Home Some remarks on formal power series and formal Laurent series
Article
Licensed
Unlicensed Requires Authentication

Some remarks on formal power series and formal Laurent series

  • Dariusz Bugajewski EMAIL logo and Xiao-Xiong Gan
Published/Copyright: June 5, 2017
Become an author with De Gruyter Brill
Author Information Explore this Subject
Mathematica Slovaca
From the journal Mathematica Slovaca Volume 67 Issue 3

Abstract

In this article we consider the topology on the set of formal Laurent series induced by the ultrametric defined via the order. In particular, we establish that the product of formal Laurent series, considered in [GAN, X. X.—BUGAJEWSKI, D.:On formal Laurent series, Bull. Braz. Math. Soc. 42 (2011), 415–437], is not continuous. We also show some applications of fixed point theorems to some nonlinear mappings defined on the space of formal power series or on the space of formal Laurent series. Finally, in the second part of the paper, we propose another approach to study of dot product and multiplication of formal Laurent series, in particular establishing integral representation of dot product and convolution representation of multiplication.

MSC 2010: Primary 13F25; 16W60; Secondary 46S10; 47H09
Keywords: convolution; dot product; formal Laurent series; formal power series; non-Archimedean valuation; nonexpansive mapping; order; product (multiplication); spherical completeness; ultrametric

(Communicated by Stanisława Kanas)

References

[1] Bugajewski, D.—Kasprzak, P.: Fixed point theorems for weakly F-contractive and strongly F-expansive mappings, J. Math. Anal. Appl. 359 (2009), 126–134.10.1016/j.jmaa.2009.05.024Search in Google Scholar

[2] Chaumat, J.—Chollet, A. M.: On composite formal power series, Trans. Amer. Math. Soc. 353 (2001), 1691–1703.10.1090/S0002-9947-01-02733-7Search in Google Scholar

[3] Cheng, C. C.—Mckay, J.—Towber, J.—Wang, S. S.—Wrigh, D.: Reversion of formal power series and the extended Raney coefficients, Trans. Amer. Math. Soc. 349 (1997), 1769–1782.10.1090/S0002-9947-97-01781-9Search in Google Scholar

[4] Cheon, G. S.—Jin, S. T: Structural properties of Riordan matrices and extending the matrices, Linear Algebra Appl. 435 (2011), 2019–2032.10.1016/j.laa.2011.04.001Search in Google Scholar

[5] Constantine, G. M.—Savits, T. H.: A multivariate Faa Di Bruno formula with applications, Trans. Amer. Math. Soc. 348 (1996), 503–520.10.1090/S0002-9947-96-01501-2Search in Google Scholar

[6] Eakin, P. M.—Harris, G. A.: When Φ (f) convergent implies f is convergent?, Math. Ann. 229 (1977), 201–210.10.1007/BF01391465Search in Google Scholar

[7] Enoch, R.: Formal power series solutions of Schroder’s equation, Aequationes Math. 74 (2007), 26–61. 10.1007/s00010-007-2875-9Search in Google Scholar

[8] Gan, X. X.: A generalized chain rule for formal power series, Comm. Math. Anal. 2 (2007), 37–44.Search in Google Scholar

[9] Gan, X. X.—Bugajewski, D.: A note on formal power series, Comment. Math. Univ. Carolin. 51 (2010), 595–604.Search in Google Scholar

[10] Gan, X. X.—Bugajewski, D.: On formal Laurent series, Bull. Braz. Math. Soc. 42 (2011), 415–437.10.1515/9783110599459-007Search in Google Scholar

[11] Gan, X. X.—Knox, N.: On composition of formal power series, Int. J. Math. Math. Sci. 30 (2002), 761–770.10.1155/S0161171202107150Search in Google Scholar

[12] He, T. X.: Riordan arrays associated with Laurent series and generalized Sheffer-type groups, Linear Algebra Appl. 435 (2011), 1241–1256.10.1016/j.laa.2011.03.004Search in Google Scholar

[13] Hedeyatian, K.: On the reflexivity of the multiplication operator on Banach spaces of formal Laurent series, Internat. J. Math. 18 (2007), 231–234.10.1142/S0129167X07004114Search in Google Scholar

[14] Henrici, P.: Applied and Computational Complex Analysis, John Wiley and Sons, 1988.Search in Google Scholar

[15] Jones, W.—Thron, W.: Sequences of meromorphic functions corresponding to a formal Laurent series, SIAM J. Math. Anal. 10 (1979), 1–17.10.1137/0510001Search in Google Scholar

[16] Li, H.: p-adic power series which commute under composition, Trans. Amer. Math. Soc. 349 (1997), 1437–1446.10.1090/S0002-9947-97-01514-6Search in Google Scholar

[17] Luzón, A.—Morón, M. A.: Riordan matrices in the reciprocation of quadratic polynomials, Linear Algebra Appl. 430 (2009), 2254–2270.10.1016/j.laa.2008.12.001Search in Google Scholar

[18] Niven, I.: Formal power series, Amer. Math. Monthly 76 (1969), 871–889.10.1080/00029890.1969.12000359Search in Google Scholar

[19] Parvica, D.—Spurr, M.: Unique summing of formal power series solutions to advanced and delayed differential equations, Discrete Cont. Dyn. Syst., suppl. vol. (2005), 730–737.Search in Google Scholar

