Séminaire du 13 octobre 2008. On Entire Solutions of Linear Difference Equations with Polynomial Coefficients. Sergei A. Abramov,
Computing Centre of the Russian Academy of Sciences, Moscow, Russia.
It is known that any linear difference equation a_d(z) y(z + d) +
... + a_1 (z) y(z + 1) + a_0(z) y(z) = 0 of order d with polynomial
coefficients (over the field C of complex numbers) has a fundamental
system of entire solutions (Praagman, 1986). We strengthen this
result: the C-linear space of sequences which are restrictions to Z
of entire solutions of the given difference equation has dimension
d. We also show that a basis for this space can be found
algorithmically. The matter is that if a segment I contains all the
integer roots of the polynomials a_0(z) and a_d(z - d), then any doubly infinite sequence which satisfies the given equation is uniquely defined by the values of the elements whose indices belong to I. We show how to construct a basis of the restrictions to I of all entire solutions.
The results were obtained by the author jointly with M. Barkatou, M. van Hoeij, M. Petkovsek.
Last modified: Thu Nov 13 18:12:34 CET 2008