Cubic Identities for Theta Series in Three Variables

We consider three-variable analogues of the theta series of Borwein and Borwein. We prove various identities involving these theta series including a generalization of the cubic identity of Borwein and Borwein.     

Cubic Identities of Theta Functions

Many remarkable cubic theorems involving theta functions can be found in Ramanujan's Lost Notebook. Using addition formulas, the Jacobi triple product identity and the quintuple product identity, we establish several theorems to prove Ramanujan's cubic identities.     

Cubic theta functions

Some new identities for the four cubic theta functions a′(q,z), a(q,z), b(q,z) and c(q,z) are given. For example, we show that

a′(q,z)³=b(q,z)³+c(q)²c(q,z).

This is a counterpart of the identity

a(q,z)³=b(q)²b(q,z³)+c(q,z)³,

which was found by Hirschhorn et al.

The Laurent series expansions of the four cubic theta functions are given. Their transformation properties are established using an elementary approach due to K. Venkatachaliengar. By applying the modular transformation to the identities given by Hirschhorn et al., several new identities in which a′(q,z) plays the role of a(q,z) are obtained.     

Modular functions on multilattices

We prove that every modular function on a multilattice L with values in a topological Abelian group generates a uniformity on L which makes the multilattice operations uniformly continuous with respect to the exponential uniformity on the power set of L.     

How to decrease the levels of modular integrals

A method to decrease the level of a modular integral with a given level is advanced. Project supported by the National Natural Science Foundation of China (Grant No. 19501009) and the Natural Science Foundation of Guangdong Province.     

Spontaneous Generation of Modular Invariants

It is possible to compute and its modular equations with no perception of its related classical group structure except at . We start by taking, for prime, an unknown ``-Newtonian' polynomial equation with arbitrary coefficients (based only on Newton's polygon requirements at for and ). We then ask which choice of coefficients of leads to some consistent Laurent series solution , (where . It is conjectured that if the same Laurent series works for -Newtonian polynomials of two or more primes , then there is only a bounded number of choices for the Laurent series (to within an additive constant). These choices are essentially from the set of ``replicable functions,' which include more classical modular invariants, particularly . A demonstration for orders and is done by computation. More remarkably, if the same series works for the -Newtonian polygons of 15 special ``Fricke-Monster' values of , then is (essentially) determined uniquely. Computationally, this process stands alone, and, in a sense, modular invariants arise ``spontaneously.'

     

The dimension of the space of Siegel–Eisenstein series of weight one

In general, it is difficult to determine the dimension of the space of Siegel modular forms of low weights. In particular, the dimensions of the spaces of cusp forms are known in only a few cases. In this paper, we calculate the dimension of the space of Siegel–Eisenstein series of weight 1, which is a certain subspace of a complement of the space of cusp forms.      

Modular Equations for Ramanujan's Cubic Continued Fraction

In this paper we present two new identities providing relations between Ramanujan's cubic continued fraction G(q) and the two continued fractions G(q⁵) and G(q⁷).     

上连通和超连通的三次Bi-Cayley图

Let G be a finite group and let S(possibly, contains the identity element) be a subset of G. The Bi-Cayley graph BC(G, S) is a bipartite graph with vertex set G×{0, 1} and edge set {(g, 0) (sg, 1) : g∈G, s ∈ S}. A graph is said to be super-connected if every minimum vertex cut isolates a vertex. A graph is said to be hyper-connected if every minimum vertex cut creates two components, one of which is an isolated vertex. In this paper, super-connected and/or hyper-connected cubic Bi-Cayley graphs are characterized.     

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

On an identity for zeros of Bessel functions

In this paper our aim is to present an elementary proof of an identity of Calogero concerning the zeros of Bessel functions of the first kind. Moreover, by using our elementary approach we present a new identity for the zeros of Bessel functions of the first kind, which in particular reduces to some other new identities. We also show that our method can be applied for the zeros of other special functions, like Struve functions of the first kind, and modified Bessel functions of the second kind.     

Finitely additive,modular, and probability functions on pre-Semirings

In this paper, we define finitely additive, probability and modular functions over semiring-like structures. We investigate finitely additive functions with the help of complemented elements of a semiring. We also generalize some classical results in probability theory such as the law of total probability, Bayes’ theorem, the equality of parallel systems, and Poincaré’s inclusion-exclusion theorem. While we prove that modular functions over a couple of known semirings are almost constant, we show it is possible to define many different modular functions over some semirings such as bottleneck algebras and the semiring (Id(D),+,?), where D is a Dedekind domain.     

A note on a Jacobian identity

An identity involving eight-fold infinite products, first derived by Jacobi in his theory of theta functions, is the subject of this note. Three similar identities, including one that implies Jacobi's identity, are presented.

     

On a unit group generated by special values of Siegel modular functions

There has been important progress in constructing units and -units associated to curves of genus 2 or 3. These approaches are based mainly on the consideration of properties of Jacobian varieties associated to hyperelliptic curves of genus 2 or 3. In this paper, we construct a unit group of the ray class field of modulo 6 with full rank by special values of Siegel modular functions and circular units. We note that . Our construction of units is number theoretic, and closely based on Shimura's work describing explicitly the Galois actions on the special values of theta functions.

     

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.     

On Eisenstein series and

In this paper, we derive some new identities satisfied by the series using Ramanujan's identities for , and . Our work is motivated by an attempt to develop a theory of elliptic functions to the septic base.

     

Elliott's identity and hypergeometric functions

Elliott's identity involving the Gaussian hypergeometric series contains, as a special case, the classical Legendre identity for complete elliptic integrals. The aim of this paper is to derive a differentiation formula for an expression involving the Gaussian hypergeometric series, which, for appropriate values of the parameters, implies Elliott's identity and which also leads to concavity/convexity properties of certain related functions. We also show that Elliott's identity is equivalent to a formula of Ramanujan on the differentiation of quotients of hypergeometric functions. Applying these results we obtain a number of identities associated with the Legendre functions of the first and the second kinds, respectively.     

一类单中心Hamilton系统在三次扰动下的Poincare分岔

使用一阶Mel‘nikov函数讨论了一类具有以抛物线与直线为边界的周期环域的单中心二次Hamilton系统的三次扰动下的Poincare分岔,得到其Poincare分岔最多可以产生两个极限环。     

Zeta functions interpolating the convolution of the Bernoulli polynomials

We prove several relations on multiple Hurwitz–Riemann zeta functions. Using analytic continuation of these multiple Hurwitz–Riemann zeta functions, we quote at negative integers Euler's nonlinear relation for generalized Bernoulli polynomials and numbers. As an application, we give a general convolution identity for Bernoulli numbers.