## Solvability of Difference Riccati Equations

J. Math. Sci. Univ. Tokyo
Vol. 17 (2010), No. 2, Page 159--178.

Nishioka, Seiji
Solvability of Difference Riccati Equations
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Abstract:
We generalize Franke's generalized Liouvillian extension and Karr's $\Pi\Sigma$-extension, and study solvability of difference Riccati equations. We define the difference field extensions of valuation ring type and prove the following. If a difference Riccati equation which does not turn out to be linear by iterations has a solution in some difference field extension of valuation ring type, then one of the iterated Riccati equations has an algebraic solution. Applying this theorem, we conclude unsolvability of the $q$-Airy equation and the $q$-Bessel equation.

Keywords: Cartan decomposition, visible action, symmetric space, herringbone stitch, spherical variety.

Mathematics Subject Classification (2010): Primary 12H10; Secondary 39A05, 39A13.