A GENERATION OF A DETERMINANTAL FAMILY OF ITERATION FUNCTIONS AND ITS CHARACTERIZATIONS

  • Ham, YoonMee (Department of Mathematics Kyonggi University) ;
  • Lee, Sang-Gu (Department of Mathematics Sungkyunkwan University) ;
  • Ridenhour, Jerry (Department of Mathematics University of Northern Iowa)
  • Received : 2008.09.19
  • Published : 2008.12.01

Abstract

Iteration functions $K_m(z)$ and $U_m(z)$, $m{\geq}2$are defined recursively using the determinant of a matrix. We show that the fixed-iterations of $K_m(z)$ and $U_m(z)$ converge to a simple zero with order of convergence m and give closed form expansions of $K_m(z)$ and $U_m(z)$: To show the convergence, we derive a recursion formula for $L_m$ and then apply the idea of Ford or Pomentale. We also find a Toeplitz matrix whose determinant is $L_m(z)/(f^{\prime})^m$, and then we adapt the well-known results of Gerlach and Kalantari et.al. to give closed form expansions.

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Acknowledgement

Supported by : KOSEF