Ordering relations for q-boson operators, continued fraction techniques and the q-CBH enigma

Abstract

Ordering properties of boson operators have been very extensively studied, and q-analogues of many of the relevant techniques have been derived. These relations have far reaching physical applications and, at the same time, provide a rich and interesting source of combinatorial identities and of their g-analogues. An interesting exception involves the transformation from symmetric to normal ordering, which, for conventional boson operators, can most simply be effected using a special case of the Campbell-Baker-Hausdorff (CBH) formula. To circumvent the lack of a suitable q-analogue of the CBH formula, two alternative procedures are proposed, based on a recurrence relation and on a double continued fraction, respectively. These procedures enrich the repertoire of techniques available in this field. For conventional bosons they result in an expression that coincides with that derived using the CBH formula.