CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
  location:  PublicationsjournalsCMB
Publications        
Abstract view

Transcendental Solutions of a Class of Minimal Functional Equations

  Published:2011-08-03
 Printed: Jun 2013
  • Michael Coons,
    University of Waterloo, Dept. of Pure Mathematics, Waterloo, ON, N2L 3G1
Format:   LaTeX   MathJax   PDF  

Abstract

We prove a result concerning power series $f(z)\in\mathbb{C}[\mkern-3mu[z]\mkern-3mu]$ satisfying a functional equation of the form $$ f(z^d)=\sum_{k=1}^n \frac{A_k(z)}{B_k(z)}f(z)^k, $$ where $A_k(z),B_k(z)\in \mathbb{C}[z]$. In particular, we show that if $f(z)$ satisfies a minimal functional equation of the above form with $n\geqslant 2$, then $f(z)$ is necessarily transcendental. Towards a more complete classification, the case $n=1$ is also considered.
Keywords: transcendence, generating functions, Mahler-type functional equation transcendence, generating functions, Mahler-type functional equation
MSC Classifications: 11B37, 11B83, 11J91 show english descriptions Recurrences {For applications to special functions, see 33-XX}
Special sequences and polynomials
Transcendence theory of other special functions
11B37 - Recurrences {For applications to special functions, see 33-XX}
11B83 - Special sequences and polynomials
11J91 - Transcendence theory of other special functions
 

© Canadian Mathematical Society, 2014 : https://cms.math.ca/