The Borweins' Cubic Theta Function Identity and Some Cubic Modular Identities of Ramanujan

In this paper the author will give new proofs of the Borweins' cubic theta function identity and a related identity relying on the properties of elliptic functions and the technique of comparing constant terms.     

Circular Summation of the 13th Powers of Ramanujan's Theta Function

An explicit formula is derived for the circular summation of the 13th power of Ramanujan's theta function in terms of Dedekind eta function.     

Modularity and Total Ellipticity of Some Multiple Series of Hypergeometric Type

We collect some new evidence for the validity of the conjecture that every totally elliptic hypergeometric series is modular invariant and briefly discuss a generalization of such series to Riemann surfaces of arbitrary genus.     

Fourier expansion of Eisenstein series on the Hilbert modular group and Hilbert class fields

In this paper we consider the Eisenstein series for the Hilbert modular group of a general number field. We compute the Fourier expansion at each cusp explicitly. The Fourier coefficients are given in terms of completed partial Hecke -series, and from their functional equations, we get the functional equation for the Eisenstein vector. That is, we identify the scattering matrix. When we compute the determinant of the scattering matrix in the principal case, the Dedekind -function of the Hilbert class field shows up. A proof in the imaginary quadratic case was given in Efrat and Sarnak, and for totally real fields with class number one a proof was given in Efrat.

     

On three differential equations associated with Chebyshev polynomials of degrees 3, 4 and 6

We shall study the differential equation y'~2= T_n(y)-(1-2μ~2);where μ~2 is a constant, T_n(x) are the Chebyshev polynomials with n = 3, 4, 6.The solutions of the differential equations will be expressed explicitly in terms of the Weierstrass elliptic function which can be used to construct theories of elliptic functions based on _2F_1(1/4, 3/4; 1; z),_2F_1(1/3, 2/3; 1; z), _2F_1(1/6, 5/6; 1; z) and provide a unified approach to a set of identities of Ramanujan involving these hypergeometric functions.     

The Vanishing of the Theta Function in the KP Direction: A Geometric Approach

We give a geometric proof of a formula, due to Segal and Wilson, which describes the order of vanishing of the Riemann theta function in the direction which corresponds to the direction of the tangent space of a Riemann surface at a marked point. While this formula appears in the work of Segal and Wilson as a by-product of some nontrivial constructions from the theory of integrable systems (loop groups, infinite-dimensional Grassmannians, tau functions, Schur polynomials, etc.) our proof only uses the classical theory of linear systems on Riemann surfaces.     

On the Products of Three Theta Functions

In this paper, we re-examine several beautiful identities of Ramanujan from the perspective of the products of three theta functions.     

Prime decomposition in the anti-cyclotomic extension

For an imaginary quadratic number field and an odd prime number , the anti-cyclotomic -extension of is defined. For primes of , decomposition laws for in the anti-cyclotomic extension are given. We show how these laws can be applied to determine if the Hilbert class field (or part of it) of is -embeddable. For some and , we find explicit polynomials whose roots generate the first step of the anti-cyclotomic extension and show how the prime decomposition laws give nice results on the splitting of these polyniomials modulo . The article contains many numerical examples.

     

Exceptional Congruences for Powers of the Partition Function

In Journal of London Math. Soc. 31 (1956), 350–359, Morris Newman studied vector spaces of functions arising from lifts to ₀(p) of certain eta-products on the group ₀(pQ), Q = p ⁿ. In this paper, the author considers vector spaces of modular functions obtained as lifts of more general eta-products from ₀(pQ) to ₀(p), (Q, p) = 1. Specifically considered are functions arising as lifts of functions of the form

A Measure of Simultaneous Approximation for Quasi-Modular Functions

We improve on a recent result of Saradha giving a transcendence measure for the quotient of a period of an elliptic curve defined over by its associated quasi-period. In an (almost successful) attempt to include in a single measure both this result and that obtained by Reyssat in 1980, we blend into the modular method ideas related to modular but also hypergeometric functions, as appearing e.g. in André's work, as well as some Galois considerations.     

