Converging to Gosper's algorithm

Chen, William Y. C.; Paule, Peter; Saad, Husam L.

doi:10.1016/j.aam.2007.11.004

Cited by 19 publications

(5 citation statements)

References 17 publications

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“…Assume that u = a(x, y)/b(x), where a, b are polynomials in x and y. We notice that one can give an estimation of d(x) by using the convergence argument introduced by Chen, Paule and Saad [12]. More precisely, we have…”

Section: Rational Solutions Of the Difference Equationsmentioning

confidence: 99%

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An algorithm for deciding the summability of bivariate rational functions

Hou

Wang

2015

Advances in Applied Mathematics

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Section: Rational Solutions Of the Difference Equationsmentioning

confidence: 99%

“…Paule [20] gave an interpretation of Gosper's algorithm in terms of the greatest factorial factorizations. Chen, Paule and Saad [12] derived an easy understanding version of Gosper's algorithm by considering the convergence of the greatest common divisors of two polynomial sequences.…”

Section: Introductionmentioning

confidence: 99%

An algorithm for deciding the summability of bivariate rational functions

Hou

Wang

2015

Advances in Applied Mathematics

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“…Now, we can use the q-Gospers algorithm [14,15,16] to verify that ( ) satisfies the qdifference equation (5.6). Setting in (5.7), the result is: ( ) ( ) ( ) If (5.7) and (5.8) are substituted into (5.2), the result will be as follows:…”

Section: The -Difference Equation and The ( ) Operatormentioning

confidence: 99%

The Operator S(a,b;θ_x ) for the Polynomials Z_n (x,y,a,b;q)

Saad

Reshem

2022

Iraqi Journal of Science

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In this work, we give an identity that leads to establishing the operator . Also, we introduce the polynomials . In addition, we provide Operator proof for the generating function with its extension and the Rogers formula for . The generating function with its extension and the Rogers formula for the bivariate Rogers-Szegö polynomials are deduced. The Rogers formula for allows to obtain the inverse linearization formula for , which allows to deduce the inverse linearization formula for . A solution to a q-difference equation is introduced and the solution is expressed in terms of the operators . The q-difference method is used to recover an identity of the operator and the generating function for the polynomials .

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“…Denominator bounds are more common than content bounds in the literature (see, for example, [1], [2], [4], [3], [6], [11], or [10]). If d is a denominator bound, then 1/d is a content bound.…”

Section: Preliminariesmentioning

confidence: 99%

“…Note that 0 is a content bound if and only if there are no non-zero rational solutions. Content bounds and denominator bounds are found in [1], [2], [4], [3], [6], [11], or [10].…”

Section: Introductionmentioning

confidence: 99%

A Family of Denominator Bounds for First Order Linear Recurrence Systems

Hoeij¹,

Barkatou²,

Middeke³

2020

Preprint

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For linear recurrence systems, the problem of finding rational solutions is reduced to the problem of computing polynomial solutions by computing a content bound or a denominator bound. There are several bounds in the literature. The sharpest bound [8] leads to polynomial solutions of lower degrees, but as shown in [7], this advantage need not compensate for the time spent on computing that bound.To strike the best balance between sharpness of the bound versus CPU time spent obtaining it, we will give a family of bounds. The J'th member of this family is similar to [2] when J = 1, similar to [8] when J is large, and novel for intermediate values of J, which give the best balance between sharpness and CPU time.The setting for our content bounds are systems τ(Y) = MY where τ is an automorphism of a unique factorization domain, and M is an invertible matrix with entries in its field of fractions. This setting includes the shift case, the q-shift case, the multi-basic case and others. We give two versions, a global version, and a version that bounds each entry separately.

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Converging to Gosper's algorithm

Cited by 19 publications

References 17 publications

An algorithm for deciding the summability of bivariate rational functions

An algorithm for deciding the summability of bivariate rational functions

The Operator S(a,b;θ_x ) for the Polynomials Z_n (x,y,a,b;q)

A Family of Denominator Bounds for First Order Linear Recurrence Systems

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