362 research outputs found
Algorithms for the indefinite and definite summation
The celebrated Zeilberger algorithm which finds holonomic recurrence
equations for definite sums of hypergeometric terms is extended to
certain nonhypergeometric terms. An expression is called a
hypergeometric term if both and are
rational functions. Typical examples are ratios of products of exponentials,
factorials, function terms, bin omial coefficients, and Pochhammer
symbols that are integer-linear with respect to and in their arguments.
We consider the more general case of ratios of products of exponentials,
factorials, function terms, binomial coefficients, and Pochhammer
symbols that are rational-linear with respect to and in their
arguments, and present an extended version of Zeilberger's algorithm for this
case, using an extended version of Gosper's algorithm for indefinite summation.
In a similar way the Wilf-Zeilberger method of rational function
certification of integer-linear hypergeometric identities is extended to
rational-linear hypergeometric identities
The q-WZ Method for Infinite Series
Motivated by the telescoping proofs of two identities of Andrews and Warnaar,
we find that infinite q-shifted factorials can be incorporated into the
implementation of the q-Zeilberger algorithm in the approach of Chen, Hou and
Mu to prove nonterminating basic hypergeometric series identities. This
observation enables us to extend the q-WZ method to identities on infinite
series. As examples, we will give the q-WZ pairs for some classical identities
such as the q-Gauss sum, the sum, Ramanujan's sum and
Bailey's sum.Comment: 17 page
The Abel-Zeilberger Algorithm
We use both Abel's lemma on summation by parts and Zeilberger's algorithm to
find recurrence relations for definite summations. The role of Abel's lemma can
be extended to the case of linear difference operators with polynomial
coefficients. This approach can be used to verify and discover identities
involving harmonic numbers and derangement numbers. As examples, we use the
Abel-Zeilberger algorithm to prove the Paule-Schneider identities, the
Apery-Schmidt-Strehl identity, Calkin's identity and some identities involving
Fibonacci numbers.Comment: 18 page
Efficient Algorithms for Mixed Creative Telescoping
Creative telescoping is a powerful computer algebra paradigm -initiated by
Doron Zeilberger in the 90's- for dealing with definite integrals and sums with
parameters. We address the mixed continuous-discrete case, and focus on the
integration of bivariate hypergeometric-hyperexponential terms. We design a new
creative telescoping algorithm operating on this class of inputs, based on a
Hermite-like reduction procedure. The new algorithm has two nice features: it
is efficient and it delivers, for a suitable representation of the input, a
minimal-order telescoper. Its analysis reveals tight bounds on the sizes of the
telescoper it produces.Comment: To be published in the proceedings of ISSAC'1
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