The Summation Package Sigma: Underlying Principles and a Rhombus Tiling Application

We give an overview of how a huge class of multisum identities can be proven and discovered with the summation package Sigma implemented in the computer algebra system Mathematica. General principles of symbolic summation are discussed. We illustrate the usage of Sigma by showing how one can find and prove a multisum identity that arose in the enumeration of rhombus tilings of a symmetric hexagon. Whereas this identity has been derived alternatively with the help of highly involved transformations of special functions, our tools enable to find and prove this identity completely automatically with the computer.

[1]  Peter A. Hendriks,et al.  Solving Difference Equations in Finite Terms , 1999, J. Symb. Comput..

[2]  Helmut Prodinger,et al.  Padé approximations to the logarithm II: Identities, recurrences, and symbolic computation , 2006 .

[3]  Peter Paule,et al.  A Mathematica q-Analogue of Zeilberger's Algorithm Based on an Algebraically Motivated Approach to q-Hypergeometric Telescoping , 1991 .

[4]  J. Davenport Editor , 1960 .

[5]  Doron Zeilberger,et al.  A fast algorithm for proving terminating hypergeometric identities , 1990, Discret. Math..

[6]  Christian Krattenthaler,et al.  The Number of Rhombus Tilings of a Symmetric Hexagon which Contain a Fixed Rhombus on the Symmetry Axis, II , 2000, Eur. J. Comb..

[7]  Marko Petkovsek,et al.  Hypergeometric Solutions of Linear Recurrences with Polynomial Coefficents , 1992, J. Symb. Comput..

[8]  Carsten Schneider Solving parameterized linear di erence equations in S-fields , 2002 .

[9]  Mark van Hoeij,et al.  Rational solutions of linear difference equations , 1998, ISSAC '98.

[10]  M. H. Protter,et al.  THE SOLUTION OF THE PROBLEM OF INTEGRATION IN FINITE TERMS , 1970 .

[11]  Axel Riese,et al.  qMultiSum--a package for proving q-hypergeometric multiple summation identities , 2003, J. Symb. Comput..

[12]  Wenchang Chu,et al.  Hypergeometric series and harmonic number identities , 2005, Adv. Appl. Math..

[13]  Sergei A. Abramov,et al.  D'Alembertian solutions of linear differential and difference equations , 1994, ISSAC '94.

[14]  Sergei A. Abramov,et al.  q-Hypergeometric solutions of q-difference equations , 1998, Discret. Math..

[15]  Christian Mallinger,et al.  Algorithmic Manipulations and Transformations of Univariate Holonomic Functions and Sequences , 2001 .

[16]  Frédéric Chyzak,et al.  An extension of Zeilberger's fast algorithm to general holonomic functions , 2000, Discret. Math..

[17]  T. Rivoal,et al.  Hypergéométrie et fonction zêta de Riemann , 2003 .

[18]  R. W. Gosper Decision procedure for indefinite hypergeometric summation. , 1978, Proceedings of the National Academy of Sciences of the United States of America.

[19]  IN ΠΣ-FIELDS,et al.  Product Representations in Πσ-fields , 2004 .

[20]  Michael Karr,et al.  Theory of Summation in Finite Terms , 1985, J. Symb. Comput..

[21]  Carsten Schneider,et al.  A Collection of Denominator Bounds To Solve Parameterized Linear Difference Equations in ΠΣ-Fields∗ , 2004 .

[22]  Michael Karr,et al.  Summation in Finite Terms , 1981, JACM.

[23]  Helmut Prodinger,et al.  Padé approximations to the logarithm III: Alternative methods and additional results , 2006 .

[24]  R. Risch The problem of integration in finite terms , 1969 .

[25]  Carsten Schneider,et al.  A Collection of Denominator Bounds to Solve Parameterized Linear Difference Equations in ΠΣ-Extensions , 2004 .

[26]  Carsten Schneider Symbolic summation with single-nested sum extensions , 2004, ISSAC '04.

[27]  Carsten Schneider,et al.  Computer proofs of a new family of harmonic number identities , 2003, Adv. Appl. Math..

[28]  Carsten Schneider,et al.  Degree Bounds to Find Polynomial Solutions of Parameterized Linear Difference Equations in ΠΣ-Fields , 2005, Applicable Algebra in Engineering, Communication and Computing.

[29]  Manuel Bronstein On Solutions of Linear Ordinary Difference Equations in their Coefficient Field , 2000, J. Symb. Comput..

[30]  Carsten Schneider,et al.  An Implementation of Karr's Summation Algorithm in Mathematica , 1999 .

[31]  Christian Krattenthaler,et al.  The number of rhombus tilings of a symmetric hexagon which contain a fixed rhombus on the symmetry axis, I , 1998 .

[32]  A. Karimi,et al.  Master‟s thesis , 2011 .

[33]  Peter Paule,et al.  A Mathematica Version of Zeilberger's Algorithm for Proving Binomial Coefficient Identities , 1995, J. Symb. Comput..

[34]  Peter Paule,et al.  SYMBOLIC SUMMATION SOME RECENT DEVELOPMENTS , 1995 .

[35]  Bruno Salvy,et al.  GFUN: a Maple package for the manipulation of generating and holonomic functions in one variable , 1994, TOMS.

[36]  Kurt Wegschaider,et al.  Computer Generated Proofs of Binomial Multi-Sum Identities , 1997 .

[37]  G. Rw Decision procedure for indefinite hypergeometric summation , 1978 .

[38]  Bruno Salvy,et al.  Non-Commutative Elimination in Ore Algebras Proves Multivariate Identities , 1998, J. Symb. Comput..