Transformation formula for a double Clausenian hypergeometric series,its q-analogue,and its invariance group

A transformation formula for a double basic hypergeometric series of type Φ_0:2;2^1:2;2 is derived. This transformation yields a double series analogue of Sears’ transformation for a terminating ₃Φ₂ series. In the limit q→1, the formula reduces to a transformation for a terminating double Clausenian hypergeometric series of unit argument (one of the proper Kampé de Fériet series, F_0:2;2^1:2;2(1,1)). This formula is a double series analogue of Whipple's terminating ₃F₂ transformation. This transformation gives rise to a transformation group (the invariance group) acting on the parameters of the double series. The invariance group is examined and shown to be a subgroup of a double copy of the symmetries of the square.     

Note on the differentiability of arithmetic Fourier series arising from Eisenstein series

The aim of this note is to present results concerning the differentiability of some Fourier series arising from Eisenstein series. Sine series exhibit different behaviours with respect to differentiability than the series with cosine function. The precise results are given for the series related to Eisenstein series of weight 2, whereas for the series arising from Eisenstein series of higher weight we conjecture the results.     

利用函数的傅里叶展开式求级数的和

利用函数的傅里叶展开式可求得级数∞∑n=11/n2+λ2及∞∑n=1(-1)m/n2+λ2的和,而通过引入复数并利用欧拉公式可求得级数∞∑n=1 1/n2+λ2及∞∑n=1(-1)m/n2+λ2的和.     

Analogues of the Hurwitz Formulas for Level 2 Eisenstein Series

In this paper, we consider certain double series of Eisenstein type involving hyperbolic functions, which can be regarded as analogues of the level 2 Eisenstein series. We prove some evaluation formulas for these series at positive integers which are analogues of both the Hurwitz formulas for the level 2 Eisenstein series and the classical results given by Cauchy, Lerch, Mellin and Ramanujan.     

Closed form expressions for some series over certain trigonometric integrals

The series (3) and (4), where T(x) denotes trigonometric integrals (2), are represented as series in terms of Riemann zeta and related functions using the sums of the series (5) and (6), whose terms involve one trigonometric function. These series can be brought in closed form in some cases, where closed form means that the series are represented by finite sums of certain integrals. By specifying the function φ(y) appearing in trigonometric integrals (2) we obtain new series for some special types of functions as well as known results.     

Transformation and summation formulas for double hypergeometric series

A number of new transformation formulas for double hypergeometric series are presented. The series appearing here are the so-called Kampé de Fériet functions of type F_1:1;2^0:3;4(1,1) and F_0:2;2^1:2;2(1,1). The transformation formulas relate such double series to a single hypergeometric series of ₄F₃(1) type. By specializing certain parameters, a list of new summation formulas for F_0:2;2^1:2;2(1,1) series is obtained. The origin of the results comes from studying symmetries of the 9-j coefficient appearing in quantum theory of angular momentum.     

Eisenstein series on quaternion half-space of degree N

In Kim [7], we studied an Eisenstein series on quaternion half-space of degree 2. By calculating the Siegel series using the method of Karel [5], we obtained the analytic continuation and functional equation of the Eisenstein series. In this note we study an Eisenstein series on quaternion half-space of degreen. By calculating the Siegel series in an analogous way as in Shimura [15] and Kitaoka [8], we obtain singular modular forms of weightk, k<2n and 4/k. Furthermore, we obtain the analytic continuation and functional equation of the Eisenstein series.     

连续切波变换反演公式的级数表示

我们主要研究连续切波变换反演公式的级数表示.首先引入两类由切波变换反演公式定义的无穷级数和有限级数,并研究了由Kittipoom等人介绍的切波生成空间,得到这个切波生成空间的一些重要性质.其次利用这些结果显示:对于这个切波生成空间,当采样密度趋于无穷时由我们定义的无穷级数按L~2-范数收敛于重构函数;对于可允许函数空间,当采样密度趋于无穷时由我们定义的有限级数按L~2-范数收敛于重构函数.     

Double zeta values, double Eisenstein series, and modular forms of level 2

We study the double shuffle relations satisfied by the double zeta values of level 2, and introduce the double Eisenstein series of level 2 which satisfy the double shuffle relations. We connect the double Eisenstein series to modular forms of level 2.     

