## Central Binomial Sums, Multiple Clausen Values and Zeta Values (2000)

Venue: | Math |

Citations: | 22 - 9 self |

### BibTeX

@ARTICLE{Borwein00centralbinomial,

author = {Jonathan Michael Borwein and David J. Broadhurst and Joel Kamnitzer},

title = {Central Binomial Sums, Multiple Clausen Values and Zeta Values},

journal = {Math},

year = {2000},

volume = {10},

pages = {25--34}

}

### OpenURL

### Abstract

We nd and prove relationships between Riemann zeta values and central binomial sums. We also investigate alternating binomial sums (also called Apery sums). The study of non-alternating sums leads to an investigation of dierent types of sums which we call multiple Clausen values. The study of alternating sums leads to a tower of experimental results involving polylogarithms in the golden ratio. AMS (1991) subject classication: Primary 40B05, 33E20, Secondary 11M99, 11Y99. Key words: binomial sums, multiple zeta values, log-sine integrals, Clausens function, multiple Clausen values, polylogarithms, Apery sums. 1 1 Introduction We shall begin by studying the central binomial sum S(k), given as: S(k) := 1 X n=1 1 n k 2n n (1) for integer k. A classical evaluation is S(4) = 17 36 (4). Using a mixture of integer relation and other computational techniques, we uncover remarkable links to values multi-dimensional polylogarithms of sixth roots of unity which we call multip...

### Citations

319 |
Polylogarithms and Associated Functions
- Lewin
- 1981
(Show Context)
Citation Context ...(3) = 2 5 (3), with its connections to Apery's proof of the irrationality of (3), (see e.g., [5]). 2 Denitions and Preliminaries We start with some denitions which for the most part follow Lewin [14], and [6, 7, 8]. A useful multi-dimensional polylogarithm is dened by Li a1 ;:::;a k (z) := X n1>>nk >0 z n1 n a1 1 : : : n ak k ; with the parameters required to be positive integers. This is a g... |

67 | Evaluations of k-fold euler/zagier sums: a compendium of results for arbitrary k
- Borwein, Bradley, et al.
(Show Context)
Citation Context ...(3), with its connections to Apery's proof of the irrationality of (3), (see e.g., [5]). 2 Denitions and Preliminaries We start with some denitions which for the most part follow Lewin [14], and [6, 7=-=-=-, 8]. A useful multi-dimensional polylogarithm is dened by Li a1 ;:::;a k (z) := X n1>>nk >0 z n1 n a1 1 : : : n ak k ; with the parameters required to be positive integers. This is a generalization o... |

45 | Computational strategies for the Riemann zeta function
- Borwein, Bradley, et al.
- 2011
(Show Context)
Citation Context ...b )x a+1 (1 x) b+1 which { when we extract the coecients of various powers of x on both sides { gives us curious innite sums of MCVs of dierent weight (reminiscent of similar rational -evaluations [9]). For example, extracting the coecient of x yields: 1 X b=0 (2; f1g b ) = i 3 : More generally, we obtain 1 X b=0 (n+1; f1g b ) (b+1)(n; f1g b )+ +( 1) n+1 b + 1 n 1 (2; f1g b ) = ( i=3... |

41 | Massive 3-loop Feynman diagrams reducible to SC* primitives of algebras of the sixth root of unity, Eur
- Broadhurst
- 1999
(Show Context)
Citation Context ...pparent that there is still more to be learned by examining all MCVs { and especially their integral representations. This is a subject we have largely ignored in this paper, but whichsgures large in =-=[11]-=-, where polylogarithms of the sixth root of unity were studied in the context of integrals arising from quantumseld theory. Experiments using linear relation algorithms suggested that the only MCVs th... |

40 | Parallel integer relation detection: Techniques and applications
- Bailey, Broadhurst
- 2000
(Show Context)
Citation Context ... and 8. Moreover, David Bailey and David Broadhurst have explicitly obtained S(n) for n 20 through a very high level application of integer relation algorithms. The result for S(20) is presented in [=-=2]-=-. 4 Multiple Clausen Values Central binomial sums naturally led us into a study of multiple Clausen values. This study proved to be quite fruitful and we were led to many striking results. To start wi... |

