On a family of polynomials related to \(\zeta (2,1)=\zeta (3)\). (English) Zbl 1455.11124
Burgos Gil, José Ignacio (ed.) et al., Periods in quantum field theory and arithmetic. Based on the presentations at the research trimester on multiple zeta values, multiple polylogarithms, and quantum field theory, ICMAT 2014, Madrid, Spain, September 15–19, 2014. Cham: Springer. Springer Proc. Math. Stat. 314, 621-630 (2020).
Summary: We give a new proof of the identity \(\zeta (\{2,1\}^l)=\zeta (\{3\}^l)\) of the multiple zeta values, where \(l=1,2,\dots \), using generating functions of the underlying generalized polylogarithms. In the course of study we arrive at (hypergeometric) polynomials satisfying 3-term recurrence relations, whose properties we examine and compare with analogous ones of polynomials originated from an (ex-)conjectural identity of Borwein, Bradley and Broadhurst [J. M. Borwein et al., Electron. J. Comb. 4, No. 2, Research paper R5, 19 p. (1997; Zbl 0884.40004)].
For the entire collection see [Zbl 1446.81002].
For the entire collection see [Zbl 1446.81002].
MSC:
| 11M32 | Multiple Dirichlet series and zeta functions and multizeta values |
| 33C47 | Other special orthogonal polynomials and functions |
Keywords:
multiple zeta values; hypergeometric function; hypergeometric polynomial; generalized orthogonalityCitations:
Zbl 0884.40004References:
| [1] | Bailey, W.N.: Generalized hypergeometric series. Cambridge Math. Tracts 32, Cambridge Univ. Press, Cambridge (1935); 2nd reprinted edn.: Stechert-Hafner, New York-London (1964) · JFM 61.0406.01 |
| [2] | Borwein, J., Bailey, D.: Mathematics by Experiment. Plausible reasoning in the 21st century, 2nd edn. A K Peters, Ltd., Wellesley, MA (2008) · Zbl 1163.00002 |
| [3] | Borwein, J.M., Bradley, D.M.: Thirty-two Goldbach variations. Intern. J. Number Theory 2(1), 65-103 (2006) · Zbl 1094.11031 · doi:10.1142/S1793042106000383 |
| [4] | Borwein, J.M., Bradley, D.M., Broadhurst, D.J.: Evaluations of \(k\)-fold Euler/Zagier sums: a compendium of results for arbitrary \(k\). Electron. J. Combin. 4, # R5 (1997); Printed version: J. Combin. 4(2), 31-49 (1997) · Zbl 0884.40004 |
| [5] | Borwein, J.M., Bradley, D.M., Broadhurst, D.J., Lisoněk, P.: Special values of multiple polylogarithms. Trans. Amer. Math. Soc. 353(3), 907-941 (2001) · Zbl 1002.11093 · doi:10.1090/S0002-9947-00-02616-7 |
| [6] | Hoffman, M.E.: Multiple harmonic series. Pacific J. Math. 152(2), 275-290 (1992) · Zbl 0763.11037 · doi:10.2140/pjm.1992.152.275 |
| [7] | Iserles, A., Nørsett, S.P.: On the theory of biorthogonal polynomials. Trans. Amer. Math. Soc. 306, 455-474 (1988) · Zbl 0662.42017 · doi:10.1090/S0002-9947-1988-0933301-8 |
| [8] | Ismail, M.E.H., Masson, D.R.: Generalized orthogonality and continued fractions. J. Approx. Theory 83, 1-40 (1995) · Zbl 0846.33005 · doi:10.1006/jath.1995.1106 |
| [9] | Mimachi, K.: Connection matrices associated with the generalized hypergeometric function \(_3F_2\). Funkcial. Ekvac. 51(1), 107-133 (2008) · Zbl 1157.33308 · doi:10.1619/fesi.51.107 |
| [10] | Spiridonov, V.P., Tsujimoto, S., Zhedanov, A.: Integrable discrete time chains for the Frobenius-Stickelberger-Thiele polynomials. Comm. Math. Phys. 272(1), 139-165 (2007) · Zbl 1136.37041 · doi:10.1007/s00220-007-0219-1 |
| [11] | Spiridonov, V., Zhedanov, A.: Spectral transformation chains and some new biorthogonal rational functions. Comm. Math. Phys. 210(1), 49-83 (2000) · Zbl 0989.33008 · doi:10.1007/s002200050772 |
| [12] | Zagier, D.: Values of zeta functions and their applications. In: Joseph, A., et al. (eds.), 1st European Congress of Mathematics (Paris, 1992), vol. II, Progr. Math. 120, pp. 497-512, Birkhäuser, Boston (1994) · Zbl 0822.11001 |
| [13] | Zhao, J.: On a conjecture of Borwein, Bradley and Broadhurst. J. Reine Angew. Math. 639, 223-233 (2010); Extended version: Double shuffle relations of Euler sums. Preprint http://arxiv.org/abs/math.AG/0412539, arXiv: 0705.2267 [math.NT] (2007) · Zbl 1236.11078 |
| [14] | Zudilin, W. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
