##
**New representations for Apéry-like sequences.**
*(English)*
Zbl 1275.11035

Focus of this paper are three sequences satisfying similar difference equations, i.e., the numbers found by R. Apéry [Astérisque 61, 11–13 (1979; Zbl 0401.10049)], the alternating version of the Domb numbers examined by H. H. Chan, S. H. Chan and Z.-G. Liu [Adv. Math. 186, No. 2, 396–410 (2004; Zbl 1122.11087)] and the AZ numbers defined by G. Almkvist and W. Zudilin [Mirror symmetry V. AMS/IP Stud. Adv. Math. 38, 481–515 (2006; Zbl 1118.14043)].

In fact, their generating series \(F_\alpha (z)\), \(F_\delta (z)\) and \(F_\xi (z)\) admit the modular parametrizations via the Hauptmoduls of the three subgroups of index two lying between \(\Gamma_0 (6)\) and its normalizer in \(\mathrm{SL}_2 (\mathbb R)\), studied by H. H. Chan and H. Verrill [Math. Res. Lett. 16, No. 2-3, 405–420 (2009; Zbl 1193.11038)], which are the main tools of the proof, whose secondary means are, e.g., the \(q\)-series expansions of the functions, the differential operator \(\theta =y(d/dy)\) and computer algebra such as Maple.

The authors establish new algebraic relations among the series \(F_\alpha (z)\), \(F_\delta (z)\), \(F_\xi (z)\) and their expressions through \({}_3 F_2\)-hypergeometric series; then they derive three new Ramanujan-type series for \( 1/\pi\) and they auspicate that achievements and methods provided in this paper could help to obtain the explicit evaluation of the three-variable Mahler measures investigated by M. D. Rogers [Ramanujan J. 18, No. 3, 327–340 (2009; Zbl 1226.11113)].

In the proof the authors employ also a congruence by F. Beukers [Journées arithmétiques, Besançon/France 1985, Astérisque 147/148, 271–283 (1987; Zbl 0613.10031)], the Kummer’s quadratic transform and the Clausen’s identity both reported in [L. J. Slater, Generalized hypergeometric functions. Cambridge: At the University Press (1966; Zbl 0135.28101)], the simplest Ramanujan’s formula proved by G. Bauer [J. Reine Angew. Math. 56, 101–121 (1859; ERAM 056.1478cj)] and the series for \( 1/\pi\) introduced by S. Ramanujan [Q. J. Math., Oxf. 45, 350–372 (1914; JFM 45.1249.01)].

Other Ramanujan-related results supplied by the first-named author {et al.} ([B. C. Berndt, H. H. Chan and S.-S. Huang, Contemp. Math. 254, 79–126 (2000; Zbl 0971.33012)]; B. C. Berndt, H. H. Chan and W.-C. Liaw [J. Number Theory 88, No.1, 129–156 (2001; Zbl 1005.33009)]) and by the second-named author [Modular forms and string duality. Proceedings of a workshop, Banff, Canada, 2006. Fields Institute Communications 54, 179–188 (2008; Zbl 1159.11053)]; [Math. Notes 81, No. 3, 297–301 (2007); translation from Mat. Zametki 81, No. 3, 335-340 (2007; Zbl 1144.33002)] are also recalled in the paper.

In fact, their generating series \(F_\alpha (z)\), \(F_\delta (z)\) and \(F_\xi (z)\) admit the modular parametrizations via the Hauptmoduls of the three subgroups of index two lying between \(\Gamma_0 (6)\) and its normalizer in \(\mathrm{SL}_2 (\mathbb R)\), studied by H. H. Chan and H. Verrill [Math. Res. Lett. 16, No. 2-3, 405–420 (2009; Zbl 1193.11038)], which are the main tools of the proof, whose secondary means are, e.g., the \(q\)-series expansions of the functions, the differential operator \(\theta =y(d/dy)\) and computer algebra such as Maple.

