By Wolfram Koepf

During this booklet glossy algorithmic strategies for summation, so much of which were brought in the final decade, are built and punctiliously carried out within the computing device algebra process Maple.

The algorithms of Gosper, Zeilberger and Petkovsek on hypergeometric summation and recurrence equations and their q-analogues are coated, and related algorithms on differential equations are thought of. An similar idea of hyperexponential integration as a result of Almkvist and Zeilberger completes the book.

the combo of all effects thought of provides paintings with orthogonal polynomials and (hypergeometric variety) targeted services an excellent algorithmic beginning. as a result, many examples from this very energetic box are given.

The ebook is designed to be used within the framework of a seminar and it's also appropriate for a sophisticated lecture path during this region.

**Read or Download Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities PDF**

**Best combinatorics books**

This revised and enlarged 5th version beneficial properties 4 new chapters, which comprise hugely unique and pleasant proofs for classics similar to the spectral theorem from linear algebra, a few more moderen jewels just like the non-existence of the Borromean earrings and different surprises. From the Reviews". .. within PFTB (Proofs from The ebook) is certainly a glimpse of mathematical heaven, the place smart insights and gorgeous principles mix in dazzling and wonderful methods.

Combinatorics and Algebraic Geometry have loved a fruitful interaction because the 19th century. Classical interactions comprise invariant idea, theta capabilities and enumerative geometry. the purpose of this quantity is to introduce contemporary advancements in combinatorial algebraic geometry and to procedure algebraic geometry with a view in the direction of functions, comparable to tensor calculus and algebraic information.

**Finite Geometry and Combinatorial Applications**

The projective and polar geometries that come up from a vector house over a finite box are relatively necessary within the building of combinatorial gadgets, resembling latin squares, designs, codes and graphs. This publication offers an creation to those geometries and their many purposes to different components of combinatorics.

- Discrete mathematics: combinatorics and graph theory with Mathematica
- Combinatorial & Computational Mathematics: Present and Future
- Polyominoes: Puzzles, Patterns, Problems, and Packings
- Combinatorics of Set Partitions (Discrete Mathematics and Its Applications)

**Extra info for Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities**

**Sample text**

B) Apply the procedure kfreerecursion to F(n, k), with I = J. 1 If this is successful, take the resulting k-free recurrence equation and replace F(n + j, k + i) symbolically by Sn+i· This generates the holonomic recurrence equation for Sn. (c) Output: The holonomic recurrence equation for Sn of the last step. 1 rationalizes the generic expression in step Proof: (c) for the given type of input. Therefore, from now on rational arithmetic applies and it is clear that the resulting expression is identical to zero if and only if its numerator is the zero polynomial with respect to k.

If I is large, this is much greater than J, and seems tobe much more than necessary to find a holonomic equation of order J having only J free coefficients. This complexity behavior in the order of the resulting recurrence equation is the most severe drawback of Fasenmyer's method. For our choice I = J we have even (J + 1) 2 variables to consider. A better algorithm of lower complexity will be considered in Chapter 7. ) are important issues, since it may happen that no k-free recurrence equation exists.

B) Apply the procedure kfreerecursion to F(n, k), with I = J. 1 If this is successful, take the resulting k-free recurrence equation and replace F(n + j, k + i) symbolically by Sn+i· This generates the holonomic recurrence equation for Sn. (c) Output: The holonomic recurrence equation for Sn of the last step. 1 rationalizes the generic expression in step Proof: (c) for the given type of input. Therefore, from now on rational arithmetic applies and it is clear that the resulting expression is identical to zero if and only if its numerator is the zero polynomial with respect to k.