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
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Authors and affiliations

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


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.


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

2010 Mathematics Subject Classification

  • 13F25
  • 11A25
  • 62E10


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Cite this paper

Haukkanen, P.  (2019).  Formal power series in several variables. Notes on Number Theory and Discrete Mathematics, 25(4), 44-57, doi: 10.7546/nntdm.2019.25.4.44-57.

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