[20] Peart, P.—Woan, W.: A divisibility property for a subgroup of Riordan matrices, Discrete Appl. Math. 98 (2000), 255–263.10.1016/S0166-218X(99)00166-3Search in Google Scholar

[21] Petelas, C.—Vidalis, T.: A fixed point theorem in non-Archimedean vector spaces, Proc. Amer. Math. Soc. 118 (1993), 819–821.10.1090/S0002-9939-1993-1132421-2Search in Google Scholar

[22] Priess-Crampe, S.: Der Banachsche Fixpunktsatz für ultrametrische Räume, Results Math. 18 (1990), 178–186.10.1007/BF03323163Search in Google Scholar

[23] Ranney, G.: Functional composition patterns and power series reversion, Trans. Amer. Math. Soc. 94 (1960), 441–451.10.1090/S0002-9947-1960-0114765-9Search in Google Scholar

[24] Schikhof, W. H.: Ultrametric Calculus. An Introduction to p-adic Analysis, Cambridge University Press, 2007.Search in Google Scholar

[25] Shapiro, L. W.—Getu, S.—Woan, W. J.— Woodson, L.: The Riordan group, Discrete Appl. Math. 34 (1991), 229–239.10.1016/0166-218X(91)90088-ESearch in Google Scholar

[26] Sibuya, Y.: Formal power series solutions in a parameter, J. Differential Equations 190 (2003), 559–578.10.1016/S0022-0396(02)00083-9Search in Google Scholar

[27] Yousefi, B.—Soltani, R.: Hypercyclicity on the Banach space of formal Laurent series, Int. J. Appl. Math. 12 (2003), 251–256.Search in Google Scholar

Received: 2014-12-14
Accepted: 2015-5-17
Published Online: 2017-6-5
Published in Print: 2017-6-27

© 2017 Mathematical Institute Slovak Academy of Sciences

Articles in the same Issue

  2. Some results in local Hilbert algebras
  4. A note on residuated po-groupoids and lattices with antitone involutions
  6. The first law of cubology for the Rubik’s Revenge
  8. Conditional associativity in orthomodular lattices
  10. A monotonicity property for generalized Fibonacci sequences
  12. Systems of conditionally independent sets
  14. On the k-th order lfsr sequence with public key cryptosystems
  16. On parametric spaces of bicentric quadrilaterals
  18. Character sums with an explicit evaluation
  20. Some remarks on formal power series and formal Laurent series
  22. The co-rank of the fundamental group: The direct product, the first Betti number, and the topology of foliations
  24. On prime and semiprime generalized hyperaction of hypermonoid
  26. On O’Malley ϱ-upper continuous functions
  28. Uniqueness theorems for finitely additive probabilities on quantum structures
  30. Some results on differential-difference analogues of Brück conjecture
  32. Oscillation of neutral second order half-linear differential equations without commutativity in deviating arguments
  34. On support, separation and decomposition theorems for t-Wright-concave functions
  36. A characterization of holomorphic bivariate functions of bounded index
  38. Asymptotic expansion of double Laplace-type integrals with a curve of minimal points and application to an exit time problem
  40. The geometry of two-valued subsets of Lp-spaces
  42. Hemi-slant submanifolds as warped products in a nearly Kaehler manifold
  44. A comparison of two tests in multivariate models
  46. Modelling prescription behaviour of general practitioners
  48. Characterization of congruence lattices of principal p-algebras
https://doi.org/10.1515/ms-2016-0297
Keywords for this article
convolution; dot product; formal Laurent series; formal power series; non-Archimedean valuation; nonexpansive mapping; order; product (multiplication); spherical completeness; ultrametric

Articles in the same Issue

  2. Some results in local Hilbert algebras
  4. A note on residuated po-groupoids and lattices with antitone involutions
  6. The first law of cubology for the Rubik’s Revenge
  8. Conditional associativity in orthomodular lattices
  10. A monotonicity property for generalized Fibonacci sequences
  12. Systems of conditionally independent sets
  14. On the k-th order lfsr sequence with public key cryptosystems
  16. On parametric spaces of bicentric quadrilaterals
  18. Character sums with an explicit evaluation
  20. Some remarks on formal power series and formal Laurent series
  22. The co-rank of the fundamental group: The direct product, the first Betti number, and the topology of foliations
  24. On prime and semiprime generalized hyperaction of hypermonoid
  26. On O’Malley ϱ-upper continuous functions
  28. Uniqueness theorems for finitely additive probabilities on quantum structures
  30. Some results on differential-difference analogues of Brück conjecture
  32. Oscillation of neutral second order half-linear differential equations without commutativity in deviating arguments
  34. On support, separation and decomposition theorems for t-Wright-concave functions
  36. A characterization of holomorphic bivariate functions of bounded index
  38. Asymptotic expansion of double Laplace-type integrals with a curve of minimal points and application to an exit time problem
  40. The geometry of two-valued subsets of Lp-spaces
  42. Hemi-slant submanifolds as warped products in a nearly Kaehler manifold
  44. A comparison of two tests in multivariate models
  46. Modelling prescription behaviour of general practitioners
  48. Characterization of congruence lattices of principal p-algebras
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 26.10.2025 from https://www.degruyterbrill.com/document/doi/10.1515/ms-2016-0297/pdf
Scroll to top button