Bailey pairs and indefinite quadratic forms

We construct classes of Bailey pairs where the exponent of q   in _αn${\alpha }_n$ is an indefinite quadratic form. As an application we obtain families of q-hypergeometric mock theta multisums.     

On minima of rational indefinite quadratic forms

Let f(x) be an indefinite quadratic form with real coefficients in n variables with nonzero determinant d. The collection of all values of $\text{v(f) = |d|}{}^{\mathrm{<mtext>?1</mtext><mtext>n</mtext>}}\text{inf}\text{|f(x)|}$, where infimum is taken over x ∈ Zⁿ such that f(x) ≠ 0 (x ≠ 0) is called the spectrum of nonzero minima (spectrum of minima) of such forms. The spectrum is said to be discrete if for every δ > 0, there are only finitely many values of v(f) > δ. It is proved that for rational quadratic forms in n ≥ 3 variables and real quadratic forms in n ≥ 21 variables the spectra of nonzero minima are discrete. Also the spectra of minima of indefinite ternary and quaternary rational quadratic forms are discrete.     

The minima of indefinite binary quadratic forms

The study of the minima of indefinite binary quadratic forms has a long history and the classical results concerning the determination of such minima are stated in terms of the continued fraction expansion of the roots. These results are recast in geometric terms. Using this, and well-known geometric properties of the modular group, some necessary and sufficient conditions for a certain class of quadratic forms to have positive unattained minima are obtained.     

Class numbers of indefinite binary quadratic forms

We determine the asymptotic average sizes of the class numbers of indefinite binary quadratic forms when ordered by the sizes of their corresponding fundamental units. The proofs make use of the Selberg trace formula.     

The distribution of hermitian indefinite quadratic forms

Hermitian quadratic forms in complex Gaussian random variables is considered in this article. Various representations of the exact densities for the central, non central, singular and nonsingular cases are given. Particular cases are shown to agree with the results obtained by others in connection with various applications in communication theory, multichannel antennas, vehicular transportations and other related problems     

Construction of Jacobi forms associated to indefinite quadratic forms

Vignéras constructs non-holomorphic theta functions according to indefinite quadratic forms with arbitrary signature. We use Vignéras’ theta functions to create examples of non-holomorphic Jacobi forms associated to indefinite theta series by two different methods.     

Class numbers of indefinite binary quadratic forms II

We obtain an asymptotic formula for the averages of class numbers of indefinite binary quadratic forms taken along certain polynomial sequences. It is found that along these sequences the class number is essentially as large as possible.     

Positive values of non-homogeneous indefinite quadratic forms II

The minimum Γ_{r, r + 1} of positive values of non-homogeneous indefinite quadratic forms of type (r, r + 1), r ≥ 2, is determined. For r = 2, an isolation theorem is proved which is used in the result for r ≥ 3. Some asymmetric inequalities for forms of type (2, 2) needed for these results are also obtained.     

On theta functions associated to indefinite quadratic forms

We introduce a family of theta functions associated to an indefinite quadratic form, and prove a modular transformation formulas by regarding each such function as a specialization of a symplectic theta function. An eighth rott of unity arises in these formulas, and it is expressly given in all cases. The theta functions feature many “translation variables,” which are useful for the study of the liftings of modular forms.