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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## Details

### 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

### References

- Apostol, T. M. (1976). Introduction to Analytic Number Theory, Springer–Verlag, New York.
- Brillinger, D. R. (1969). The calculation of cumulants via conditioning, Ann. Inst. Statist. Math., 21, 215–218.
- Haukkanen, P. (2018). Derivation of arithmetical functions under the Dirichlet convolution, Int. J. Number Theory, 14 (05), 1257–1264.
- Kendall, M. G. & Stuart, A. (1977). The Advanced Theory of Statistics, Volume 1, 4th edition, Griffin.
- 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.
- 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.
- Leonov, V. P. & Shiryaev, A. N. (1959). On a methods of calculation of semi-invariants, Theor. Prob. Appl., 4, 319–329.
- Niven, I. (1969). Formal power series, Amer. Math. Monthly, 76, 871–889.
- Speed, T. P. (1983). Cumulants and partition lattices, Austral. J. Statist., 25, 378–388.
- Stanley, R. P. (1986). Enumerative Combinatorics, Vol. 1, Wadsworth and Brooks/Cole.
- 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.
- Vaidyanathaswamy, R. (1931). The theory of multiplicative arithmetic functions, Trans.Amer. Math. Soc., 33, 579–662.
- Wilf, H. S. (2005). Generating functionology, AK Peters/CRC Press.

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## Cite this paper

APAHaukkanen, 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.

ChicagoHaukkanen 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.

MLAHaukkanen 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.