A formula for a quartic integral: a survey of old proofs and some new ones

Amdeberhan, Tewodros; Moll, Victor H.

doi:10.1007/s11139-007-9041-9

Cited by 36 publications

(43 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This identity has been proved rigorously in [1,2]. A simpler, albeit heuristic, proof can be achieved by exploiting umbral method [18] can be found in [4,9].…”

Section: Ramanujan Master Theorem and Its Implicationmentioning

confidence: 89%

The Umbral operator and the integration involving generalized Bessel-type functions

et al. 2015

View full text Add to dashboard Cite

show abstract

“…This identity has been proved rigorously in [1,2]. A simpler, albeit heuristic, proof can be achieved by exploiting umbral method [18] can be found in [4,9].…”

Section: Ramanujan Master Theorem and Its Implicationmentioning

confidence: 89%

The Umbral operator and the integration involving generalized Bessel-type functions

et al. 2015

View full text Add to dashboard Cite

show abstract

“…Many proofs of how to tackle this integral are collected in [8]. We want to sketch how to prove it with computer algebra and present two strategies.…”

Section: Univariate Versus Multivariatementioning

confidence: 99%

Advanced applications of the holonomic systems approach

Koutschan

2010

ACM Commun. Comput. Algebra

107

174

View full text Add to dashboard Cite

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...

show abstract

“…A variety of proofs of (1.3) can be found in [3] and properties of the coefficients {d (m)} have been reviewed in [26].…”

Section: Introductionmentioning

confidence: 99%

A Hypergeometric Inequality

2015

Self Cite

View full text Add to dashboard Cite

show abstract

A formula for a quartic integral: a survey of old proofs and some new ones

Abstract: Abstract. We discuss several existing proofs of the value of a quartic integral and present a new proof that evolved from rational Landen transformations. We include our personal renditions as related to the history of these particular computations.

Cited by 36 publications

References 16 publications

The Umbral operator and the integration involving generalized Bessel-type functions

The Umbral operator and the integration involving generalized Bessel-type functions

Advanced applications of the holonomic systems approach

A Hypergeometric Inequality

Contact Info

Product

Resources

About