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19
Trading order for degree in creative telescoping
, 2012
"... We analyze the differential equations produced by the method of creative telescoping applied to a hyperexponential term in two variables. We show that equations of low order have high degree, and that higher order equations have lower degree. More precisely, we derive degree bounding formulas which ..."
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We analyze the differential equations produced by the method of creative telescoping applied to a hyperexponential term in two variables. We show that equations of low order have high degree, and that higher order equations have lower degree. More precisely, we derive degree bounding formulas which allow to estimate the degree of the output equations from creative telescoping as a function of the order. As an application, we show how the knowledge of these formulas can be used to improve, at least in principle, the performance of creative telescoping implementations, and we deduce bounds on the asymptotic complexity of creative telescoping for hyperexponential terms.
Creative Telescoping for Holonomic Functions
"... Abstract The aim of this article is twofold: on the one hand it is intended to serve as a gentle introduction to the topic of creative telescoping, from a practical point of view; for this purpose its application to several problems is exemplified. On the other hand, this chapter has the flavour of ..."
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Abstract The aim of this article is twofold: on the one hand it is intended to serve as a gentle introduction to the topic of creative telescoping, from a practical point of view; for this purpose its application to several problems is exemplified. On the other hand, this chapter has the flavour of a survey article: the developments in this area during the last two decades are sketched and a selection of references is compiled in order to highlight the impact of creative telescoping in numerous contexts. 1
The Holonomic Toolkit
"... This is an overview over standard techniques for holonomic functions, written for readers who are new to the subject. We state the definition for holonomy in a couple of different ways, including some concrete special cases as well as a more abstract and more general version. We give a collection ..."
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Cited by 4 (3 self)
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This is an overview over standard techniques for holonomic functions, written for readers who are new to the subject. We state the definition for holonomy in a couple of different ways, including some concrete special cases as well as a more abstract and more general version. We give a collection of standard examples and state several fundamental properties of holonomic objects. Two techniques which are most useful in applications are explained in some more detail: closure properties, which can be used to prove identities among holonomic functions, and guessing, which can be used to generate plausible conjectures for equations satisfied by a given function.
The Cfinite Ansatz
"... While this article is dedicated to both Mourad Ismail and Dennis Stanton, it does not directly reference any of their works. The main reason is that I only talk about the most trivial kind of recurrences: linear and constant coefficients. But try searching Ismail AND Recurrences or Stanton AND Recur ..."
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Cited by 3 (3 self)
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While this article is dedicated to both Mourad Ismail and Dennis Stanton, it does not directly reference any of their works. The main reason is that I only talk about the most trivial kind of recurrences: linear and constant coefficients. But try searching Ismail AND Recurrences or Stanton AND Recurrences in the database MathSciNet (or Google Scholar) and you would see that both Mourad and Dennis are great gurus in recurrences, so the subject matter of this paper is not entirely inappropriate as a tribute to them. The present work is also largely experimental, and Dennis Stanton is a great pioneer in computer experimentations!
Automatic solution of Richard Stanley’s Amer. Math. Monthly Problem #11610 and ANY problem of that type
, 2011
"... Suppose you toss a (fair) coin n times. If n is large, the law of large numbers promises you that (with high probability) you would roughly get as many Heads as Tails. But what is the exact probability that you would have exactly as many Heads as Tails? If n is odd, the answer is easy (you do it!). ..."
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Suppose you toss a (fair) coin n times. If n is large, the law of large numbers promises you that (with high probability) you would roughly get as many Heads as Tails. But what is the exact probability that you would have exactly as many Heads as Tails? If n is odd, the answer is easy (you do it!). If n is even, then it is almost as easy, and there is a nice, “closedform ” formula for that probability, namely n!/((n/2)! 22n). Richard Stanley [St1] proposed the problem of finding, a(n), the number of nletter words in the alphabet {H, T} where there are as many occurrences of “HT ” (i.e. Head immediately followed by Tail) as there are occurrences of “TT ” (two Tails in a row). He didn’t give a “closed form ” formula, but he gave something almost as good, an explicit formula as an (algebraic, as it turned out, in fact quadratic) formal power series for the (ordinary) generating function P (t): = ∑ ∞ n=0 a(n)tn. The fact that the generating function, P (t), is an algebraic generating function is not at all surprising! This can be seen in (at least) two ways. One way is to show that the “language ” of words with as many occurrences of “HT ” as “HH ” is contextfree (type 2) with an unambiguous grammar, and hence its weightenumerator is algebraic. It is possible to (automatically!) generate its grammar, and then automatically generate a system of algebraic equations one of whose unknowns is the desired generating function, and solving that system would (presumbly, we didn’t do it) yield Stanley’s proposed expression. A better way is to find (automatically, of course!), the rational generating function F (t; z[HT], z[T T]) that is the weightenumerator of all words in the alphabet {H, T} according to the weight W eight(w) = tlength(w) z[HT]#HT (w) z[T T]#T T (w). This can be done in several ways, including the GouldenJackson method, beautifully surveyed in [NZ], and efficiently implemented in the Maple package
2178 AND ALL THAT
"... Abstract. For integers g ≥ 2, k ≥ 2, call a number N a (g, k)reverse multiple if the reversal of N in base g is equal to k times N. The numbers 1089 and 2178 are the two smallest (10, k)reverse multiples, their reversals being 9801 = 9 · 1089 and 8712 = 4 · 2178. In 1992, A. L. Young introduced cer ..."
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Abstract. For integers g ≥ 2, k ≥ 2, call a number N a (g, k)reverse multiple if the reversal of N in base g is equal to k times N. The numbers 1089 and 2178 are the two smallest (10, k)reverse multiples, their reversals being 9801 = 9 · 1089 and 8712 = 4 · 2178. In 1992, A. L. Young introduced certain trees in order to study the problem of finding all (g, k)reverse multiples. By using modified versions of her trees, which we call Young graphs, we determine the possible values of k for bases g = 2 through 100, and then show how to apply the transfermatrix method to enumerate the (g, k)reverse multiples with a given number of baseg digits. These Young graphs are interesting finite directed graphs, whose structure is not at all well understood. 1.
Bounds for Dfinite closure properties
, 2014
"... We provide bounds on the size of operators obtained by algorithms for executing Dfinite closure properties. For operators of small order, we give bounds on the degree and on the height (bitsize). For higher order operators, we give degree bounds that are parameterized with respect to the order and ..."
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We provide bounds on the size of operators obtained by algorithms for executing Dfinite closure properties. For operators of small order, we give bounds on the degree and on the height (bitsize). For higher order operators, we give degree bounds that are parameterized with respect to the order and reflect the phenomenon that higher order operators may have lower degrees (orderdegree curves).