Partition identities arising from Ramanujan’s formulas for multipliers

We find new partition identities arising from Ramanujan’s formulas of multipliers. Several of the identities are for overpartitions, overpartition pairs, and \(\ell \)-regular partitions.     

Exploring multivariate Padé approximants for multiple hypergeometric series

We investigate the approximation of some hypergeometric functions of two variables, namely the Appell functions F _i , i = 1,...,4, by multivariate Padé approximants. Section 1 reviews the results that exist for the projection of the F _i onto ϰ=0 or y=0, namely, the Gauss function ₂ F ₁(a, b; c; z), since a great deal is known about Padé approximants for this hypergeometric series. Section 2 summarizes the definitions of both homogeneous and general multivariate Padé approximants. In section 3 we prove that the table of homogeneous multivariate Padé approximants is normal under similar conditions to those that hold in the univariate case. In contrast, in section 4, theorems are given which indicate that, already for the special case F ₁(a, b, b′; c; x; y) with a = b = b′ = 1 and c = 2, there is a high degree of degeneracy in the table of general multivariate Padé approximants. Section 5 presents some concluding remarks, highlighting the difference between the two types of multivariate Padé approximants in this context and discussing directions for future work. This revised version was published online in June 2006 with corrections to the Cover Date.     

The Saalschütz chain reactions and bilateral basic hypergeometric series

By iterating recursively the q-Saalschütz summation formula, we introduce the Saalschütz chain reactions. A general series transform, which expresses a nonterminating bilateral series in terms of a finite multiple unilateral sum, will be established. As applications we derive, by means of Bailey’s 6ψ6 -series identity, several bilateral transformations including one due to Milne [12]. These transformations further yield a number of closed formulas of very well-poised bilateral basic hypergeometric series; which are closely related to the identities obtained by Minton [13], Karlsson [11], Gasper [8], and Chu [5], [6], [7] through the partial fraction method and divided differences.     

Blossoming and Hermite-Padé approximation for hypergeometric series

Numerical Algorithms - Based on the blossoming theory, in this work we develop a new method for deriving Hermite-Padé approximants of certain hypergeometric series. Its general principle...     

Heat Kernel and Hardy’s Theorem for Jacobi Transform

In this paper, the authors obtain sharp upper and lower bounds for the heat kernel associated with Jacobi transform, and get some analogues of Hardy's Theorem for Jacobi transform by using the sharp estimate of the heat kernel.     

Heat Kernel and Hardy’s Theorem for Jacobi Transform

In this paper, the authors obtain sharp upper and lower hounds for the heat kernel associated with Jacobi transform, and get some analogues of Hardy‘s Theorem for Jacobi transform by using the sharp estimate of the heat kernel.     

Heat Kernel and Hardy’s Theorem for Jacobi Transform

In this paper, the authors obtain sharp upper and lower bounds for the heat kernel associated with Jacobi transform, and get some analogues of Hardy’s Theorem for Jacobi transform by using the sharp estimate for the heat kernel.     

Bibasic analystic functions and discrete ‘bibasic’ hypergeometric series

This paper deals with a study of a class of functions called bibasis analytic functions. Using discrete powerz ⁽ⁿ⁾, discrete bibasic hypergeometric functions have been introduced.     

q-Generalizations of Mortenson’s identities and further identities

By means of partial fraction decomposition, we give simple proofs of Mortenson’s identities first. Then, inspired by them, we derive their q-generalizations and explore further identities of similar type.     

Fischer Decomposition and Cauchy Kernel for Dunkl–Dirac operators

In this paper we present a Fischer decomposition for Dirac operator and an explicit construction of a Cauchy kernel for Dunkl-monogenic functions.     

Regev’s and Amitsur’s conjectures for codimensions of generalized polynomial identities

Let A be a finite-dimensional associative algebra over a field of characteristic 0. Then there exist C ∈ ℚ₊ and t ∈ ℤ₊ such that gc _n (A) ∼ Cn ^t d ⁿ as n → ∞, where d = PIexp(A). In particular, Amitsur’s and Regev’s conjectures hold for the codimensions gc _n (A) of generalized polynomial identities.     

Some extensions for Ramanujan’s circular summation formulas and applications

The Ramanujan Journal - In this paper, we give some extensions for Ramanujan’s circular summation formulas with the mixed products of two Jacobi’s theta functions. As applications, we...     

Thue’s inequalities and the hypergeometric method

We establish upper bounds for the number of primitive integer solutions to inequalities of the shape \(0<|F(x, y)| \le h\), where \(F(x , y) =(\alpha x + \beta y)^r -(\gamma x + \delta y)^r \in \mathbb {Z}[x ,y]\), \(\alpha \), \(\beta \), \(\gamma \) and \(\delta \) are algebraic constants with \(\alpha \delta -\beta \gamma \ne 0\), and \(r \ge 5\) and h are integers. As an important application, we pay special attention to binomial Thue’s inequalities \(|ax^r - by^r| \le c\). The proofs are based on the hypergeometric method of Thue and Siegel and its refinement by Evertse.     

Whitney’s formulas for curves on surfaces

The classical Whitney formula relates the number of times an oriented plane curve cuts itself to its rotation number and the index of a base point. In this paper we generalize Whitney’s formula to curves on an oriented punctured surface Σ _{m, n} , obtaining a family of identities indexed by elements of π ₁(Σ _{m, n} ). To define analogs of the rotation number and the index of a base point of a curve γ, we fix an arbitrary vector field on Σ _{m, n} . Similar formulas are obtained for non-based curves.     

A note on q-difference equations for Ramanujan’s integrals

The Ramanujan Journal - This short paper derives the relationship between solutions of q-difference equations and generating functions for q-orthogonal polynomials. The key of the method is to...     

Evaluation of Gaussian hypergeometric series using Huff’s models of elliptic curves

The Ramanujan Journal - A Huff curve over a field K is an elliptic curve defined by the equation $$ax(y^2-1)=by(x^2-1)$$ where $$a,bin K$$ are such that $$a^2ne b^2$$ . In a similar fashion, a...     

Wiener’s and Lévy’s theorems for some weighted power series

We obtain weighted version of the classical theorems of Wiener and Lévy on absolutely convergent power series.     

Boundary integral operators and boundary value problems for Laplace’s equation

In this paper, we define boundary single and double layer potentials for Laplace’s equation in certain bounded domains with d-Ahlfors regular boundary, considerably more general than Lipschitz domains. We show that these layer potentials are invertible as mappings between certain Besov spaces and thus obtain layer potential solutions to the regularity, Neumann, and Dirichlet problems with boundary data in these spaces.