M. van Hoeij. Rational Solutions of Linear Difference Equations. to appear in ISSAC'98 Proceedings, 1998. Available from http://www.math.fsu.edu/~hoeij/papers.html  Home/Search   Document Details and Download   Summary   Related Articles   Check

....The notion of solvable in terms of liouvillian sequences generalizes the notion of solvable in hypergeometric closed form of ( 15] p. 141) The algorithm presented here generalizes algorithms that find hypergeometric solutions (e.g. Hyper in [14] 15] or the algorithm presented in [7] [8]) The paper is organized as follows. In Section 2, we review the basics of the Galois theory of difference equations. In Section 3 we discuss rings of sequences, define liouvillian sequences and give the Galois theoretic characterization of solvability in terms of liouvillian sequences. In ....

.... find a set H = fh 1 ; h t g ae C(x) such that any hypergeometric solution of L(y) 0 is a solution of L 1 (y) 0 where L 1 = LCLMfOE Gamma hg h2H Proof: An algorithm for this was presented in [14] see also [15] Recent improvements (and other references) are contained in [1] 7] [8]. Note that these algorithms either produce (or can be modified to produce) an operator L 1 as above that divides L and a basis for the solution space of L 1 . Lemma 5.3 Let L be a linear difference operator of order n with coefficients in k. For each m = 1; n one can find a set Hm = fh 1 ....

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M. van Hoeij. Rational solutions of linear difference equations. Technical report, Department of Mathematics, Florida State University, 1998.

....in order to reduce intermediate expression swell. We can combine this with modular arithmetic to eliminate expression swell. The ffl valuations of these u i and v i at the points between q l and q r can be used to bound the denominators of rational solutions (this statement is the content of [12]) The valuation growth g p has been defined for (solutions of) operators L of order 1 (c.f. definition 9) and corresponds to the valuation growth g p;ffl of solutions of L ffl . For higher order L, if u 2 V (L) is hypergeometric, g p (u) has been defined as the valuation growth of the ....

....first need to bound the denominators of rational solutions; for each point q we need a lower bound for the valuation of these rational solutions at q. Such a bound can be obtained from the ffl valuation of the u i (q) and v i (q) This idea is treated in more detail for the case of systems in [12]. The following remarks are topics for a subsequent paper. The set g p can also be defined and computed for systems of equations (Y ) AY where A 2 GLn (k) without having to use cyclic vectors, so our algorithm works for systems as well. In fact the definition for systems when p is finite is ....

M. van Hoeij. Rational Solutions of Linear Difference Equations. to appear in ISSAC'98 Proceedings, 1998. Available from http://www.math.fsu.edu/~hoeij/papers.html

....for example intermediate results in the computations of the g p (L) can also be used to find rational solutions in step 3b in an efficient way. Furthermore g p can also be defined and computed for systems of equations (Y ) AY where A 2 GLn (k) without having to use cyclic vectors. See also [8] where these ideas are used to compute rational solutions of systems. The methods in this paper can be applied for q difference equations as well. ....

M. van Hoeij. Rational Solutions of Linear Difference Equations. submitted to ISSAC'98. Available from http://klein.math.fsu.edu/~hoeij/papers.html

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