XI.—On <i>q</i>-Functions and a certain Difference Operator

In this paper my object is, primarily, to investigate the properties of a certain operative symbol which appears to be of great utility in discussing q -functions. The first part of the paper will consist of an investigation into the various forms of and the nature of the inverse operations symbolised by Δ − n . With certain restrictions as to continuity, etc., φ( x ) will denote an arbitrary function of x .

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