Some infinite series related to Feynman diagrams

Results are presented for some infinite series appearing in Feynman diagram calculations, many of which are similar to the Euler series. These include both one-, two- and three-dimensional series. The sums of these series can be evaluated with the help of various integral representations for hypergeometric functions, and expressed in terms of ζ(2),ζ(3), the Catalan constant G and Cl₂(π/3) where Cl₂(θ) is Clausen's function.     

Poincaré series of semi-invariants of 2×2 matrices

The Poincaré series of the algebra of -invariants of m-tuples of 2×2 matrices is presented both as a rational function and as a series of Schur functions. We show that this algebra of invariants is generated by the determinants, the mixed discriminants and the discriminants of 2×2 matrices. Consequences on invariants of three-dimensional matrices of the shape 2×2×m are discussed. For arbitrary n2, we prove an explicit functional equation for the Poincaré series of the -invariants of m-tuples of n×n matrices.     

多维平稳时间序列积分谱图的不变原理

用积分周期图估计平稳时间序列的谱函数，不论是高斯序列还是非高斯序列，其误差过程的不变原理都已被证明，许重光采用自迥归谱估计的积分估计谱函数也证明了误差过程的不变原理成立。本文讨论多维平稳时间序列，引进积分谱图来估计谱函数，证明了误差过程的不变原理在较弱的条件下成立。积分周期图是积分谱图的特例，因而［１］［３］［６］的一些结果是本文有关定理的特例。     

用黎曼求和法(R,k)求广义富理埃级数之和时的基卜斯现象

<正> 假设级数,■满足下面两个条件,就是:(i)在原点的某邻域内,对于 h(≠0)的一切值级数     

On convergence of ultraspherical lacunary series

Summary Kolmogoroff's classical result on the convergence of lacunary Fourier trigonometric series corresponding to a function of L² class has been extended to the convergence of the Fourier Ultraspherical series possessing lacunae similar to those supposed in Kolmogoroff's theorem for the trigonometric series.     

Pointwise and uniform error estimates associated with Abel-Whittaker interpolation series and its dual

BIT Numerical Mathematics - Abel-Whittaker series (1934) is viewed as a two-point Taylor series. It solves the interpolation problem $$f^{(2n)}(1) =a_n$$ , $$f^{(2n+1)}(0) =b_n,\,n\ge 0,$$ for an...     

Hypergeometric approach to Apéry-like series

By reformulating four hypergeometric series formulae, we derive 36 Apéry-like series expressions for the Riemann zeta function, including a couple of identities conjectured by Sun [New series for some special values of L-functions. Nanjing Univ J Math. 2015;32(2): 189–218].     

用黎曼求和法(R,1)求和時的基卜斯現象

<正> 假設級數滿足下面兩個條件,即:(甲)在原點的某一鄰域內,對於h(≠0)的一切值級數收斂;     

Geometry of curves with exceptional secant planes: linear series along the general curve

We study linear series on a general curve of genus g, whose images are exceptional with regard to their secant planes. Working in the framework of an extension of Brill?CNoether theory to pairs of linear series, we prove that a general curve has no linear series with exceptional secant planes, in a very precise sense, whenever the total number of series is finite. Next, we partially solve the problem of computing the number of linear series with exceptional secant planes in a one-parameter family in terms of tautological classes associated with the family, by evaluating our hypothetical formula along judiciously-chosen test families. As an application, we compute the number of linear series with exceptional secant planes on a general curve equipped with a one-dimensional family of linear series. We pay special attention to the extremal case of d-secant (d ? 2)-planes to (2d ? 1)-dimensional series, which appears in the study of Hilbert schemes of points on surfaces. In that case, our formula may be rewritten in terms of hypergeometric series, which allows us both to prove that it is nonzero and to deduce its asymptotics in d.     

Editorial Note

This issue the third of a series, contains papers honouring Professor Ky Fan. The dedication for the entire series may be found in Volume 11, No. 2, page 105 (1982). The previous (second) issue in the series was Volume 12, No. 3, (1982), beginning at page 157.     

What is an infinite series?

<正>Finite sequences and series have defined first and last terms,otherwise,infinite sequences and series continue indefinitely.Infinite series need tools from mathematical analysis,which are also widely used in other quantitative disciplines such as physics,computer science,and finance.Now we have an infinite sequence a_1,a_2,