Lifting Bailey pairs to WP-Bailey pairs

A pair of sequences such that and

     

A reciprocity relation for WP-Bailey pairs

We derive a new general transformation for WP-Bailey pairs by considering a certain limiting case of a WP-Bailey chain previously found by the authors, and examine several consequences of this new transformation. These consequences include new summation formulae involving WP-Bailey pairs. Other consequences include new proofs of some classical identities due to Jacobi, Ramanujan and others, and indeed extend these identities to identities involving particular specializations of arbitrary WP-Bailey pairs.     

Some new Steffensen pairs

Using some basic properties of the extended mean values and the generalized weighted mean values, an integral expression of the function (b ^x -a ^x)/x, and analytic techniques, some new Steffensen pairs are established.     

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.     

Ramanujan-type partial theta identities and conjugate Bailey pairs

Residual identities of Ramanujan-type partial theta identities are tailor-made for producing conjugate Bailey pairs. This is carried out for partial theta identities in Ramanujan??s lost notebook, a number of the partial theta identities of Warnaar, and for some new ones as well.     

Some new q-series identities through an extended Bailey transform

In this note, a variety of new formulas for basic hypergeometric series will be derived by an application of a single Bailey-type transform to some q-series formulas which appeared recently in the literature.

     

Hessenberg pairs of linear transformations

Let denote a field and V denote a nonzero finite-dimensional vector space over . We consider an ordered pair of linear transformations A:V→V and A^*:V→V that satisfy (i)–(iii) below.

1. [(i)]Each of A,A^* is diagonalizable on V.

2. [(ii)]There exists an ordering of the eigenspaces of A such that

where V_-1=0, V_d+1=0.

3. [(iii)]There exists an ordering of the eigenspaces of A^* such that

where , .

We call such a pair a Hessenberg pair on V. In this paper we obtain some characterizations of Hessenberg pairs. We also explain how Hessenberg pairs are related to tridiagonal pairs.

On pointwise complete pairs of linear transformations

We prove the pointwise completeness of the order n system with constant coefficients under the assumption that the matrices of the system split into square blocks of the same size so that the collection of all blocks embeds into a finite dimensional associative division algebra; the block rank of the passive matrix is at most 2.     

Ramanujan-type partial theta identities and conjugate Bailey pairs,II. Multisums

In the first paper of this series, we described how to find conjugate Bailey pairs from residual identities of Ramanujan-type partial theta identities. Here we carry this out for four multisum residual identities of Warnaar and two more due to the authors. Applying known Bailey pairs gives expressions in the algebra of modular forms and indefinite theta functions.     

Some new invariant pairs (t,3) for projective hjelmslev planes

Associated with every finite projective Hjelmslev plane is an invariant pair (t,r): t is the number of neighbours of a given point on a given line passing through it and r is the order of the underlying projective plane. The Drake-Lenz method [2],[3] of using auxiliary matrices for the constructions of projective Hjelmslev planes has become standard by now. This paper is intended to give some new constructions of projective Hjelmslev planes with invariant pairs (t,3) by making use of the generalization and improvement of the Drake-Lenz theorem [3] obtained by the author in [6] and [7]. The results of this paper add 8 new values to the list ([5], example 3.7(ii)) of invariant pairs (t,3) with t 1,000 for projective Hjelmslev planes.     

Some transformations and reduction formulas for multipleq-hypergeometric series

Summary Some simple ideas of G. E. Andrews ([2], [3]) are used here to derive a transformation formula for a general multiple q-series with essentially arbitrary terms. As applications of (or motivated by) this q-series transformation, several transformation and reduction formulas for q-hypergeometric series in two and more variables are presented. Relevant connections of the various q-identities considered here with a number of known results are also indicated.Supported, in part, by N.S.E.R.C. (Canada) Grant A-7353.     

G-度量空间中两对非相容映象的几个新的公共不动点定理

在G-度量空间的框架下,利用自映象对的非相容性和（Ag）型R-弱交换性条件,在既不要求空间的完备性,也不要求映象连续的条件下,建立了一类？-压缩条件下的4个映象的几个新的公共不动点定理,此定理在很大程度上改进和发展了已有文献的相关结果.     

A $$$$ WP-Bailey lattice and its applications

In this paper, we establish a \(U(n+1)\) WP-Bailey lattice. By iterating this method, we obtain a \(U(n+1)\) transformation formula for \(U(n+1)\) WP-Bailey pairs. By considering the \(U(n+1)\) unit WP-Bailey pair, several new \(U(n+1)\) transformation formulas are given. In particular, we give a new \(U(n+1)\) Andrews–Gordon identity.     

On birational transformations of pairs in the complex plane

This article deals with the study of the birational transformations of the projective complex plane which leave invariant an irreducible algebraic curve. We try to describe the state of the art and provide some new results on this subject.      

拉格朗日反演的q模拟与Bailey引理的注记

本文主要揭示了Gessel Ira.等给出的拉格朗日反演的q—模拟形式与An-drews G.E.的Bailey引理之间的相互转化的联系,做为例证,给出了利用这些关系得到的古典超几何级数(hypergeometric series)变换和求和公式的新证明,同时得到了模5、7、9、27四个新的Roger’s-Ramanujan类型的恒等式,其具有十分重要的组合意义。     

Some explicit badly approximable pairs

I consider the Diophantine approximation problem of sup-norm simultaneous rational approximation with common denominator of a pair of irrational numbers, and compute explicitly some pairs with large approximation constant. One of these pairs is the most badly approximable pair yet computed.     

Extensions of pairs of linear transformations between infinite-dimensional vector spaces

To a pair A, B:V→W of linear maps between complex vector spaces attach the pair (V, W) endowed with the operation (α, β)υ = (αA + βB)(υ), α,β ∈ C, υ ∈ V. A concept of rank, similar to the torsion-free rank of abelian groups, is definable for the systems (V, W). With appropriate morphisms, the systems from an abelian category and Ext¹ can be construed as a vector space valued functor. We find all the cases in which Ext¹ ((V, W), (X, Y)), with (X, Y), (V, W) indecomposable systems of rank 0 or 1, is finite-dimensional, and compute its dimension in these cases. This extends a former computation for finite-dimensional systems.     

Some matrix transformations and measures of noncompactness

In this paper we shall compute the Hausdorff measure of noncompactness for any bounded subset of \(r_{p}^{q}\) and also study the evaluation of the Hausdorff measure of noncompactness of an operator on the space \(r_{p}^{q}\ (1 \leq p < \infty)\) associated with some matrix transformation.