%0 Journal Article
%T Generalized Powers of Substitution with Pre-Function Operators
%A Laurent Poinsot
%J Applied Mathematics
%P 12-17
%@ 2152-7393
%D 2013
%I Scientific Research Publishing
%R 10.4236/am.2013.47A004
%X <p>
	An operator on formal power series of the form <em>S</em> <img alt=\"\" src=\"Edit_2e9ead22-c8af-4903-8709-f3231d46573c.bmp\" /> <em>¦ÌS</em> , where <em>¦Ì</em> is an invertible power series, and <span id=\"__kindeditor_bookmark_start_16__\" style=\"display:none;\"></span> <span style=\"font-family:;\" times=\"\" new=\"\" roman?,?serif?;font-size:10.5pt;?=\"\">¦Ò</span> is a series of the form&nbsp;t+<img alt=\"\" src=\"Edit_fce1af4a-6c0e-4c4e-9ef6-17cccbe9abe8.bmp\" />£¨t<sup>2</sup>£©&nbsp;is called a unipotent substitution with pre-function. Such operators, denoted by a pair (<em>¦Ì </em>£¬<span style=\"font-family:;\" times=\"\" new=\"\" roman?,?serif?;font-size:10.5pt;?=\"\">¦Ò</span> £©&nbsp; , 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 <img alt=\"\" src=\"Edit_f46f03a5-0468-44ab-a37e-2c056cb90d92.bmp\" /><span style=\"font-family:;\" times=\"\" new=\"\" roman?,?serif?;font-size:10.5pt;?=\"\">¦Ò</span> for every<img alt=\"\" src=\"Edit_81b10fa6-109b-429a-ac75-d1923fa40a98.bmp\" /> .
</p>
%K Formal Power Series
%K Formal Substitution
%K Riordan Group
%K Generalized Powers
%K Sheffer Sequences
%K Umbral Calculus
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=33983