Formal power series in several variables

Pentti Haukkanen
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 25, 2019, Number 4, Pages 44-57
DOI: 10.7546/nntdm.2019.25.4.44-57
Full paper (PDF, 219 Kb)

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Authors and affiliations

Pentti Haukkanen
Faculty of Information Technology and Communication Sciences
FI-33014 Tampere University, Finland

Abstract

We present the theory of formal power series in several variables in an elementary way. This is a generalization of Niven’s theory of formal power series in one variable. We refer to a formal power series in  n variables as an n-way array of complex or real numbers and investigate its algebraic properties without analytic tools. We also consider the formal derivative, logarithm and exponential of a formal power series in n variables. Applications to multiplicative arithmetical functions in several variables and cumulants in statistics are presented.

Keywords

• Formal power series
• Derivative
• Logarithm
• Exponential function
• Arithmetical functions in several variables
• Cumulants

2010 Mathematics Subject Classification

• 13F25
• 11A25
• 62E10

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