Application of unspecified sequences in symbolic summation

We consider symbolic sums which contain subexpressions representing unspecified sequences. Existing symbolic summation technology is extended to sums of this kind. We show how this can be applied in the systematic search for general summation identities. Both, results about the non-existence of identities of a certain form, and examples of general families of identities which we have discovered automatically are included in the paper.

[1]  Zhizheng Zhang.  A kind of binomial identity , 1999, Discret. Math..

[2]  G. Egorychev.  Integral representation and the computation of combinatorial sums , 1984 .

[3]  Doron Zeilberger,et al.  The Method of Creative Telescoping , 1991, J. Symb. Comput..

[4]  Peter Kirschenhofer,et al.  A note on alternating sums , 1995, Electron. J. Comb..

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

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

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

[8]  Sergei A. Abramov,et al.  Gosper's algorithm, accurate summation, and the discrete Newton-Leibniz formula , 2005, ISSAC '05.

[9]  Carsten Schneider,et al.  Solving parameterized linear difference equations in terms of indefinite nested sums and products , 2005 .

[10]  Donald E. Knuth,et al.  The art of computer programming: V.1.: Fundamental algorithms , 1997 .

[11]  Peter Paule,et al.  Greatest Factorial Factorization and Symbolic Summation , 1995, J. Symb. Comput..

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

[13]  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.

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

[15]  Donald E. Knuth,et al.  The Art of Computer Programming, Volume I: Fundamental Algorithms, 2nd Edition , 1997 .

[16]  Carsten Schneider,et al.  The Summation Package Sigma: Underlying Principles and a Rhombus Tiling Application , 2004, Discret. Math. Theor. Comput. Sci..

[17]  David Thomas,et al.  The Art in Computer Programming , 2001 .

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

[19]  Manuel Kauers,et al.  Computer proofs for polynomial identities in arbitrary many variables , 2004, ISSAC '04.

[20]  Keith O. Geddes,et al.  Telescoping in the context of symbolic summation in Maple , 2004, J. Symb. Comput..

[21]  Manuel Kauers,et al.  Indefinite summation with unspecified sequences , 2004 .

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

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

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

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

[26]  Carsten Schneider,et al.  Finding telescopers with minimal depth for indefinite nested sum and product expressions , 2005, ISSAC.