TAILIEUCHUNG - Báo cáo toán học: "The degree of a q-holonomic sequence is a quadratic quasi-polynomial"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: The degree of a q-holonomic sequence is a quadratic quasi-polynomial. | The degree of a q-holonomic sequence is a quadratic quasi-polynomial Stavros Garoufalidis School of Mathematics Georgia Institute of Technology Atlanta GA 30332-0160 USA stavros@ http stavros Submitted Jun 30 2010 Accepted Mar 5 2011 Published Mar 15 2011 Mathematics Subject Classification 05C88 Abstract A sequence of rational functions in a variable q is q-holonomic if it satisfies a linear recursion with coefficients polynomials in q and qn. We prove that the degree of a q-holonomic sequence is eventually a quadratic quasi-polynomial and that the leading term satisfies a linear recursion relation with constant coefficients. Our proof uses differential Galois theory adapting proofs regarding holonomic D-modules to the case of q-holonomic D-modules combined with the Lech-Mahler-Skolem theorem from number theory. En route we use the Newton polygon of a linear q-difference equation and introduce the notion of regular-singular q-difference equation and a WKB basis of solutions of a linear q-difference equation at q 0. We then use the Skolem-Mahler-Lech theorem to study the vanishing of their leading term. Unlike the case of q 1 there are no analytic problems regarding convergence of the WKB solutions. Our proofs are constructive and they are illustrated by an explicit example. Contents 1 Introduction 2 History. 2 The degree and the leading term of a q-holonomic sequence. 3 The Newton polygon of a linear q-difference equation. 4 WKB sums . 5 An example. 7 Plan of the paper. 10 To Doron Zeilberger on the occasion of his 60th birthday THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2 2011 P4 1 2 Proof of Theorem 11 Reduction to the case of a single slope . 11 Reduction to the case of a single eigenvalue. 12 First order linear q-difference equation. 13 Proof of Theorem . 14 The regular-singular non-resonant case. 14 3 Proof of Theorem 16 Generalized power sums . 16 Proof of .

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