Elementary derivations of identities for bilateral basic hypergeometric series

We give elementary derivations of several classical and some new summation and transformation formulae for bilateral basic hypergeometric series. For purpose of motivation, we review our previous simple proof ("A simple proof of Bailey's very-well-poised 6-psi-6 summation", Proc. Amer. Math. Soc., to appear) of Bailey's very-well-poised 6-psi-6 summation. Using a similar but different method, we now give elementary derivations of some transformations for bilateral basic hypergeometric series. In particular, these include M. Jackson's very-well-poised 8-psi-8 transformation, a very-well-poised 10-psi-10 transformation, by induction, Slater's general transformation for very-well-poised 2r-psi-2r series, and Slater's transformation for general r-psi-r series. Finally, we derive some new transformations for bilateral basic hypergeometric series of Chu-Gasper-Karlsson-Minton-type.Comment: LaTeX2e, 35 pages, revised abstract and introductio

A simple proof of Bailey's very-well-poised 6-psi-6 summation

We give elementary derivations of some classical summation formulae for bilateral (basic) hypergeometric series. In particular, we apply Gauss' 2-F-1 summation and elementary series manipulations to give a simple proof of Dougall's 2-H-2 summation. Similarly, we apply Rogers' nonterminating 6-phi-5 summation and elementary series manipulations to give a simple proof of Bailey's very-well-poised 6-psi-6 summation. Our method of proof extends M. Jackson's first elementary proof of Ramanujan's 1-psi-1 summation.Comment: LaTeX2e, 10 pages, submitted to Proc. AMS, revised version, proofs of 1-psi-1 and 2-H-2 summations include

A new A_n extension of Ramanujan's 1-psi-1 summation with applications to multilateral A_n series

In this article, we derive some identities for multilateral basic hypergeometric series associated to the root system A_n. First, we apply Ismail's argument to an A_n q-binomial theorem of Milne and derive a new A_n generalization of Ramanujan's 1-psi-1 summation theorem. From this new A_n 1-psi-1 summation and from an A_n 1-psi-1 summation of Gustafson we deduce two lemmas for deriving simple A_n generalizations of bilateral basic hypergeometric series identities. These lemmas are closely related to the Macdonald identities for A_n. As samples for possible applications of these lemmas, we provide several A_n extensions of Bailey's 2-psi-2 transformations, and several A_n extensions of a particular 2-psi-2 summation.Comment: LaTeX2e, 26 pages, submitted to Rocky Mount. J. Math., spec. vol., conference proceedings of SF2000, Tempe, Arizona, May 29 - June 9, 200

Detection And Mapping (DAM) package. Volume 2a: Software user manual (part 1)

For abstract, see N77-18516

There’s just huge anxiety: ontological security, moral panic, and the decline in young people’s mental health and well-being in the UK

This study aims to critically discuss factors associated with a recent dramatic rise in recorded mental health issues amongst UK youth. It draws from interviews and focus groups undertaken with young people, parents and professionals. We offer valuable new insights into significant issues affecting young people’s mental health and well-being that are grounded in their lived experiences and in those who care for and work with them. By means of a thematic analysis of the data, we identified an increase in anxiety related to: future orientation, social media use, education, austerity, and normalization of mental distress and self-harm. We apply the notion of ontological security in our interpretation of how socio-cultural and political changes have increased anxiety amongst young people and consequent uncertainty about the self, the world and the future, leading to mental health problems. There are also problems conceptualizing and managing adolescent mental health, including increased awareness, increased acceptance of these problems, and stigmatisation. We relate this to the tendency for moral panic and widespread dissemination of problems in a risk society. In our conclusion, we highlight implications for future research, policy and practice

A new multivariable 6-psi-6 summation formula

By multidimensional matrix inversion, combined with an A_r extension of Jackson's 8-phi-7 summation formula by Milne, a new multivariable 8-phi-7 summation is derived. By a polynomial argument this 8-phi-7 summation is transformed to another multivariable 8-phi-7 summation which, by taking a suitable limit, is reduced to a new multivariable extension of the nonterminating 6-phi-5 summation. The latter is then extended, by analytic continuation, to a new multivariable extension of Bailey's very-well-poised 6-psi-6 summation formula.Comment: 16 page

Multilateral inversion of A_r, C_r and D_r basic hypergeometric series

In [Electron. J. Combin. 10 (2003), #R10], the author presented a new basic hypergeometric matrix inverse with applications to bilateral basic hypergeometric series. This matrix inversion result was directly extracted from an instance of Bailey's very-well-poised 6-psi-6 summation theorem, and involves two infinite matrices which are not lower-triangular. The present paper features three different multivariable generalizations of the above result. These are extracted from Gustafson's A_r and C_r extensions and of the author's recent A_r extension of Bailey's 6-psi-6 summation formula. By combining these new multidimensional matrix inverses with A_r and D_r extensions of Jackson's 8-phi-7 summation theorem three balanced very-well-poised 8-psi-8 summation theorems associated with the root systems A_r and C_r are derived.Comment: 24 page

Beats of the Magnetocapacitance Oscillations in Lateral Semiconductor Superlattices

We present calculations on the magnetocapacitance of the two-dimensional electron gas in a lateral semiconductor superlattice under two-dimensional weak periodic potential modulation in the presence of a perpendicular magnetic field. Adopting a Gaussian broadening of magnetic-field-dependent width in the density of states, we present explicit and simple expressions for the magnetocapacitance, valid for the relevant weak magnetic fields and modulation strengths. As the modulation strength in both directions increase, beats of the magnetocapacitance oscillations are observed, in the low magnetic field range (Weiss-oscillations regime), which are absent in the one-dimensional weak modulation case.Comment: 11 pages, 7 figures, accepted by Mod. Phys. Lett. B (March 2007