scholarly journals Infinite families of accelerated series for some classical constants by the Markov-WZ Method

2005 ◽  
Vol Vol. 7 ◽  
Author(s):  
Mohamud Mohammed
Keyword(s):  
Hypergeometric Series ◽  
Series Representations ◽  
International Audience ◽  
Wz Method

International audience In this article we show the Markov-WZ Method in action as it finds rapidly converging series representations for a given hypergeometric series. We demonstrate the method by finding new representations for log(2),ζ (2) and ζ (3).

scholarly journals Matrix coefficients of the large discrete series representations of Sp(2; R) as hypergeometric series of two variables

2012 ◽  
Vol 208 ◽  
pp. 201-263
Author(s):  
Takayuki Oda
Keyword(s):  
Differential Equations ◽  
Hypergeometric Series ◽  
Discrete Series ◽  
Higher Order ◽  
Holonomic System ◽  
Radial Part ◽  
Matrix Coefficients ◽  
Series Representations ◽  
The Matrix ◽  
Discrete Series Representations

AbstractWe investigate the radial part of the matrix coefficients with minimal K-types of the large discrete series representations of Sp(2; R). They satisfy certain difference-differential equations derived from Schmid operators. This system is reduced to a holonomic system of rank 4, which is finally found to be equivalent to higher-order hypergeometric series in the sense of Appell and Kampé de Fériet.

scholarly journals An extension to overpartitions of Rogers-Ramanujan identities for even moduli

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Sylvie Corteel ◽  
Jeremy Lovejoy ◽  
Olivier Mallet
Keyword(s):  
Generating Functions ◽  
Hypergeometric Series ◽  
Basic Hypergeometric Series ◽  
Lattice Paths ◽  
Infinite Products ◽  
International Audience

International audience We investigate class of well-poised basic hypergeometric series $\tilde{J}_{k,i}(a;x;q)$, interpreting these series as generating functions for overpartitions defined by multiplicity conditions. We also show how to interpret the $\tilde{J}_{k,i}(a;1;q)$ as generating functions for overpartitions whose successive ranks are bounded, for overpartitions that are invariant under a certain class of conjugations, and for special restricted lattice paths. We highlight the cases $(a,q) \to (1/q,q)$, $(1/q,q^2)$, and $(0,q)$, where some of the functions $\tilde{J}_{k,i}(a;x;q)$ become infinite products. The latter case corresponds to Bressoud's family of Rogers-Ramanujan identities for even moduli.

scholarly journals Accelerated series for universal constants, by the WZ method

1999 ◽  
Vol Vol. 3 no. 4 ◽  
Author(s):  
Herbert S. Wilf
Keyword(s):  
Wz Theory ◽  
International Audience ◽  
Wz Method

International audience In this paper, the author presents a method, based on WZ theory, for finding rapidly converging series for universal constants. This method is analogous but different from Amdeberhan and Zeilberger's method.

scholarly journals In Praise of an Elementary Identity of Euler

10.37236/2009 ◽  
2011 ◽  
Vol 18 (2) ◽  
Author(s):  
Gaurav Bhatnagar
Keyword(s):  
Hypergeometric Series ◽  
Fibonacci Numbers ◽  
Basic Hypergeometric Series ◽  
Binomial Theorem ◽  
Pell Numbers ◽  
Generalized Hypergeometric Series ◽  
Pentagonal Number Theorem ◽  
Wz Method ◽  
Elementary Identity

We survey the applications of an elementary identity used by Euler in one of his proofs of the Pentagonal Number Theorem. Using a suitably reformulated version of this identity that we call Euler's Telescoping Lemma, we give alternate proofs of all the key summation theorems for terminating Hypergeometric Series and Basic Hypergeometric Series, including the terminating Binomial Theorem, the Chu–Vandermonde sum, the Pfaff–Saalschütz sum, and their $q$-analogues. We also give a proof of Jackson's $q$-analog of Dougall's sum, the sum of a terminating, balanced, very-well-poised $_8\phi_7$ sum. Our proofs are conceptually the same as those obtained by the WZ method, but done without using a computer. We survey identities for Generalized Hypergeometric Series given by Macdonald, and prove several identities for $q$-analogs of Fibonacci numbers and polynomials and Pell numbers that have appeared in combinatorial contexts. Some of these identities appear to be new.

scholarly journals $n$-color overpartitions, lattice paths, and multiple basic hypergeometric series

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Olivier Mallet
Keyword(s):  
Generating Functions ◽  
Hypergeometric Series ◽  
Basic Hypergeometric Series ◽  
Combinatorial Identities ◽  
Lattice Paths ◽  
Multiple Series ◽  
Infinite Products ◽  
Nous Montrons ◽  
International Audience ◽  
Multiple Basic Hypergeometric Series

