$F_{p}$ classes and hypergeometric series
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- by Daniel S. Moak PDF
- Proc. Amer. Math. Soc. 87 (1983), 634-636 Request permission
Abstract:
For $\operatorname {Re} (p) \geqslant 0$, let ${F_p} = {\text { \{ }}f:f(z) = f{(1 - xz)^{ - p}}d\mu (x),\;\left | z \right | < 1,\;\mu$ a probability measure on $\left | x \right | = 1\}$, 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.References
- Kent Pearce, A product theorem for ${\cal F}_{p}$ classes and an application, Proc. Amer. Math. Soc. 84 (1982), no.Β 4, 509β515. MR 643739, DOI 10.1090/S0002-9939-1982-0643739-4
- Earl D. Rainville, Special functions, The Macmillan Company, New York, 1960. MR 0107725
- Lucy Joan Slater, Generalized hypergeometric functions, Cambridge University Press, Cambridge, 1966. MR 0201688
Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 87 (1983), 634-636
- MSC: Primary 33A30
- DOI: https://doi.org/10.1090/S0002-9939-1983-0687631-9
- MathSciNet review: 687631