scholarly journals Creative Telescoping on Multiple Sums

2021 ◽  
Author(s):  
Christoph Koutschan ◽  
Elaine Wong
Keyword(s):  
Creative Telescoping

scholarly journals Efficient rational creative telescoping

2021 ◽  
Author(s):  
Mark Giesbrecht ◽  
Hui Huang ◽  
George Labahn ◽  
Eugene Zima
Keyword(s):  
Creative Telescoping

scholarly journals THE NON-COMMUTATIVE A-POLYNOMIAL OF TWIST KNOTS

2010 ◽  
Vol 19 (12) ◽  
pp. 1571-1595 ◽  
Author(s):  
STAVROS GAROUFALIDIS ◽  
XINYU SUN
Keyword(s):  
Difference Equation ◽  
Recursion Relation ◽  
Jones Polynomial ◽  
Minimal Order ◽  
Quantum Topology ◽  
And Topology ◽  
Jones Polynomials ◽  
Creative Telescoping ◽  
Colored Jones Polynomials ◽  
Telescoping Method

The purpose of the paper is two-fold: to introduce a multivariable creative telescoping method, and to apply it in a problem of Quantum Topology: namely the computation of the non-commutative A-polynomial of twist knots. Our multivariable creative telescoping method allows us to compute linear recursions for sums of the form [Formula: see text] given a recursion relation for [Formula: see text] and the hypergeometric kernel c(n, k). As an application of our method, we explicitly compute the non-commutative A-polynomial for twist knots with -15 and 15 crossings. The non-commutative A-polynomial of a knot encodes the monic, linear, minimal order q-difference equation satisfied by the sequence of colored Jones polynomials of the knot. Its specialization to q = 1 is conjectured to be the better-known A-polynomial of a knot, which encodes important information about the geometry and topology of the knot complement. Unlike the case of the Jones polynomial, which is easily computable for knots with 50 crossings, the A-polynomial is harder to compute and already unknown for some knots with 12 crossings.

scholarly journals Reduction-based creative telescoping for fuchsian D-finite functions

2018 ◽  
Vol 85 ◽  
pp. 108-127 ◽  
Author(s):  
Shaoshi Chen ◽  
Mark van Hoeij ◽  
Manuel Kauers ◽  
Christoph Koutschan
Keyword(s):  
Creative Telescoping

scholarly journals Some open problems related to creative telescoping

2017 ◽  
Vol 30 (1) ◽  
pp. 154-172 ◽  
Author(s):  
Shaoshi Chen ◽  
Manuel Kauers
Keyword(s):  
Open Problems ◽  
Creative Telescoping

scholarly journals An Apéry-like Difference Equation for Catalan's Constant

10.37236/1707 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
W. Zudilin
Keyword(s):  
Difference Equation ◽  
Continued Fraction ◽  
Hypergeometric Series ◽  
Second Order ◽  
Fast Calculation ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
Catalan’S Constant ◽  
Creative Telescoping

Applying Zeilberger's algorithm of creative telescoping to a family of certain very-well-poised hypergeometric series involving linear forms in Catalan's constant with rational coefficients, we obtain a second-order difference equation for these forms and their coefficients. As a consequence we derive a new way of fast calculation of Catalan's constant as well as a new continued-fraction expansion for it. Similar arguments are put forward to deduce a second-order difference equation and a new continued fraction for $\zeta(4)=\pi^4/90$.

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