On Non-Congruence Subgroups of the Analogue of the Modular Group in Characteristic p

Let k[t] be the polynomial ring over a finite field k. The group SL ₂(k[t]) is often referred to as the analogue, in characteristic p, of the classical modular group SL ₂( ), where is the ring of rational integers. It is well-known that the smallest index of a non-congruence subgroup of SL ₂( ) is 7. Here we compute this index for SL ₂(k[t]). (In all but 6 cases it turns out to be 1 + q, where q is the order of k.)     

模李超代数研究的若干进展

近几年，模李超代数（即素特征域上的李超代数）的研究取得了一些进展。本文分析以下6个方面介绍近几年在模李超代数研究中取得的一些新结果。1，限制李超代数，2，Cartan型模李超代数；3，深度1的Z－阶化李超代数；4，深度1的滤过李超代数。5；Cartan型模李超代数的某些内蕴性质；6，公开的问题。     

Some families of generating functions for the extended Srivastava polynomials

Almost four decades ago, H.M. Srivastava considered a general family of univariate polynomials, the Srivastava polynomials, and initiated a systematic investigation for this family [10]. In 2001, B. González, J. Matera and H.M. Srivastava extended the Srivastava polynomials by inserting one more parameter [4]. In this study we obtain a family of linear generating functions for these extended polynomials. Some illustrative results including Jacobi, Laguerre and Bessel polynomials are also presented. Furthermore, mixed multilateral and multilinear generating functions are derived for these polynomials.     

An asymptotic expansion for the semi‐infinite sum of Dirac‐δ functions

In this paper, we derive an asymptotic expansion for the semi‐infinite sum of Dirac‐δ functions centered at discrete equidistant points defined by the set . The method relies on the Laplace transform of the semi‐infinite sum of Dirac‐δ functions. The derived series distribution takes the form of the Euler‐Maclaurin summation when the distributions are defined for complex or real‐valued continuous functions over the interval . For n=1, the series expansion contributes with a term equal to δ(x)/2, which survives in the limit when a→0⁺. This term represents a correction term, which is in general omitted in calculations of the density of states of quantum confined systems by finite‐size effects.     

函数域常值域扩张的类数整除性

设K/F_q是亏格大于0的整体函数域,K_n:=KF_(q~n)是K上的n次常值域扩张.利用整体函数域zeta函数的整系数多项式的有理表达式,结合函数域常值域扩张的基本性质,对于满足特定条件的素数l,本文讨论了使得除子类群Pic~0(K_n)的Sylow-l子群为非平凡群的常值域扩张K_n的存在性.     

Certain Modular Functions Similar to the Dedekind eta Function

In this paper, we study certain modular functions η_ψ(z) similar to the Dedekind eta function η(z). The functions are given by an analogue of Borcherds type liftings. It turns out that the functions η_ψ(z) have some good properties, similarly as in the case of the Dedekind eta function.     

随机变量和的模函数分布及其应用

介绍了一个在计算机科学、信息科学等学科中具有广泛应用的随机变量和的模函数,计算了其分布,并提供了该函数在图像信息安全领域的一个应用例子,验证了理论结果.     

Computation of certain infinite series of the form for arbitrary real-valued k

We provide a general method from which one may obtain more arbitrary closed form solutions for infinite series of the form ∑f(n)n^k, for and sufficiently well-behaved f(n). We illustrate the methods for the special case of f(n)=aⁿ/n!, obtaining generalizations to the Bell numbers and polynomials.     

Estimate of an Entire Function in the Union of Two Angles

We obtain an estimate of an entire function expressed as the limit of a sequence of polynomials in exponentials in the union of two angles via an estimate of the function and its derivatives on a set lying on a horizontal strip.     

On Finite Forms of Certain Bilateral Basic Hypergeometric Series and Their Applications

In this paper we derive finite forms of the summation formulas for bilateral basic hypergeometric series 3ψ3,4ψ4 and 5ψ5.We therefrom obtain the summation formulae obtained recently by Wenchang CHU and Xiaoxia WANG.As applications of these summation formulae,we deduce the well-known Jacobi's two and four square theorems,a formula for the number of representations of an integer n as sum of four triangular numbers and some theta function identities.