No abstract

In this we paper we prove several new identities of the Rogers-RamanujanSlater type. These identities were found as the result of computer searches. The proofs involve a variety of techniques, including series-series identities, Bailey pairs, a theorem of Watson on basic hypergeometric series, generating functions and miscellaneous methods.

In this survey article, we present an expanded version of Lucy Slater's famous list of identities of the Rogers-Ramanujan type, including identities of similar type, which were discovered after the publication of Slater's papers, and older identities (such as those in Ramanujan's lost notebook) which were not included in Slater's papers. We attempt to supply the earliest known reference for each identity. Also included are identities of false theta functions, along with their relationship to Rogers-Ramanujan type identities. We also describe several ways in which pairs/larger sets of identities may be related, as well as dependence relationships between identities.

Since its discovery by Dorn in 1900, studies of radon and its progeny have contributed to such diverse scientific fields as meteorology, geophysics, mineral exploration and radiation health effects. In addition to terrestrial scientific studies of radon, NASA missions in recent decades have yielded data on the behaviour of radon and its progeny on the Moon and on Mars. Radon has been used therapeutically for ∼100 y in the form of radon seeds for the irradiation of malignant tumours. It is, however, for its negative health effects that radon is better and more justifiably known. The causal role of radon and, in particular, its progeny in the elevated incidence of lung cancer in underground uranium miners was established in the 1950s. It is of historical interest to note that the fatal lung disease of silver miners in Saxony and Bohemia in the 16th century, was undoubtedly lung cancer caused by the high levels of radon in the mines. In recent decades there has been an ever-growing interest in the public health effects of exposure to radon in homes. Extensive radon epidemiological studies both of underground miners and of the general public in recent decades have quantified the lung cancer risks from radon exposure. Radon was classified in 1988 by International Agency for Research on Cancer as a human carcinogen and in 2009 the World Health Organization identified radon as the second cause of lung cancer globally after smoking. Radon control strategies are used by many governments to control and reduce the risk to public health from radon.

Motivated by a recent paper of Liu and Ma, we describe a number of general WP-Bailey chains. We show that many of the existing WP-Bailey chains (or branches of the WP-Bailey tree), including chains found by Andrews, Warnaar and Liu and Ma, arise as special cases of these general WP-Bailey chains. We exhibit three new branches of the WP-Bailey tree, branches which also follow as special cases of these general WP-Bailey chains. Finally, we describe a number of new transformation formulae for basic hypergeometric series which arise as consequences of these new WP-Bailey chains.