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


  1. Apostol, T. M. (1976). Introduction to Analytic Number Theory, Springer–Verlag, New York.
  2. Brillinger, D. R. (1969). The calculation of cumulants via conditioning, Ann. Inst. Statist. Math., 21, 215–218.
  3. Haukkanen, P. (2018). Derivation of arithmetical functions under the Dirichlet convolution, Int. J. Number Theory, 14 (05), 1257–1264.
  4. Kendall, M. G. & Stuart, A. (1977). The Advanced Theory of Statistics, Volume 1, 4th edition, Griffin.
  5. Kuich, W. (1997). Semirings and formal power series: Their relevance to formal languages and automata theory. In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages, Volume 1, Chapter 9, pages 609–677. Springer–Verlag, Berlin.
  6. Laohakosol, V. & Tangsupphathawat, P. (2018). An identical equation for arithmetic functions of several variables and applications. Notes on Number Theory and Discrete Mathematics, 24 (4), 11–17.
  7. Leonov, V. P. & Shiryaev, A. N. (1959). On a methods of calculation of semi-invariants, Theor. Prob. Appl., 4, 319–329.
  8. Niven, I. (1969). Formal power series, Amer. Math. Monthly, 76, 871–889.
  9. Speed, T. P. (1983). Cumulants and partition lattices, Austral. J. Statist., 25, 378–388.
  10.  Stanley, R. P. (1986). Enumerative Combinatorics, Vol. 1, Wadsworth and Brooks/Cole.
  11. Tóth, L. (2014). Multiplicative arithmetic functions of several variables: A survey, in vol. Mathematics Without Boundaries, Surveys in Pure Mathematics, T. M. Rassias, P. M. Pardalos (eds.), Springer–Verlag, pages 483–514.
  12. Vaidyanathaswamy, R. (1931). The theory of multiplicative arithmetic functions, Trans.Amer. Math. Soc., 33, 579–662.
  13. Wilf, H. S. (2005). Generating functionology, AK Peters/CRC Press.

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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.


Haukkanen Pentti.”Formal Power Series in Several Variables.” Notes on Number Theory and Discrete Mathematics 25, no. 4 (2019): 44-57, doi: 10.7546/nntdm.2019.25.4.44-57.


Haukkanen Pentti. “Formal Power Series in Several Variables.” Notes on Number Theory and Discrete Mathematics 25.4 (2019): 44-57. Print, doi: 10.7546/nntdm.2019.25.4.44-57.

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