© 1992 by London Mathematical Society
Sub-Exponential Growth of Solutions of Difference Equations
Department of Mathematics, Witwatersrand University PO Wits 2050, Johannesburg, South Africa e-mail 036dor.{at}witsvma.wits.ac.za
Department of Mathematics The Ohio State University PO Box 3341 Columbus Ohio 43210–0341 USA e-mail nevai{at}mps.ohio-state.edu
We consider the growth of solutions 
of a difference equation 
where 
and 
are sequences of elements of a normed space X, whereas 
and 
x are sequences of linear operators acting on X. Assuming that lim 
, lim 
, and Sup1 
m < n < 
||An An–1...Am|| < , we establish sub-exponential growth of , that is, for each 0 < p < ,
As a consequence, we apply this to higher-order scalar difference equations, to orthogonal polynomials associated with weights on finite and infinite intervals, and to Gauss-Jacobi quadrature processes associated with them. For instance, given a sequence of orthonormal polynomials 
satisfying the recurrence
where p–1 = 0, p0 = const > 0, and if lim$${lim}_{n\to \infty }{a}_{n+1}/{a}_{n}$$ and$${lim}_{n\to \infty }{b}_{n}/{a}_{n}$$ then for each fixed 0 < s < 1 and 0 < p ,
.