q-Hypergeometric Functions and Irrationality Measures

Ville Merilä

Bollettino dell'Unione Matematica Italiana (2010)

  • Volume: 3, Issue: 1, page 137-148
  • ISSN: 0392-4041

Abstract

top
We present a q-analogue of the Rhin-Viola method for the analysis of Φ -adic valuations of the q-gamma factors occurring in the basic Euler-Pochhammer integral representation of the Heine series 2 ⁢ ϕ 1 . Moreover, we show that this approach yields the best known irrationality measures for log q ⁡ ( z ) , log q ⁡ ( 2 ) and ζ q ⁢ ( 1 ) .

How to cite

top

Merilä, Ville. "q-Hypergeometric Functions and Irrationality Measures." Bollettino dell'Unione Matematica Italiana 3.1 (2010): 137-148. <http://eudml.org/doc/290639>.

@article{Merilä2010,
abstract = {We present a q-analogue of the Rhin-Viola method for the analysis of $\Phi$-adic valuations of the q-gamma factors occurring in the basic Euler-Pochhammer integral representation of the Heine series $2\phi_\{1\}$. Moreover, we show that this approach yields the best known irrationality measures for $\log_\{q\}(z)$, $\log_\{q\}(2)$ and $\zeta_\{q\}(1)$.},
author = {Merilä, Ville},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {2},
number = {1},
pages = {137-148},
publisher = {Unione Matematica Italiana},
title = {q-Hypergeometric Functions and Irrationality Measures},
url = {http://eudml.org/doc/290639},
volume = {3},
year = {2010},
}

TY - JOUR
AU - Merilä, Ville
TI - q-Hypergeometric Functions and Irrationality Measures
JO - Bollettino dell'Unione Matematica Italiana
DA - 2010/2//
PB - Unione Matematica Italiana
VL - 3
IS - 1
SP - 137
EP - 148
AB - We present a q-analogue of the Rhin-Viola method for the analysis of $\Phi$-adic valuations of the q-gamma factors occurring in the basic Euler-Pochhammer integral representation of the Heine series $2\phi_{1}$. Moreover, we show that this approach yields the best known irrationality measures for $\log_{q}(z)$, $\log_{q}(2)$ and $\zeta_{q}(1)$.
LA - eng
UR - http://eudml.org/doc/290639
ER -

References

top
  1. ANDREWS, G. E. - ASKEY, R. - ROY, R., Special functions, Encyclopedia of Mathematics and Its Applications, 71 (Cambridge University Press, Cambridge, 1999), xvi+664. MR1688958DOI10.1017/CBO9781107325937
  2. CHUDNOVSKY, G. V., Hermite-Padé approximations to exponential functions and elementary estimates of the measure of irrationality of π , Lect. Notes in Math., 925 (1982), 299-322. MR659875
  3. HATA, M., Legendre type polynomials and irrationality measures, J. Reine Angew. Math., 407 (1990), 99-125. MR1048530DOI10.1515/crll.1990.407.99
  4. MATALA-AHO, T. - VÄÄNÄNEN, K. - ZUDILIN, W., New irrationality measures for q-logarithms, Math.Comp., 75, no. 254 (2006), 879-889. Zbl1158.11033MR2196997DOI10.1090/S0025-5718-05-01812-0
  5. MERILÄ, V., On arithmetical properties of certain q-series, Results Math., 53, no. 1-2 (2009), 129-151. Zbl1192.11041MR2481409DOI10.1007/s00025-008-0297-1
  6. RHIN, G. - VIOLA, C., On a permutation group related to ζ ⁢ ( 2 ) , Acta Arith., 77, no. 1 (1996), 23-56. Zbl0864.11037MR1404975DOI10.4064/aa-77-1-23-56
  7. VIOLA, C., Hypergeometric functions and irrationality measures, Analytic number theory (Kyoto, 1996), London Math. Soc. Lecture Note Ser., 247 (Cambridge Univ. Press, Cambridge, 1997), 353-360. Zbl0904.11020MR1695002DOI10.1017/CBO9780511666179.024
  8. ZUDILIN, W., Remarks on irrationality of q-harmonic series, Manuscripta Math., 107, no. 4 (2002), 463-477. Zbl1044.11068MR1906771DOI10.1007/s002290200249
  9. ZUDILIN, W., Heine's basic transform and a permutation group for q-harmonic series, Acta Arith., 111, no. 2 (2004), 153-164. Zbl1052.11053MR2039419DOI10.4064/aa111-2-4

NotesEmbed ?

top

You must be logged in to post comments.