A Mathematica package for q-holonomic sequences and power series

Kauers, Manuel; Koutschan, Christoph

doi:10.1007/s11139-008-9132-2

Cited by 34 publications

(40 citation statements)

References 17 publications

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“…In Section 8, we show that our method provides an algorithm for determining these numbers of free subgroups of Γ m (3) modulo any given 2-power in the case when m is odd. The corresponding results (see Theorems 19 and 20) go far beyond the previous result [29, Theorem 1] on the behaviour of the number of free subgroups of P SL 2 (Z) modulo 16. Our method provides as well an algorithm for determining the number of all subgroups of index n in P SL 2 (Z) modulo powers of 2, as we demonstrate in Section 9.…”

Section: Introductionmentioning

confidence: 51%

“…We overcome this obstacle by instead tuning our computations with the target of obtaining results modulo 16 = 2 4 . Indeed, this leads to the determination of the number of subgroups of index n in SL 2 (Z) and in Γ 3 (3) modulo 8 (see Theorems 26 and 30), but direct application of our method does not produce corresponding results modulo 16. Only by an enhancement of the method, which we outline in Appendix D, we are able to produce descriptions of the subgroup numbers of SL 2 (Z) modulo 16, see Theorem 28.…”

Section: Introductionmentioning

confidence: 99%

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A Method for Determining the Mod-$2^k$ Behaviour of Recursive Sequences, with Applications to Subgroup Counting

Kauers¹,

Krattenthaler²,

Müller³

2012

Electron. J. Combin.

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We present a method to obtain congruences modulo powers of 2 for sequences given by recurrences of finite depth with polynomial coefficients. We apply this method to Catalan numbers, Fuß-Catalan numbers, and to subgroup counting functions associated with Hecke groups and their lifts. This leads to numerous new results, including many extensions of known results to higher powers of 2.2000 Mathematics Subject Classification. Primary 20E06; Secondary 05A15 05E99 11A07 20E07 33F10 68W30.

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Section: Introductionmentioning

confidence: 51%

Section: Introductionmentioning

confidence: 99%

A Method for Determining the Mod-$2^k$ Behaviour of Recursive Sequences, with Applications to Subgroup Counting

Kauers¹,

Krattenthaler²,

Müller³

2012

Electron. J. Combin.

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“…List of the related sizes and partitions, where 1 and 2 modulo 3 parts are colored differently. n π n π n π n π 16 (11, 5) 28 (17,8,3) 34 (17,12,5) 40 (18,14,8) 19 (14,5) (17, 11) (18,11,5) 43 (17,12,9,5) 22 (14,8) 31 (15,11,5) (17,11,5,1) 40 (17,11,7,5) 49 (18,15,11,5) 28 (14,9,5) (17, 11, 6) (17,12,8,3) 52 (19,17,11,5) One example of this corollary is presented in Table 1.…”

Section: Q-trinomial Identitiesmentioning

confidence: 99%

Polynomial identities implying Capparelli's partition theorems

Berkovich

Uncu

2019

Journal of Number Theory

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We propose and recursively prove polynomial identities which imply Capparelli's partition theorems. We also find perfect companions to the results of Andrews, and Alladi, Andrews and Gordon involving q-trinomial coefficients. We follow Kurşungöz's ideas to provide direct combinatorial interpretations of some of our expressions. We make use of the trinomial analogue of Bailey's lemma to derive new identities. These identities relate certain triple sums and products. A couple of new Slater type identities involving bases q 2 , q 3 , q 6 , and q 12 are also proven. We also discuss a new infinite hierarchy containing these Slater type identities.

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“…For univariate closure properties in the q-world, the package qGeneratingFunctions [47] is available. Since both sums have finite support we can perform creative telescoping without having to care about the delta part (recall that J and L replace the expressions q j and q l , respectively: …”

Section: Q-identitiesmentioning

confidence: 99%

Advanced applications of the holonomic systems approach

Koutschan

2010

ACM Commun. Comput. Algebra

Self Cite

107

174

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Ich erkläre an Eides statt, dass ich die vorliegende Dissertation selbstständig und ohne fremde Hilfe verfasst, andere als die angegebenen Quellen und Hilfsmittel nicht benutzt bzw. die wörtlich oder sinngemäß entnommenen Stellen als solche kenntlich gemacht habe. Linz, September 2009Christoph Koutschan AbstractThe holonomic systems approach was proposed in the early 1990s by Doron Zeilberger. It laid a foundation for the algorithmic treatment of holonomic function identities. Frédéric Chyzak later extended this framework by introducing the closely related notion of ∂-finite functions and by placing their manipulation on solid algorithmic grounds. For practical purposes it is convenient to take advantage of both concepts which is not too much of a restriction: The class of functions that are holonomic and ∂-finite contains many elementary functions (such as rational functions, algebraic functions, logarithms, exponentials, sine function, etc.) as well as a multitude of special functions (like classical orthogonal polynomials, elliptic integrals, Airy, Bessel, and Kelvin functions, etc.). In short, it is composed of functions that can be characterized by sufficiently many partial differential and difference equations, both linear and with polynomial coefficients. An important ingredient is the ability to execute closure properties algorithmically, for example addition, multiplication, and certain substitutions. But the central technique is called creative telescoping which allows to deal with summation and integration problems in a completely automatized fashion. Part of this thesis is our Mathematica package HolonomicFunctions in which the above mentioned algorithms are implemented, including more basic functionality such as noncommutative operator algebras, the computation of Gröbner bases in them, and finding rational solutions of parameterized systems of linear differential or difference equations.Besides standard applications like proving special function identities, the focus of this thesis is on three advanced applications that are interesting in their own right as well as for their computational challenge. First, we contributed to translating Takayama's algorithm into a new context, in order to apply it to an until then open problem, the proof of Ira Gessel's lattice path conjecture. The computations that completed the proof were of a nontrivial size and have been performed with our software. Second, investigating basis functions in finite element methods, we were able to extend the existing algorithms in a way that allowed us to derive various relations which generated a considerable speed-up in the subsequent numerical simulations, in this case of the propagation of electromagnetic waves. The third application concerns a computer proof of the enumeration formula for totally symmetric plane partitions, also known as Stembridge's theorem. To make the underlying computations feasible we employed a new approach for finding creative telescoping operators. KurzzusammenfassungIn den frühen 1990er Jahren beschr...

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A Mathematica package for q-holonomic sequences and power series

Cited by 34 publications

References 17 publications

A Method for Determining the Mod-$2^k$ Behaviour of Recursive Sequences, with Applications to Subgroup Counting

A Method for Determining the Mod-$2^k$ Behaviour of Recursive Sequences, with Applications to Subgroup Counting

Polynomial identities implying Capparelli's partition theorems

Advanced applications of the holonomic systems approach

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