14,939 research outputs found
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
Embodied cognition and temporally extended agency
According to radical versions of embodied cognition, human cognition and agency should be explained without the ascription of representational mental states. According to a standard reply, accounts of embodied cognition can explain only instances of cognition and agency that are not “representation-hungry”. Two main types of such representation-hungry phenomena have been discussed: cognition about “the absent” and about “the abstract”. Proponents of representationalism have maintained that a satisfactory account of such phenomena requires the ascription of mental representations. Opponents have denied this. I will argue that there is another important representation-hungry phenomenon that has been overlooked in this debate: temporally extended planning agency. In particular, I will argue that it is very difficult to see how planning agency can be explained without the ascription of mental representations, even if we grant, for the sake of argument, that cognition about the absent and abstract can. We will see that this is a serious challenge for the radical as well as the more modest anti-representationalist versions of embodied cognition, and we will see that modest anti-representationalism is an unstable position
Dual-system theory and the role of consciousness in intentional action
According to the standard view in philosophy, intentionality is the mark of genuine action. In psychology, human cognition and agency are now widely explained in terms of the workings of two distinct systems (or types of processes), and intentionality is not a central notion in this dual-system theory. Further, it is often claimed, in psychology, that most human actions are automatic, rather than consciously controlled. This raises pressing questions. Does the dual-system theory preserve the philosophical account of intentional action? How much of our behavior is intentional according to this view? And what is the role of consciousness? I will propose here a revised account of intentional action within the dual-system framework, and we will see that most of our behavior can qualify as intentional, even if most of it is automatic. An important lesson will be that philosophical accounts of intentional action need to pay more attention to the role of consciousness in action
Bilateral identities of the Rogers-Ramanujan type
We derive by analytic means a number of bilateral identities of the
Rogers-Ramanujan type. Our results include bilateral extensions of the
Rogers-Ramanujan and the G\"ollnitz-Gordon identities, and of related
identities by Ramanujan, Jackson, and Slater. We give corresponding results for
multiseries including multilateral extensions of the Andrews-Gordon identities,
of Bressoud's even modulus identities, and other identities. The here revealed
closed form bilateral and multilateral summations appear to be the very first
of their kind. Given that the classical Rogers-Ramanujan identities have
well-established connections to various areas in mathematics and in physics, it
is natural to expect that the new bilateral and multilateral identities can be
similarly connected to those areas. This is supported by concrete combinatorial
interpretations for a collection of four bilateral companions to the classical
Rogers-Ramanujan identities.Comment: 25 page
On Warnaar's elliptic matrix inversion and Karlsson-Minton-type elliptic hypergeometric series
Using Krattenthaler's operator method, we give a new proof of Warnaar's
recent elliptic extension of Krattenthaler's matrix inversion. Further, using a
theta function identity closely related to Warnaar's inversion, we derive
summation and transformation formulas for elliptic hypergeometric series of
Karlsson-Minton-type. A special case yields a particular summation that was
used by Warnaar to derive quadratic, cubic and quartic transformations for
elliptic hypergeometric series. Starting from another theta function identity,
we derive yet different summation and transformation formulas for elliptic
hypergeometric series of Karlsson-Minton-type. These latter identities seem
quite unusual and appear to be new already in the trigonometric (i.e., p=0)
case.Comment: 16 page
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