21 | Polylogarithmic ladders, hypergeometric series and the ten millionth digits of ζ(3) and ζ(5
- Broadhurst
- 1998
(Show Context)
Citation Context ...0000 L 9 ( 12 ) 19014912 L 9 ( 10 ) 206671500 L 9 ( 8 ) + 1295616000 L 9 ( 6 ) 3180657375 L 9 ( 4 ) + 4907952000 L 9 ( 2 ) 52537600 log 9 (): This still falls short of the ladder for (11), in [12]. The current record is set by the ladder for (17) in [3], which extends the weight-16 analysis of Henri Cohen, Leonard Lewin and Don Zagier [13], in the numberseld of the Lehmer polynomial of conjec... |

20 | Experimental mathematics: Recent developments and future outlook
- Bailey, Borwein
- 2001
(Show Context)
Citation Context ...k+1 k 5 2k k = 2(5) 4 3 log() 5 + 8 3 log() 3 (2) + 4 log() 2 (3) + 80 X n>0 1 (2n) 5 log() (2n) 4 2n ; (23) 14 which, along with previous integer relation exclusion bounds (see, e.g., [4], [10]), helps explain why no `simple' evaluation for A(5) such as those for S(2); A(3); S(4) has ever been found. At k = 6 we found the empirical relation 11 A(6) 2 5 2 (3) = 144 e L 6 () 64 9 e... |

16 |
Parallel Integer Relation Detection
- Bailey, Broadhurst
(Show Context)
Citation Context ... and 8. Moreover, David Bailey and David Broadhurst have explicitly obtained S(n) for n ≤ 20 through a very high level application of integer relation algorithms. The result for S(20) is presented in =-=[2]-=-. 4 Multiple Clausen Values Central binomial sums naturally led us into a study of multiple Clausen values. This study proved to be quite fruitful and we were led to many striking results. To start wi... |

10 | A Seventeenth-Order Polylogarithm Ladder
- Bailey, Broadhurst
- 1999
(Show Context)
Citation Context ...+ 1295616000 L 9 ( 6 ) 3180657375 L 9 ( 4 ) + 4907952000 L 9 ( 2 ) 52537600 log 9 (): This still falls short of the ladder for (11), in [12]. The current record is set by the ladder for (17) in [3], which extends the weight-16 analysis of Henri Cohen, Leonard Lewin and Don Zagier [13], in the numberseld of the Lehmer polynomial of conjecturally smallest Mahler measure. Acknowledgements. This re... |

10 |
and Petr Lisoněk, Combinatorial Aspects of Multiple Zeta Values
- Borwein, Bradley, et al.
- 1998
(Show Context)
Citation Context ...(3), with its connections to Apery's proof of the irrationality of (3), (see e.g., [5]). 2 Denitions and Preliminaries We start with some denitions which for the most part follow Lewin [14], and [6, 7=-=-=-, 8]. A useful multi-dimensional polylogarithm is dened by Li a1 ;:::;a k (z) := X n1>>nk >0 z n1 n a1 1 : : : n ak k ; with the parameters required to be positive integers. This is a generalization o... |

9 | A sixteenth-order polylogarithm ladder
- Cohen, Lewin, et al.
- 1992
(Show Context)
Citation Context ...): This still falls short of the ladder for (11), in [12]. The current record is set by the ladder for (17) in [3], which extends the weight-16 analysis of Henri Cohen, Leonard Lewin and Don Zagier [1=-=3]-=-, in the numberseld of the Lehmer polynomial of conjecturally smallest Mahler measure. Acknowledgements. This research was in part performed while Joel Kamnitzer was an NSERC Undergraduate Research Fe... |

3 |
Borwein and Petr Lisonek, \Applications of Integer Relation Algorithms," Discrete Mathematics (Special issue for FPSAC
- Jonathan
- 1997
(Show Context)
Citation Context ... we call multiple Clausen values. We are thence able to prove some surprising identities { and empirically determine many more. Our experimental integer relation tools are described in some detail in =-=[10-=-]. We shallsnish by discussing the corresponding alternating sum: A(k) := 1 X n=1 ( 1) n+1 n k 2n n : (2) These are related to polylogarithmic ladders in the golden ratio p 5 1 2 . A classical evalua... |

1 |
Borwein and Peter Borwein, Pi and the AGM
- Jonathan
- 1987
(Show Context)
Citation Context ...) These are related to polylogarithmic ladders in the golden ratio p 5 1 2 . A classical evaluation is A(3) = 2 5 (3), with its connections to Apery's proof of the irrationality of (3), (see e.g., [5]). 2 Denitions and Preliminaries We start with some denitions which for the most part follow Lewin [14], and [6, 7, 8]. A useful multi-dimensional polylogarithm is dened by Li a1 ;:::;a k (z) := X ... |