Some Eighth Order Mock Theta Functions

A method is developed for obtaining Ramanujan's mock theta functionsfrom ordinary theta functions by performing certain operationson their q-series expansions. The method is then used to constructseveral new mock theta functions, including the first ones ofeighth order. Summation and transformation formulae for basichypergeometric series are used to prove that the new functionsactually have the mock theta property. The modular transformationformulae for these functions are obtained.     

Spin Verlinde Spaces and Prym Theta Functions

It is shown that the theta functions of level n on the principallypolarised Prym varieties of an algebraic curve are dual to thesections of the orthogonal theta line bundle on the moduli spaceof Spin(n)-bundles over the curve. As a by-product of our computations,we also note that when n is odd, the Pfaffian line bundle onmoduli space has a basis of sections labelled by the even thetacharacteristics of the curve. 1991 Mathematics Subject Classification:14D20, 14H42, 14H60, 14K25.     

A Duality for Spin Verlinde Spaces and Prym Theta Functions

The paper proves canonical isomorphisms between Spin Verlindespaces, that is, spaces of global sections of a determinantline bundle over the moduli space of semistable Spin_n-bundlesover a smooth projective curve C, and the dual spaces of thetafunctions over Prym varieties of unramified double covers ofC.     

Identities Involving Partial Mock Theta Functions of Order Five

In this paper we have given transformations for the partial mock theta functions of order five and also some identities between these partial mock theta functions analogous to the identities given by Ramanujan.     

Finite and Infinite Order Differential Relations for Theta Functions

We give different linear and nonlinear differential relations on Jacobi theta functions with more emphasis on the nonlinear differential equation of the third order of Jacobi. We present different points of view with a special attention to the role played by the second order linear differential equations, and their link to the Riccati equation and the Schwarzian equation. We also study an identity for theta functions resulting from the action of certain infinite order differential operators.     

On Diophantine Approximations of Mock Theta Functions of Third Order

The paper gives bounds for the approximation of the values of Ramanujan's Mock Theta functions of third order and more generally of some q-hypergeometric functions by the elements of an algebraic number field. Simultaneous approximations for the values of q-exponential function are also obtained. All the results are given both in the archimedean and p-adic case.     

二阶周期线性微分方程解的几个性质

In this paper,the zeros of solutions of periodic second order linear differential equation y ＋ Ay = 0,where A（z） = B（e z ）,B（ζ） = g（ζ） ＋ p j=1 b ？j ζ ？j ,g（ζ） is a transcendental entire function of lower order no more than 1/2,and p is an odd positive integer,are studied.It is shown that every non-trivial solution of above equation satisfies the exponent of convergence of zeros equals to infinity.     

Some Lambert Series Expansions of Products of Theta Functions

Let q be a complex number satisfying |q| < 1. The theta function (q) is defined by (q) = . Ramanujan has given a number of Lambert series expansions such as

Stability Properties in Some Classes of Second Order Partial Differential Equations

The main purpose of this note is to investigate the stability problem in a certain class of partial differential equations.     

Modular Transformations of Ramanujan's Fifth and Seventh Order Mock Theta Functions

In his last letter to Hardy, Ramanujan defined 17 functions F(q), where |q| < 1. He called them mock theta functions, because as q radially approaches any point e ^2ir (r rational), there is a theta function F _r(q) with F(q) – F _r(q) = O(1). In this paper we obtain the transformations of Ramanujan's fifth and seventh order mock theta functions under the modular group generators + 1 and –1/, where q = e ⁱ. The transformation formulas are more complex than those of ordinary theta functions. A definition of the order of a mock theta function is also given.     

On Nonlinear Equations Integrable in Theta Functions of Nonprincipally Polarized Abelian Varieties

Siberian Mathematical Journal -     

Second Order Differential-functional Inequalities for Bounded Functions

We prove a differential-functional version of the Lemma of Nagumo and Westphal which can be applied to differential-functional inequalities for bounded C -functions on having bounded derivatives. As an application an existence and uniqueness result for bounded solutions of linear second order differential-functional equations is proved.     

Superparabolic Functions Related to Second Order Hypoelliptic Operators

In this paper, we consider a wide class of second order hypoelliptic partial differential operators with nonnegative characteristic form. We prove a monotone smoothing theorem and a representation formula for superparabolic functions.     

On Combinatorial Extensions of Some Ramanujan’s Mock Theta Functions

Ukrainian Mathematical Journal - Five mock theta functions of S. Ramanujan are combinatorially interpreted by means of certain associated lattice path functions and antihook differences. These...     

Some Properties of Bent Functions

First, this paper discusses and sums up some properties of a pair of functions p（x）, q（x） that makes （y ＋ 1）p（x） ＋ yq（x） into a bent function. Then it discusses the properties of bent functions. Also, the upper and lower bounds of the number of bent functions on GF（2）2k are discussed.     

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.     

Second Order Linear Differential Polynomials and Real Meromorphic Functions

The main result determines all real meromorphic functions f of finite lower order in the plane such that f has finitely many zeros and non-real poles, while f′′ + a ₁ f′ + a ₀ f has finitely many non-real zeros, where a ₁ and a ₀ are real rational functions which satisfy a ₁(∞) = 0 and a ₀(x) ≥ 0 for all real x with |x| sufficiently large. This is accomplished by refining some earlier results on the zeros in a neighbourhood of infinity of meromorphic functions and second order linear differential polynomials. Examples are provided illustrating the results.     

Oscillatory Properties of Second Order Half-Linear Difference Equations

We study oscillatory properties of the second order half-linear difference equation