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