Generalized Powers of Substitution with Pre-Function Operators


An operator on formal power series of the form S μS , where μ is an invertible power series, and σ is a series of the form t+(t2) is called a unipotent substitution with pre-function. Such operators, denoted by a pair (μ σ )  , form a group. The objective of this contribution is to show that it is possible to define a generalized powers for such operators, as for instance fractional powers σ for every .

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L. Poinsot, "Generalized Powers of Substitution with Pre-Function Operators," Applied Mathematics, Vol. 4 No. 7A, 2013, pp. 12-17. doi: 10.4236/am.2013.47A004.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] A. Benhissi, “Rings of Formal Power Series,” Queen’s Papers in Pure and Applied Mathematics, Queen’s University, Kingsone, 2003.
[2] R. P. Stanley, “Enumerative Combinatorics—Volume 1, Volume 49 of Cambridge Studies in Advanced Mathematics,” Cambridge University Press, Cambridge, 2000.
[3] G. H. E. Duchamp, K. A. Penson, A. I. Solomon, A. Horzeal and P. Blasiak, “One-Parameter Groups and Combinatorial Physics,” Proceedings of the Symposium COPROMAPH3: Contemporary Problems in Mathematical Physics, Cotonou, 2004, pp. 436-449.
[4] L. Poinsot and G. H. E. Duchamp, “A Formal Calculus on the Riordan near Algebra,” Advances and Applications in Discrete Mathematics, Vol. 6, No. 1, 2010, pp. 11-44.
[5] L. W. Shapiro, S. Getu, W.-J. Woan and L. C. Woodson, “The Riordan Group,” Discrete Applied Mathematics, Vol. 34, No. 1-3, 1991, pp. 229-239. doi:10.1016/0166-218X(91)90088-E
[6] G. Markowsky, “Differential Operators and the Theory of Binomial Enumeration,” Journal of Mathematical Analysis and Application, Vol. 63, No. 1, 1978, pp. 145-155. doi:10.1016/0022-247X(78)90111-7
[7] G. H. E. Duchamp, L. Poinsot, A. I. Solomon, K. A. Penson, P. Blasiak and A. Horzela, “Ladder Operators and Endomorphisms in Combinatorial Physics,” Discrete Mathematics and Theoretical Computer Science, Vol. 12, No. 2, 2010, pp. 23-46.
[8] G.-C. Rota, D. Kahaner and A. Odlyzko, “Finite Operator Calculus,” Journal of Mathematical Analysis and Its Applications, Vol. 42, No. 3, 1973, pp. 684-760.
[9] I. M. Sheffer, “Some Properties of Polynomial Sets of Type Zero,” Duke Mathematical Journal, Vol. 5, No. 3, 1939, pp. 590-622. doi:10.1215/S0012-7094-39-00549-1
[10] J. F. Steffensen, “The Poweroid, an Extension of the Mathematical Notion of Power,” Acta Mathematica, Vol. 73, No. 1, 1941, pp. 333-366. doi:10.1007/BF02392231
[11] S. Roman, “The Umbral Calculus,” Dover Publications, New York, 1984.
[12] T.-X. He, L.C. Hsu and P.J.-S. Shiue, “The Sheffer Group and the Riordan Group,” Discrete Applied Mathematics, Vol. 155, No. 15, 2007, pp. 1895-1909. doi:10.1016/j.dam.2007.04.006

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