International audience We define two classes of multiple basic hypergeometric series $V_{k,t}(a,q)$ and $W_{k,t}(a,q)$ which generalize multiple series studied by Agarwal, Andrews, and Bressoud. We show how to interpret these series as generating functions for special restricted lattice paths and for $n$-color overpartitions with weighted difference conditions. We also point out that some specializations of our series can be written as infinite products, which leads to combinatorial identities linking $n$-color overpartitions with ordinary partitions or overpartitions. Nous définissons deux classes de séries hypergéométriques basiques multiples $V_{k,t}(a,q)$ et $W_{k,t}(a,q)$ qui généralisent des séries multiples étudiées par Agarwal, Andrews et Bressoud. Nous montrons comment interpréter ces séries comme les fonctions génératrices de chemins avec certaines restrictions et de surpartitions $n$-colorées vérifiant des conditions de différences pondérées. Nous remarquons aussi que certaines spécialisations de nos séries peuvent s'écrire comme des produits infinis, ce qui conduit à des identités combinatoires reliant les surpartitions $n$-colorées aux partitions ou surpartitions ordinaires.

scholarly journals Simultaneous generation for zeta values by the Markov-WZ method

2008 ◽  
Vol Vol. 10 no. 3 (Combinatorics) ◽  
Author(s):  
Khodabakhsh Hessami Pilehrood ◽  
Tatiana Hessami Pilehrood
Keyword(s):  
Generating Function ◽  
Simultaneous Generation ◽  
Function Identity ◽  
Zeta Values ◽  
International Audience ◽  
Bivariate Generating Function ◽  
Wz Method

Combinatorics International audience By application of the Markov-WZ method, we prove a more general form of a bivariate generating function identity containing, as particular cases, Koecher's and Almkvist-Granville's Apéry-like formulae for odd zeta values. As a consequence, we get a new identity producing Apéry-like series for all ζ(2n+4m+3),n,m ≥ 0, convergent at the geometric rate with ratio 2−10.

scholarly journals Hypergeometric Series Acceleration Via the WZ method

10.37236/1318 ◽  
1996 ◽  
Vol 4 (2) ◽  
Author(s):  
Tewodros Amdeberhan ◽  
Doron Zeilberger
Keyword(s):  
Zeta Function ◽  
Hypergeometric Series ◽  
Infinite Family ◽  
Series Acceleration ◽  
Wz Method

Based on the WZ method, some series acceleration formulas are given. These formulas allow us to write down an infinite family of parameterized identities from any given identity of WZ type. Further, this family, in the case of the Zeta function, gives rise to many accelerated expressions for $\zeta(3)$.

Somethin’ about Scandinavia: Danish Shorts on the Post-war International Scene

2018 ◽  
Author(s):  
C. Claire Thomson
Keyword(s):  
United Nations ◽  
International Collaboration ◽  
Marshall Plan ◽  
The Third ◽  
Animated Film ◽  
International Audience ◽  
Post War ◽  
Nordic Cooperation

Building on the picture of post-war Anglo-Danish documentary collaboration established in the previous chapter, this chapter examines three cases of international collaboration in which Dansk Kulturfilm and Ministeriernes Filmudvalg were involved in the late 1940s and 1950s. They Guide You Across (Ingolf Boisen, 1949) was commissioned to showcase Scandinavian cooperation in the realm of aviation (SAS) and was adopted by the newly-established United Nations Film Board. The complexities of this film’s production, funding and distribution are illustrative of the activities of the UN Film Board in its first years of operation. The second case study considers Alle mine Skibe (All My Ships, Theodor Christensen, 1951) as an example of a film commissioned and funded under the auspices of the Marshall Plan. This US initiative sponsored informational films across Europe, emphasising national solutions to post-war reconstruction. The third case study, Bent Barfod’s animated film Noget om Norden (Somethin’ about Scandinavia, 1956) explains Nordic cooperation for an international audience, but ironically exposed some gaps in inter-Nordic collaboration in the realm of film.

Conclusion

2018 ◽  
Author(s):  
Alistair Fox
Keyword(s):  
National Identity ◽  
New Zealand ◽  
Identity Formation ◽  
Essential Role ◽  
Coming Of Age ◽  
Financial Returns ◽  
Self Recognition ◽  
International Audience ◽  
National Identity Formation

The conclusion reaffirms the essential role played by cinema generally, and the coming-of-age genre in particular, in the process of national identity formation, because of its effectiveness in facilitating self-recognition and self-experience through a process of triangulation made possible, for the most part, by a dialogue with some of the nation’s most iconic works of literature. This section concludes by point out the danger posed, however, by an observable trend toward generic standardization in New Zealand films motivated by a desire to appeal to an international audience out of consideration for the financial returns expected by funding bodies under current regimes.

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