GENERALIZED UMEMURA POLYNOMIALS

We introduce and study generalized Umemura polynomials $U_{n,m}^{\left( k \right)}\left( {z,w:a,b} \right)$ which are a natural generalization of the Umemura polynomials Un(z, w; a, b) related to Painlevé VI equation. We will show that if a = b or a = 0 or b = 0, then polynomials $U_{n.m}^{\left( 0...

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Bibliographic Details
Published in:The Rocky Mountain journal of mathematics 2002-06, Vol.32 (2), p.691-702
Main Authors: KIRILLOV, ANATOL N., TANEDA, MAKOTO
Format: Article
Language:English
Subjects:
Algebra
Integers
Mathematical functions
Mathematical theorems
Polynomials
Recurrence relations
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Summary:We introduce and study generalized Umemura polynomials $U_{n,m}^{\left( k \right)}\left( {z,w:a,b} \right)$ which are a natural generalization of the Umemura polynomials Un(z, w; a, b) related to Painlevé VI equation. We will show that if a = b or a = 0 or b = 0, then polynomials $U_{n.m}^{\left( 0 \right)}\left( {z,w;a,b} \right)$ generate solutions to Painlevé VI. We will describe a connection between polynomials $U_{n.m}^{\left( 0 \right)}\left( {z,w;a,0} \right)$ and certain Umemura polynomials Uk(z, w; α, β).
ISSN:0035-7596
1945-3795
DOI:10.1216/rmjm/1030539693