We prove these recurrences using the WZ method, and we apply these recurrences to obtain series acceleration identities. We introduce a family of summations generalizing a Ramanujan-type series for $\frac{1}{π^2}$ due to Guillera, and a family…

与 Appell 超几何函数相关的轨迹的完备性

Completeness of trajectories associated to Appell hypergeometric functions

本文证明在 Dirichlet 边界条件下，非线性 Schrödinger 方程及其与 Appell 超几何函数相关的完全非线性版本的轨迹均保持线性完备性，重新审视了 Fraenkel 提出的非线性完备性框架。

关于一类通过递归和乘积二项式 - 调和恒等式的超几何和

Recurrences for Hypergeometric Moments and Binomial-Harmonic Sums

研究系数特定的超几何级数，通过在分母引入移位参数，利用欧拉超几何微分方程推导递归，求解得到普通和与交错和化简及正分母幂公式，还应用于乘积二项式情形与调和数和，给出处理多种相关和式的统一框架。

Mixed moment of $GL(2)$ and $GL(3)$ $L$-functions

Let $ \mathfrak{f} $ run over the space $ H_{4k} $ of primitive cusp forms of level one and weight $ 4k $, $ k \in N $. We prove an explicit formula for the mixed moment of the Hecke $ L $-function $ L(\mathfrak{f}, 1/2) $ and the symmetric square $L$-function $ L(sym^2\mathfrak{f}, 1/2)$, relating it to the dual mixed moment of the double Dirichlet series and the Riemann zeta function weighted by the ${}_3F_{2}$ hypergeometric function. Analysing the corresponding special functions by the means of the Liouville-Green approximation followed by the saddle point method, we prove that the initial…

Two General Series Identities Involving Modified Bessel Functions and a Class of Arithmetical Functions

We consider two sequences $a(n)$ and $b(n)$, $1\leq n<\infty$, generated by Dirichlet series $$\sum_{n=1}^{\infty}\frac{a(n)}{λ_n^{s}}\qquad\text{and}\qquad \sum_{n=1}^{\infty}\frac{b(n)}{μ_n^{s}},$$ satisfying a familiar functional equation involving the gamma function $Γ(s)$. Two general identities are established. The first involves the modified Bessel function $K_μ(z)$, and can be thought of as a 'modular' or 'theta' relation wherein modified Bessel functions, instead of exponential functions, appear. Appearing in the second identity are $K_μ(z)$, the Bessel functions of imaginary argument…