The authors establish new algebraic relations among the series \(F_\alpha (z)\), \(F_\delta (z)\), \(F_\xi (z)\) and their expressions through \({}_3 F_2\)-hypergeometric series; then they derive three new Ramanujan-type series for \( 1/\pi\) and they auspicate that achievements and methods provided in this paper could help to obtain the explicit evaluation of the three-variable Mahler measures investigated by M. D. Rogers [Ramanujan J. 18, No. 3, 327–340 (2009; Zbl 1226.11113)].

In the proof the authors employ also a congruence by F. Beukers [Journées arithmétiques, Besançon/France 1985, Astérisque 147/148, 271–283 (1987; Zbl 0613.10031)], the Kummer’s quadratic transform and the Clausen’s identity both reported in [L. J. Slater, Generalized hypergeometric functions. Cambridge: At the University Press (1966; Zbl 0135.28101)], the simplest Ramanujan’s formula proved by G. Bauer [J. Reine Angew. Math. 56, 101–121 (1859; ERAM 056.1478cj)] and the series for \( 1/\pi\) introduced by S. Ramanujan [Q. J. Math., Oxf. 45, 350–372 (1914; JFM 45.1249.01)].

Other Ramanujan-related results supplied by the first-named author {et al.} ([B. C. Berndt, H. H. Chan and S.-S. Huang, Contemp. Math. 254, 79–126 (2000; Zbl 0971.33012)]; B. C. Berndt, H. H. Chan and W.-C. Liaw [J. Number Theory 88, No.1, 129–156 (2001; Zbl 1005.33009)]) and by the second-named author [Modular forms and string duality. Proceedings of a workshop, Banff, Canada, 2006. Fields Institute Communications 54, 179–188 (2008; Zbl 1159.11053)]; [Math. Notes 81, No. 3, 297–301 (2007); translation from Mat. Zametki 81, No. 3, 335-340 (2007; Zbl 1144.33002)] are also recalled in the paper.

Reviewer: Enzo Bonacci (Latina)

### MSC:

11B65 | Binomial coefficients; factorials; \(q\)-identities |

11F11 | Holomorphic modular forms of integral weight |

11F20 | Dedekind eta function, Dedekind sums |

33C20 | Generalized hypergeometric series, \({}_pF_q\) |

### Citations:

Zbl 0401.10049; Zbl 1122.11087; Zbl 1118.14043; Zbl 1193.11038; Zbl 1226.11113; Zbl 0613.10031; Zbl 0135.28101; Zbl 0971.33012; Zbl 1005.33009; Zbl 1159.11053; Zbl 1144.33002; JFM 45.1249.01; ERAM 056.1478cj### Software:

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\textit{H. H. Chan} and \textit{W. Zudilin}, Mathematika 56, No. 1, 107--117 (2010; Zbl 1275.11035)

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### References:

[1] | Ramanujan, Q. J. Math. Oxford Ser. (2) 45 pp 350– (1914) |

[2] | Chan, Math. Res. Lett. 16 pp 405– (2009) · Zbl 1193.11038 |

[3] | DOI: 10.1016/j.aim.2003.07.012 · Zbl 1122.11087 |

[4] | DOI: 10.1006/jnth.2000.2615 · Zbl 1005.33009 |

[5] | Berndt, Contemp. Math. 254 pp 79– (2000) |

[6] | DOI: 10.1007/s11139-007-9040-x · Zbl 1226.11113 |

[7] | Apéry, Astérisque 61 pp 11– (1979) |

[8] | Almkvist, Mirror Symmetry V pp 481– (2007) |

[9] | Zudilin, Modular Forms and String Duality (Banff, June 2006) pp 179– (2008) |

[10] | DOI: 10.1134/S0001434607030030 · Zbl 1144.33002 |

[11] | Slater, Generalized Hypergeometric Functions (1966) · Zbl 0135.28101 |

[12] | Bauer, J. für Math. (Crelles J.) 56 pp 101– (1859) |

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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.