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491 results on '"Paule, Peter"'

1. Creative Telescoping for Hypergeometric Double Sums

Author

Paule, Peter and Schneider, Carsten

Subjects
Computer Science - Symbolic Computation
Abstract

We present efficient methods for calculating linear recurrences of hypergeometric double sums and, more generally, of multiple sums. In particular, we supplement this approach with the algorithmic theory of contiguous relations, which guarantees the applicability of our method for many input sums. In addition, we elaborate new techniques to optimize the underlying key task of our method to compute rational solutions of parameterized linear recurrences.

Published
2024

2. Error bounds for the asymptotic expansion of the partition function

Author

Banerje, Koustav, Paule, Peter, Radu, Cristian-Silviu, and Schneider, Carsten

Subjects
Mathematics - Number Theory, Computer Science - Symbolic Computation, Mathematics - Combinatorics, 05A16, 11P82, 68W30
Abstract

Asymptotic study on the partition function $p(n)$ began with the work of Hardy and Ramanujan. Later Rademacher obtained a convergent series for $p(n)$ and an error bound was given by Lehmer. Despite having this, a full asymptotic expansion for $p(n)$ with an explicit error bound is not known. Recently O'Sullivan studied the asymptotic expansion of $p^{k}(n)$-partitions into $k$th powers, initiated by Wright, and consequently obtained an asymptotic expansion for $p(n)$ along with a concise description of the coefficients involved in the expansion but without any estimation of the error term. Here we consider a detailed and comprehensive analysis on an estimation of the error term obtained by truncating the asymptotic expansion for $p(n)$ at any positive integer $n$. This gives rise to an infinite family of inequalities for $p(n)$ which finally answers to a question proposed by Chen. Our error term estimation predominantly relies on applications of algorithmic methods from symbolic summation.

Published
2022

9. Towards a symbolic summation theory for unspecified sequences

Author

Paule, Peter and Schneider, Carsten

Subjects
Computer Science - Symbolic Computation, Mathematics - Combinatorics
Abstract

The article addresses the problem whether indefinite double sums involving a generic sequence can be simplified in terms of indefinite single sums. Depending on the structure of the double sum, the proposed summation machinery may provide such a simplification without exceptions. If it fails, it may suggest a more advanced simplification introducing in addition a single nested sum where the summand has to satisfy a particular constraint. More precisely, an explicitly given parameterized telescoping equation must hold. Restricting to the case that the arising unspecified sequences are specialized to the class of indefinite nested sums defined over hypergeometric, multi-basic or mixed hypergeometric products, it can be shown that this constraint is not only sufficient but also necessary.

Published
2018

13. Holonomic Tools for Basic Hypergeometric Functions

Author

Koutschan, Christoph and Paule, Peter

Subjects
Computer Science - Symbolic Computation, Mathematics - Combinatorics
Abstract

With the exception of q-hypergeometric summation, the use of computer algebra packages implementing Zeilberger's "holonomic systems approach" in a broader mathematical sense is less common in the field of q-series and basic hypergeometric functions. A major objective of this article is to popularize the usage of such tools also in these domains. Concrete case studies showing software in action introduce to the basic techniques. An application highlight is a new computer-assisted proof of the celebrated Ismail-Zhang formula, an important q-analog of a classical expansion formula of plane waves in terms of Gegenbauer polynomials.

Published
2016

14. Relativistic Coulomb Integrals and Zeilberger's Holonomic Systems Approach II

Author

Koutschan, Christoph, Paule, Peter, and Suslov, Sergei K.

Subjects
Quantum Physics, Computer Science - Symbolic Computation, Mathematical Physics
Abstract

We derive the recurrence relations for relativistic Coulomb integrals directly from the integral representations with the help of computer algebra methods. In order to manage the computational complexity of this problem, we employ holonomic closure properties in a sophisticated way.

Published
2013
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15. Computer-Assisted Proofs of Some Identities for Bessel Functions of Fractional Order

Author

Gerhold, Stefan, Kauers, Manuel, Koutschan, Christoph, Paule, Peter, Schneider, Carsten, and Zimmermann, Burkhard

Subjects
Computer Science - Symbolic Computation
Abstract

We employ computer algebra algorithms to prove a collection of identities involving Bessel functions with half-integer orders and other special functions. These identities appear in the famous Handbook of Mathematical Functions, as well as in its successor, the DLMF, but their proofs were lost. We use generating functions and symbolic summation techniques to produce new proofs for them., Comment: Final version, some typos were corrected. 21 pages, uses svmult.cls

Published
2013
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19. Relativistic Coulomb Integrals and Zeilberger's Holonomic Systems Approach. I

Author

Paule, Peter and Suslov, Sergei K.

