It is shown that, if f is a meromorphic function of order zero andfor all r on a set of logarithmic density 1. The remainder of the paper consist of applications of identity ( ‡) to the study of value distribution of zero-order meromorphic functions, and, in particular, zero-order meromorphic solutions of q-difference equations. The results obtained include q-shift analogues of the Second Main Theorem of Nevanlinna theory, Picard's theorem, and Clunie and Mohon'ko lemmas.

A single-copy extensin gene (atExt1) has been isolated from Arabidopsis thaliana (L.) Heynh. The deduced amino acid sequence consists of 374 amino acids which are organised into highly ordered repeating blocks in which Ser(Pro)4 and Ser(Pro)3 motifs alternate. Two copies of the Tyr-X-Tyr-Lys motif and 13 copies of the Val-Tyr-Lys motif are present, showing that this extensin may be highly cross-linked, possessing the capacity for both intra and inter-molecular bond formation. The gene atExt1 is normally expressed in the root and is silent in the leaf; wounding reverses this pattern, turning on the gene in the leaf and repressing it in the root. The promoter contains motifs which have been found to activate plant defence genes in response to salicylic acid, abscisic acid and methyl jasmonate; when these compounds are applied to the roots, the atExt1 gene is activated in the leaf.

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A class of Schlömilch-type series, which are functions of two spatial parameters x and y, are transformed into series which are more suitable for numerical computation when x is small. The resulting formulae involve a free parameter a, which does not affect the value of the original series, but does affect the convergence characteristics of the new representation. By choosing a appropriately we can improve the convergence of the series significantly and calculations are presented which demonstrate the efficiency of the new representations when evaluating the series numerically. Mellin transforms are used to derive the asymptotic form of the series in the limit as x → 0 (y constant) and this also leads to efficient methods for computing certain 'image series'.

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