Leonhard Euler and a q-analogue of the logarithm

Proceedings of the American Mathematical Society

Volumn 137, Issue 5, 2009, Pages 1663-1676

  Koelink, Erik ^a   Van Assche, Walter ^b  

^a RADBOUD UNIVERSITY NIJMEGEN   (Netherlands)

^b UNIVERSITY OF LEUVEN   (Belgium)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 75649117749     PISSN: 00029939     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0002-9939-08-09374-X     Document Type: Article

References (32)

1. G. E. Andrews, R. Askey, R. Roy, 5Special Functions, Encyclopedia of Mathematics and its Applications 71, Cambridge University Press, 1999. MR1688958 (2000g:33001)
2. R. P. Boas, Jr., Entire Functions, Academic Press, New York, 1954. MR0068627 (16:914f)
3. n+r, J. Number Theory 37 (1991), 253-259. MR1096442 (92b:11046)
4. P. Borwein, On the irrationality of certain series, Math. Proc. Cambridge Philos. Soc. 112 (1992), 141-146. MR1162938 (93g:11074)
5. K.-S. Chung, W.-S. Chung, S.-T. Nam, H.-J. Kang, New q-derivative and q-logarithm, Internat. J. Theor. Phys. 33 (1994), 2019-2029. MR1306795 (95i:33018)
6. A. Duval, Une remarque sur les "logarithmes" associés à certains caractères, Aequationes Math. 68 (2004), no. 1-2, 88-97. MR2167011 (2006e:39029) (Pubitemid 39305716)
7. P. Erdos, On arithmetical properties of Lambert series, J. Indian Math. Soc. (N. S.) 12 (1948), 63-66. MR0029405 (10:594c)
8. L. Euler, Consideratio quarumdam serierum quae singularibus proprietatibus sunt praeditae, Novi Commentarii Academiae Scientiarum Petropolitanae 3 (1750-1751), pp. 10-12, 86-108;
9. Opera Omnia, Ser. I, Vol. 14, B. G. Teubner, Leipzig, 1925, pp. 516-541.
10. V. V. Fock, A. B. Goncharov, The quantum dilogarithm and representations of quantum cluster varieties, arXiv:math/0702397v6, to appear in Invent. Math.
11. G. Gasper, M. Rahman, Basic Hypergeometric Series, 2nd ed., Encyclopedia of Mathematics and its Applications 96, Cambridge University Press, 2004. MR2128719 (2006d:33028)
12. W. Gautschi, On Euler's attempt to compute logarithms by interpolation: A commentary to his letter of February 17, 1734, to Daniel Bernoulli, J. Comput. Appl. Math. 219 (2008), no. 2, 408-415.
13. 0, 5, arXiv:0706.4054v2
14. G. Kaniadakis, M. Lissia, Editorial, Physica A 340 (2004), xv-xix.
15. A. N. Kirillov, Dilogarithm Identities, Progr. Theoret. Phys. Suppl. 118 (1995), 61-142 (Lectures in Math. Sci. 7, Univ. of Tokyo, 1995). MR1356515 (96h:11102)
16. K. Knopp, Theorie und Anwendung der Unendlichen Reihen, 4th ed., Springer-Verlag, Berlin- Heidelberg, 1947. MR0028430 (10:446a)
17. T. H. Koornwinder, Special functions and q-commuting variables, pp. 131-166 in Special Functions, q-Series and Related Topics (eds. M. E. H. Ismail, D. R. Masson and M. Rahman), Fields Inst. Commun. 14, Amer. Math. Soc., Providence, RI, 1997. MR1448685 (99e:33024)
18. C. Krattenthaler, T. Rivoal, W. Zudilin, Séries hypergéométriques basiques, q-analogues des valeurs de la fonction zêta et séries d'Eisenstein, J. Inst. Math. Jussieu 5 (2006), 53-79. MR2195945 (2006k:11137)
19. L. Lewin, Dilogarithms and Associated Functions, Macdonald, London, 1958. MR0105524 (21:4264)
20. T. Matala-Aho, K. Väänänen, W. Zudilin, New irrationality measures for q-logarithms, Math. Comp. 75 (2006), no. 254, 879-889. MR2196997 (2007e:11082) (Pubitemid 43681477)
21. q (z)}, J. Phys. A 29 (1996), no. 24, 8099-8115. MR1446909 (98d:33007) (Pubitemid 126345185)
22. G. Pólya, G. Szego, Problems and Theorems in Analysis, Volume II, Springer-Verlag, New York-Heidelberg, 1976 (revised and enlarged translation of Aufgaben und Lehrsätze aus der Analysis II, 4th edition, 1971). MR0465631 (57:5529)
23. q (2), J. Number Theory 126 (2007), 119-154. MR2348015
24. S. N. M. Ruijsenaars, First order analytic difference equations and integrable quantum systems, J. Math. Phys. 38 (1997), no. 2, 1069-1146. MR1434226 (98m:58065) (Pubitemid 127025193)
25. J. Sauloy, Systèmes aux q-différences singuliers réguliers: Classification, matrice de connexion et monodromie, Ann. Inst. Fourier (Grenoble) 50 (2000), no. 4, 1021-1071. MR1799737 (2001m:39043)
26. C. Tsallis, Possible generalization of Boltzmann-Gibbs statistics, J. Stat. Phys. 52 (1988), 479-487. MR968597 (89i:82119)
27. K. Ueno, M. Nishizawa, Quantum groups and zeta-functions, pp. 115-126 in Quantum Groups (eds. J. Lukierski, Z. Popowicz and J. Sobczyk), PWN, Warsaw, 1995. MR1647965 (99k:11136)
28. W. Van Assche, Little q-Legendre polynomials and irrationality of certain Lambert series, The Ramanujan Journal 5 (2001), 295-310. MR1876702 (2002k:11124) (Pubitemid 33574691)
29. V. V. Zudilin, On the irrationality measure of the q-analogue of ζ (2), Mat. Sbornik 193 (2002), no. 8, 49-70 (in Russian) ;
30. Sbornik Math. 193 (2002), no. 7-8, 1151-1172. MR1934544 (2003g:11077)
31. W. Zudilin, Approximations to q-logarithms and q-dilogarithms, with applications to q-zeta values, Zap. Nauchn. Sem. POMI 322 (2005), 107-124 (in Russian) ;
32. J. Math. Sci. (N. Y.) 137 (2006), no. 2, 4673-4683. MR2138454 (2006h:11088)