Subjects
Quantum Physics
Abstract

With the help of computer algebra we study the diagonal matrix elements , where O are the standard Dirac matrix operators and the angular brackets denote the quantum-mechanical average for the relativistic Coulomb problem. Using Zeilberger's extension of Gosper's algorithm and a variant to it, three-term recurrence relations for each of these expectation values are derived together with some transformation formulas for the corresponding generalized hypergeometric series. In addition, the virial recurrence relations for these integrals are also found and proved algorithmically., Comment: 16 pages, no figures

Published
2012

20. Converging to Gosper's Algorithm

Author

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

Subjects
Mathematics - Classical Analysis and ODEs, Mathematics - Combinatorics, 33F10, 05A19
Abstract

Given two polynomials, we find a convergence property of the GCD of the rising factorial and the falling factorial. Based on this property, we present a unified approach to computing the universal denominators as given by Gosper's algorithm and Abramov's algorithm for finding rational solutions to linear difference equations with polynomial coefficients., Comment: 13 pages

Published
2007

23. Lattice paths, q-multinomials and two variants of the Andrews-Gordon identities

Author

Berkovich, Alexander and Paule, Peter

Subjects
Mathematics - Combinatorics, Mathematics - Number Theory, Mathematics - Quantum Algebra, 05A10, 05A19, 11B65, 11P82
Abstract

A few years ago Foda, Quano, Kirillov and Warnaar proposed and proved various finite analogs of the celebrated Andrews-Gordon identities. In this paper we use these polynomial identities along with the combinatorial techniques introduced in our recent paper to derive Garrett, Ismail, Stanton type formulas for two variants of the Andrews-Gordon identities., Comment: 15 pages, 9 figures, added clarification comments, stylistic changes, submitted to Ramanujan Journal

Published
2001

26. Holonomic Sequences and Power Series

Author

Kauers, Manuel, Paule, Peter, Corless, Robert, editor, Hong, Hoon, editor, Ida, Tetsuo, editor, Kreuzer, Martin, editor, Salvy, Bruno, editor, Wang, Dongming, editor, Paule, Peter, editor, and Kauers, Manuel

Published
2011
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27. Formal Power Series

Author

Kauers, Manuel, Paule, Peter, Corless, Robert, editor, Hong, Hoon, editor, Ida, Tetsuo, editor, Kreuzer, Martin, editor, Salvy, Bruno, editor, Wang, Dongming, editor, Paule, Peter, editor, and Kauers, Manuel

Published
2011
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28. C-Finite Sequences

Author

Kauers, Manuel, Paule, Peter, Corless, Robert, editor, Hong, Hoon, editor, Ida, Tetsuo, editor, Kreuzer, Martin, editor, Salvy, Bruno, editor, Wang, Dongming, editor, Paule, Peter, editor, and Kauers, Manuel

Published
2011
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29. Algebraic Functions

Author

Kauers, Manuel, Paule, Peter, Corless, Robert, editor, Hong, Hoon, editor, Ida, Tetsuo, editor, Kreuzer, Martin, editor, Salvy, Bruno, editor, Wang, Dongming, editor, Paule, Peter, editor, and Kauers, Manuel

Published
2011
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30. Hypergeometric Series

Author

Kauers, Manuel, Paule, Peter, Corless, Robert, editor, Hong, Hoon, editor, Ida, Tetsuo, editor, Kreuzer, Martin, editor, Salvy, Bruno, editor, Wang, Dongming, editor, Paule, Peter, editor, and Kauers, Manuel

Published
2011
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31. Introduction

Author

Kauers, Manuel, Paule, Peter, Corless, Robert, editor, Hong, Hoon, editor, Ida, Tetsuo, editor, Kreuzer, Martin, editor, Salvy, Bruno, editor, Wang, Dongming, editor, Paule, Peter, editor, and Kauers, Manuel

Published
2011
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32. Polynomials

Author

Kauers, Manuel, Paule, Peter, Corless, Robert, editor, Hong, Hoon, editor, Ida, Tetsuo, editor, Kreuzer, Martin, editor, Salvy, Bruno, editor, Wang, Dongming, editor, Paule, Peter, editor, and Kauers, Manuel

Published
2011
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33. Algorithms in Symbolic Computation

Author

Paule, Peter, Buchberger, Bruno, Kartashova, Lena, Kauers, Manuel, Schneider, Carsten, Winkler, Franz, Buchberger, Bruno, editor, Affenzeller, Michael, editor, Ferscha, Alois, editor, Haller, Michael, editor, Jebelean, Tudor, editor, Klement, Erich Peter, editor, Paule, Peter, editor, Pomberger, Gustav, editor, Schreiner, Wolfgang, editor, Stubenrauch, Robert, editor, Wagner, Roland, editor, Weiß, Gerhard, editor, and Windsteiger, Wolfgang, editor

Published
2009
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34. Relativistic Coulomb Integrals and Zeilberger’s Holonomic Systems Approach II

