CONTENTS -- Preface -- Hypergeometric Summation Revisited S. A. Abramov, M. Petkovsek -- 1. Introduction -- 2. Validity conditions of the discrete Newton-Leibniz formula -- 2.1. A criterion -- 2.2. Summation of proper hypergeometric sequences -- 2.3. When the interval I contains no leading integer singularity of L -- 3. The spaces VI(L) and WI(R(k), L) -- 3.1. The structure of WI(R(k), L) -- 3.2. When a rational solution of Gosper's equation is not unique -- 3.3. If Gosper's equation has a rational solution R(k) then WI(R, L) = 0 -- References -- Five Applications of Wilf-Zeilberger Theory to Enumeration and Probability M. Apagodu, D. Zeilberger -- Explicit Formulas vs. Algorithms -- The Holonomic Ansatz -- Why this Paper? -- The Maple packages AppsWZ and AppsWZmulti -- Asymptotics -- First Application: Rolling a Die -- Second Application: How many ways to have r people chip in to pay a bill of n cents -- Third Application: Hidden Markov Models -- Fourth Application: Lattice Paths Counting -- References -- Factoring Systems of Linear Functional Equations Using Eigenrings M. A. Barkatou -- 1. Introduction and notations -- 2. Preliminaries -- 3. Eigenrings and reduction of pseudo-linear equations -- Maximal Decompsition -- 4. Spaces of homomorphisms and factorization -- Appendix A. K[X -- φ, δ].modules and matrix pseudo-linear equations -- Appendix A.1. Pseudo-linear operators -- Appendix A.2. Similarity, reducibility, decomposability and complete reducibility -- Appendix A.3. The ring of endomorphisms of a pseudo-linear operator -- References -- Modular Computation for Matrices of Ore Polynomials H. Cheng, G. Labahn -- 1. Introduction -- 2. Preliminaries -- 2.1. Notation -- 2.2. Definitions -- 2.3. The FFreduce Elimination Algorithm -- 3. Linear Algebra Formulation -- 4. Reduction to Zp[t][Z] -- 4.1. Lucky Homomorphisms -- 4.2. Termination.

5. Reduction to Zp -- 5.1. Applying Evaluation Homomorphisms and Computation in Zp -- 5.2. Lucky Homomorphisms and Termination -- 6. Complexity Analysis -- 7. Implementation Considerations and Experimental Results -- 8. Concluding Remarks -- References -- Beta-Expansions of Pisot and Salem Numbers K. G. Hare -- 1. Introduction and History -- 2. Univoque Pisot Numbers -- 3. Algorithms and Implementation Issues -- 4. Conclusions and Open Questions -- References -- Logarithmic Functional and the Weil Reciprocity Law A. Khovanskii -- 1. Introduction -- 1.1. The Weil reciprocity law -- 1.2. Topological explanation of the reciprocity law over the field C -- 1.3. Multi-dimensional reciprocity laws -- 1.4. The logarithmic functional -- 1.5. Organization of material -- 2. Formulation of the Weil reciprocity law -- 3. LB-functional of the pair of complex valued functions of the segment on real variable -- 4. LB-functional of the pair of complex valued functions and one-dimensional cycle on real manifold -- 5. Topological proof of the Weil reciprocity law -- 6. Generalized LB-functional -- 7. Logarithmic function and logarithmic functional -- 7.1. Zero-dimensional logarithmic functional and logarithm -- 7.2. Properties of one-dimensional logarithmic functional -- 7.3. Prove of properties of logarithmic functional -- 8. Logarithmic functional and generalized LB-functional -- References -- On Solutions of Linear Functional Systems and Factorization of Laurent-Ore Modules M. Wu, Z. Li -- 1. Introduction -- 2. Preliminaries -- 3. Fully integrable systems -- 4. -finite systems -- 4.1. Generic solutions of linear algebraic equations -- 4.2. Laurent-Ore algebras -- 4.3. Modules of formal solutions -- 4.4. Fundamental matrices and Picard-Vessiot extensions -- 5. Computing linear dimension -- 6. Factorization of Laurent-Ore modules.

6.1. Constructions with modules over Laurent-Ore algebras -- 6.2. A module-theoretic approach to factorization -- 6.3. Eigenrings and decomposition of Laurent-Ore modules -- References -- The Vector Rational Function Reconstruction Problem Z. Olesh, A. Storjohann -- 1. Introduction -- 2. Reduced bases -- 3. Minimal approximant bases -- 3.1. An algorithm for simultaneous Padé approximation -- 4. Vector rational function reconstruction -- 5. Application to linear solving -- 6. Conclusion -- References -- Fast Algorithm for Computing Multipole Matrix Elements with Legendre Polynomials V. Yu. Papshev, S. Yu. Slavyanov -- Introduction -- 1. Hilbert transform for solutions of the equation for the product -- 2. Equation for the product of Legendre polynomials and its Hilbert transform -- 3. Calculation of particular cases of Clebsh-Gordon coefficients -- References -- Recurrence Relations for the Coe.cients in Hypergeometric Series Expansions L. Rebillard, H. Zakraj.sek -- 1. Introduction -- 2. Notations and basic properties -- 3. Associated families -- 4. Depression of the order -- 5. Normal form of a di.erential operator -- 5.1. σ has a double root ξ1 = ξ2 -- 5.2. σ is of degree one -- 6. Conclusion -- Acknowledgments -- References -- On Factorization and Solution of Multidimensional Linear Partial Di.erential Equations S. P. Tsarev -- 1. Introduction -- 2. Laplace and generalized Laplace transformations -- 3. Dini transformation: an example -- 4. Dini transformation: a general result for dim = 3, ord = 2 -- 5. Open problems -- Acknowledgments -- References -- Two Families of Algorithms for Symbolic Polynomials S. M. Watt -- 1. Introduction -- 2. Symbolic Polynomials -- 3. Multiplicative Properties -- 4. Extension Algorithms -- 5. Projection Methods -- 6. Finding Corresponding Terms -- 7. Generalizations -- 8. Conclusions -- References -- Author Index.