Power Series Representations of Hypergeometric Type Functions

Tabuguia, Bertrand Teguia; Koepf, Wolfram

doi:10.1007/978-3-030-81698-8_25

Cited by 3 publications

(2 citation statements)

References 30 publications

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“…The thesis deals with the holonomic, hypergeometric type, and non-holonomic functions. For holonomic and hypergeometric type functions, see the references [23], [22], and [24]. Our software is available for download for Maple 2021 and Maxima 5.44 users at its dedicated web page http://www.mathematik.unikassel.de/~bteguia/FPS_webpage/FPS.htm.…”

Section: Examplementioning

confidence: 99%

On the Representation of Non-Holonomic Power Series

Tabuguia

Koepf

2022

Maple Trans.

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Holonomic functions play an essential role in Computer Algebra since they allow the application of many symbolic algorithms. Among all algorithmic attempts to find formulas for power series, the holonomic property remains the most important requirement to be satisfied by the function under consideration. The targeted functions mainly summarize that of meromorphic functions. However, expressions like tan(z), z/(exp(z)-1), sec(z), etc. are not holonomic, therefore their power series are inaccessible by non-pattern matching implementations like the current Maple convert/FormalPowerSeries up to Maple 2021. From the mathematical dictionaries, one can observe that most of the known closed-form formulas of non-holonomic power series involve another sequence whose evaluation depends on some finite summations. In the case of tan(z) and sec(z) the corresponding sequences are the Bernoulli and Euler numbers, respectively. Thus providing a symbolic approach that yields complete representations when linear summations for power series coefficients of non-holonomic functions appear, might be seen as a step forward towards the representation of non-holonomic power series. By adapting the method of ansatz with undetermined coefficients, we build an algorithm that computes least-order quadratic differential equations with polynomial coefficients for a large class of non-holonomic functions. A differential equation resulting from this procedure is converted into a recurrence equation by applying the Cauchy product formula and rewriting powers into polynomials and derivatives into shifts. Finally, using enough initial values we are able to give normal form representations (Geddes et al. 1992) to characterize several non-holonomic power series. As a consequence of the defined normal relation, it turns out that our algorithm is able to detect identities between non-holonomic functions that were not accessible in the past. We discuss this algorithm and its implementation for Maple 2022. Our Maple and Maxima implementations are available under the FPS software which can be downloaded at http://www.mathematik.uni-kassel.de/~bteguia/FPS_webpage/FPS.htm.

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Section: Examplementioning

confidence: 99%

On the Representation of Non-Holonomic Power Series

Tabuguia

Koepf

2022

Maple Trans.

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“…The computation of m-fold hypergeometric term solutions of holonomic REs is effective (see [9,11,14]). On this poster, we use the algorithm mfoldHyper (available in Maple 2022 as LREtools-mhypergeomsols), developed in [14] for the efficient computation of hypergeometric type power series (see also [13,12]). It is worth mentioning that mfoldHyper extends the algorithms by Petkovšek and Mark van Hoeij (see [7,15,1]) and has a much better performance than some previous approaches in the same direction (see [8,3]).…”

Section: Introductionmentioning

confidence: 99%

FPS In Action: An Easy Way To Find Explicit Formulas For Interlaced Hypergeometric Sequences

Tabuguia¹,

Koepf²

2022

Preprint

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Linear recurrence equations with constant coefficients define the power series coefficients of rational functions. However, one usually prefers to have an explicit formula for the sequence of coefficients, provided that such a formula is "simple" enough. Simplicity is related to the compactness of the formula due to the presence of algebraic numbers: "the smaller, the simpler". This poster showcases the capacity of recent updates on the Formal Power Series (FPS) algorithm, implemented in Maxima and Maple (convert/FormalPowerSeries), to find simple formulas for sequences like those from https:// oeis.org/A307717, https://oeis.org/A226782, or https://oeis.org/A226784 by computing power series representations of their correctly guessed generating functions. We designed the algorithm for the more general context of univariate P -recursive sequences. Our implementations are available at http://www.mathematik.uni-kassel.de/ ~bteguia/FPS_webpage/FPS.htm.

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Hypergeometric-type sequences

Teguia Tabuguia

2024

Journal of Symbolic Computation

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Power Series Representations of Hypergeometric Type Functions

Cited by 3 publications

References 30 publications

On the Representation of Non-Holonomic Power Series

On the Representation of Non-Holonomic Power Series

FPS In Action: An Easy Way To Find Explicit Formulas For Interlaced Hypergeometric Sequences

Hypergeometric-type sequences

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