Author

Koutschan, Christoph, Paule, Peter, Suslov, Sergei K., Hutchison, David, editor, Kanade, Takeo, editor, Kittler, Josef, editor, Kleinberg, Jon M., editor, Mattern, Friedemann, editor, Mitchell, John C., editor, Naor, Moni, editor, Nierstrasz, Oscar, editor, Pandu Rangan, C., editor, Steffen, Bernhard, editor, Sudan, Madhu, editor, Terzopoulos, Demetri, editor, Tygar, Doug, editor, Vardi, Moshe Y., editor, Weikum, Gerhard, editor, Barkatou, Moulay, editor, Cluzeau, Thomas, editor, Regensburger, Georg, editor, and Rosenkranz, Markus, editor

Published
2014
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42. Topics of Computer Algebra

Author

Loos, Rüdiger, Giusti, M., Pardo, L. M., Niklasch, Gerhard, Dentzer, Ralf, Landau, Susan, Roy, Marie-Françoise, Pohst, Michael E., Grabmeier, Johannes, Weispfenning, Volker, von zur Gathen, Joachim, Kaltofen, Erich, Gutierrez, Jaime, Kozen, Dexter, Kemper, Gregor, Saunders, B. David, Giesbrecht, Mark, Storjohann, Arne, Villard, Gilles, Buchmann, Johannes, Müller, Volker, Pfahler, Thomas, Zayer, Jörg, Klüners, Jürgen, Heinrich Matzat, B., Zimmer, Horst G., Greuel, Gert-Martin, Becker, Eberhard, Apel, Joachim, Cohen, Arjeh M., Dräxler, Peter, Laßner, Wolfgang, Madlener, Klaus E., Reinert, Birgit, Sims, Charles C., Hiss, Gerhard, Kerber, Adalbert, Laue, Reinhard, Paule, Peter, Strehl, Voker, Bronstein, Manuel, Seiler, W. M., Ulmer, F., Schwarz, F., Hereman, W., Gerdt, V. P., Ollivier, F., Gatermann, K., Hantzschmann, Karl, Rump, Siegfried M., Corless, Robert M., Watt, Stephen M., Bürgisser, Peter, Clausen, Michael, Shokrollahi, M. Amin, Beth, Thomas, Egner, Sebastian, Geiselmann, Willi, Grassl, Markus, Lazic, Dejan, Müller-Quade, Jörn, Schaefer-Lorinser, Frank, Bündgen, Reinhard, Calmet, Jacques, Fortenbacher, Albrecht, Schupp, Sibylle, Gautier, Thierry, Hong, Hoon, Roch, Jean-Louis, Schreiner, Wolfgang, Dewar, Michael, Klappenecker, Andreas, Nückel, Armin, Grabmeier, Johannes, editor, Kaltofen, Erich, editor, and Weispfenning, Volker, editor

Published
2003
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43. Groups

Author

Mikhalev, Alexander V., Leedham-Green, Charles R., Lausch, Hans, Cameron, Peter J., Hilgert, Joachim, Olver, Peter J., Solomon, Ronald, Baumslag, Gilbert, Holt, Derek, Grigorchuk, Rostislav I., Lysenok, Igor G., Paule, Peter, Meldrum, John D. P., Plotkin, Boris I., Fleischmann, Peter, Maxson, Carlton J., Vogt, Frank, Iskander, Awad A., Karzel, Helmut, Gudrun, Kalmbach H. E., Mikhalev, Alexander V., and Pilz, Günter F.

Published
2002
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49. MacMahon's Partition Analysis XIV : partitions with n copies of n : RISC Report Series 22-14

Author

Andrews, George E. and Paule, Peter

Subjects
partitions, Radu's Ramanujan-Kolberg algorithm, partitions with $n$ copies of $n$, overpartitions, partition analysis, $q$-series, modular forms and partition congruences
Abstract

We apply the methods of partition analysis to partitions with~$n$ copies of~$n$. This allows us to obtain multivariable generating functions related to classical Rogers-Ramanujan type identities. Also, partitions with $n$ copies of $n$ are extended to partition diamonds yielding numerous new results including a natural connection to overpartitions and a variety of partition congruences.

Published
2022

50. The Ramanujan Journal / New inequalities for p(n) and logp(n)

Author

Radu, Christian-Silviu, Banerjee, Koustav, Paule, Peter, and Zeng, WenHuan

Subjects
The partition function asymptotics, Hardy–Ramanujan–Rademacher formula, Log-concavity
Abstract

Let p(n) denote the number of partitions of n. A new infinite family of inequalities for p(n) is presented. This generalizes a result by William Chen et al. From this infinite family, another infinite family of inequalities for logp(n) is derived. As an application of the latter family one, for instance obtains that for n≥120, p(n)2>(1+π24−−√n3/2−1n2)p(n−1)p(n+1). Version of record

Published
2022

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