Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

$ F\sb{p}$ classes and hypergeometric series


Author: Daniel S. Moak
Journal: Proc. Amer. Math. Soc. 87 (1983), 634-636
MSC: Primary 33A30
MathSciNet review: 687631
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Abstract: For $ \operatorname{Re} (p) \geqslant 0$, let $ {F_p} = {\text{ \{ }}f:f(z) = f{(1 - xz)^{ - p}}d\mu (x),\;\left\vert z \right\vert < 1,\;\mu $ a probability measure on $ \left\vert x \right\vert = 1\} $, and let $ {F_p} \cdot {F_q} = \left\{ {f \cdot g:f\;\operatorname{in} {F_p},g\;{\text{in}}\;{F_q}} \right\}$. Brickman, Hallenbeck, MacGregor and Wilken proved that $ p > 0$ and $ q > 0$, then $ {F_p} \cdot {F_p} \subseteq {F_{p + q}}$. Kent Pearce recently proved a converse result: if $ {F_p} \cdot {F_q} \subseteq {F_{p + q}}$, then $ p > 0$ and $ q > 0$, or $ p = q = 1 + it$ for some real $ t$. The case $ p = q = 1 + it$, $ t \ne 0$, will be excluded. Consequently a full converse of the above theorem holds.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1983-0687631-9
Keywords: $ {F_p}$ spaces, hypergeometric functions
Article copyright: © Copyright 1983 American Mathematical Society