IJPAM: Volume 97, No. 2 (2014)

FACTORIZATIONS OF ANALYTIC FUNCTIONS VIA
GENERALIZED INVERSE POWER PRODUCT EXPANSIONS

H. Gingold$^1$, J. Quaintance$^2$
$^1$Department of Mathematics
West Virginia University
Morgantown WV 26506, USA
$^2$Department of Mathematics
Rutgers University-Hill Center for the Mathematical Sciences
Piscataway, NJ 08854-8019, USA


Abstract. Given an arbitrary sequence of complex numbers $\{a_n\}_{n=1}^{\infty}$ and an arbitrary nonzero sequence of complex numbers $\{r_n\}_{n=1}^{\infty}$, we study the expansion of the Taylor series $1 + \sum_{n=1}^{\infty}a_nx^n$ into infinite products of the form $\prod_{n=1}^{\infty}(1-h_nx^n)^{-r_n}$. Algebraic properties, convergence criteria, and combinatorial interpretations of the infinite products are investigated. We also provide an asymptotic formula for the majorizing product expansion associated with $1 - \sum_{n=1}^{\infty}s^nx^n$, $s:=\sup_{n\geq1}{\vert a_{n}\vert^{\frac{1}{n}}}$.

Received: January 20, 2014

AMS Subject Classification: 41A10, 30E10, 11P81, 05A17

Key Words and Phrases: power series, expansions, analytic functions, power products, generalized power products, generalized inverse power products, convergence, asymptotics, multi-sets, partitions, compositions

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DOI: 10.12732/ijpam.v97i2.2 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2014
Volume: 97
Issue: 2
Pages: 115 - 146


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