Leonhard Euler and a 𝑞-analogue of the logarithm

Koelink, Erik; Van Assche, Walter

doi:10.1090/S0002-9939-08-09374-X

Leonhard Euler and a $q$-analogue of the logarithm
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by Erik Koelink and Walter Van Assche PDF

Proc. Amer. Math. Soc. 137 (2009), 1663-1676 Request permission

Abstract:

We study a $q$-logarithm which was introduced by Euler and give some of its properties. This $q$-logarithm has not received much attention in the recent literature. We derive basic properties, some of which were already given by Euler in a 1751 paper and in a 1734 letter to Daniel Bernoulli. The corresponding $q$-analogue of the dilogarithm is introduced. The relation to the values at $1$ and $2$ of a $q$-analogue of the zeta function is given. We briefly describe some other $q$-logarithms that have appeared in the recent literature.

References

George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958, DOI 10.1017/CBO9781107325937
Ralph Philip Boas Jr., Entire functions, Academic Press, Inc., New York, 1954. MR 0068627
Peter B. Borwein, On the irrationality of $\sum (1/(q^n+r))$, J. Number Theory 37 (1991), no. 3, 253–259. MR 1096442, DOI 10.1016/S0022-314X(05)80041-1
Peter B. Borwein, On the irrationality of certain series, Math. Proc. Cambridge Philos. Soc. 112 (1992), no. 1, 141–146. MR 1162938, DOI 10.1017/S030500410007081X
Ki-Soo Chung, Won-Sang Chung, Sang-Tack Nam, and Hye-Jung Kang, New $q$-derivative and $q$-logarithm, Internat. J. Theoret. Phys. 33 (1994), no. 10, 2019–2029. MR 1306795, DOI 10.1007/BF00675167
Anne Duval, Une remarque sur les “logarithmes” associés à certains caractères, Aequationes Math. 68 (2004), no. 1-2, 88–97 (French, with English summary). MR 2167011, DOI 10.1007/s00010-003-2721-7
P. Erdös, On arithmetical properties of Lambert series, J. Indian Math. Soc. (N.S.) 12 (1948), 63–66. MR 29405
L. Euler, Consideratio quarumdam serierum quae singularibus proprietatibus sunt praeditae, Novi Commentarii Academiae Scientiarum Petropolitanae 3 (1750–1751), pp. 10–12, 86–108; Opera Omnia, Ser. I, Vol. 14, B.G. Teubner, Leipzig, 1925, pp. 516–541.
V.V. Fock, A.B. Goncharov, The quantum dilogarithm and representations of quantum cluster varieties, arXiv:math/0702397v6, to appear in Invent. Math.
George Gasper and Mizan Rahman, Basic hypergeometric series, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 96, Cambridge University Press, Cambridge, 2004. With a foreword by Richard Askey. MR 2128719, DOI 10.1017/CBO9780511526251
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.
A.B. Goncharov, The pentagon relation for the quantum dilogarithm and quantized $M_{0,5}$, arXiv:0706.4054v2
G. Kaniadakis, M. Lissia, Editorial, Physica A 340 (2004), xv–xix.
Anatol N. Kirillov, Dilogarithm identities, Progr. Theoret. Phys. Suppl. 118 (1995), 61–142. Quantum field theory, integrable models and beyond (Kyoto, 1994). MR 1356515, DOI 10.1143/PTPS.118.61
Konrad Knopp, Theorie und Anwendung der Unendlichen Reihen, Springer-Verlag, Berlin-Heidelberg, 1947 (German). 4th ed. MR 0028430, DOI 10.1007/978-3-662-01232-1
Tom H. Koornwinder, Special functions and $q$-commuting variables, Special functions, $q$-series and related topics (Toronto, ON, 1995) Fields Inst. Commun., vol. 14, Amer. Math. Soc., Providence, RI, 1997, pp. 131–166. MR 1448685, DOI 10.1016/s0898-1221(96)90020-6
C. Krattenthaler, T. Rivoal, and W. Zudilin, Séries hypergéométriques basiques, $q$-analogues des valeurs de la fonction zêta et séries d’Eisenstein, J. Inst. Math. Jussieu 5 (2006), no. 1, 53–79 (French, with English and French summaries). MR 2195945, DOI 10.1017/S1474748005000149
L. Lewin, Dilogarithms and associated functions, Macdonald, London, 1958. Foreword by J. C. P. Miller. MR 0105524
Tapani Matala-Aho, Keijo Väänänen, and Wadim Zudilin, New irrationality measures for $q$-logarithms, Math. Comp. 75 (2006), no. 254, 879–889. MR 2196997, DOI 10.1090/S0025-5718-05-01812-0
Charles A. Nelson and Michael G. Gartley, On the two $q$-analogue logarithmic functions: $\ln _q(w)$, $\ln \{e_q(z)\}$, J. Phys. A 29 (1996), no. 24, 8099–8115. MR 1446909, DOI 10.1088/0305-4470/29/24/031
G. Pólya and G. Szegő, Problems and theorems in analysis. Vol. II, Revised and enlarged translation by C. E. Billigheimer of the fourth German edition, Springer Study Edition, Springer-Verlag, New York-Heidelberg, 1976. Theory of functions, zeros, polynomials, determinants, number theory, geometry. MR 0465631, DOI 10.1007/978-1-4757-6292-1
Kelly Postelmans and Walter Van Assche, Irrationality of $\zeta _q(1)$ and $\zeta _q(2)$, J. Number Theory 126 (2007), no. 1, 119–154. MR 2348015, DOI 10.1016/j.jnt.2006.11.011
S. N. M. Ruijsenaars, First order analytic difference equations and integrable quantum systems, J. Math. Phys. 38 (1997), no. 2, 1069–1146. MR 1434226, DOI 10.1063/1.531809
Jacques Sauloy, Systèmes aux $q$-différences singuliers réguliers: classification, matrice de connexion et monodromie, Ann. Inst. Fourier (Grenoble) 50 (2000), no. 4, 1021–1071 (French, with English and French summaries). MR 1799737, DOI 10.5802/aif.1784
Constantino Tsallis, Possible generalization of Boltzmann-Gibbs statistics, J. Statist. Phys. 52 (1988), no. 1-2, 479–487. MR 968597, DOI 10.1007/BF01016429
K. Ueno and M. Nishizawa, Quantum groups and zeta-functions, Quantum groups (Karpacz, 1994) PWN, Warsaw, 1995, pp. 115–126. MR 1647965
Walter Van Assche, Little $q$-Legendre polynomials and irrationality of certain Lambert series, Ramanujan J. 5 (2001), no. 3, 295–310. MR 1876702, DOI 10.1023/A:1012930828917
V. V. Zudilin, On the irrationality measure of the $q$-analogue of $\zeta (2)$, Mat. Sb. 193 (2002), no. 8, 49–70 (Russian, with Russian summary); English transl., Sb. Math. 193 (2002), no. 7-8, 1151–1172. MR 1934544, DOI 10.1070/SM2002v193n08ABEH000674
W. Zudilin, Approximations to $q$-logarithms and $q$-dilogarithms, with applications to $q$-zeta values, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 322 (2005), no. Trudy po Teorii Chisel, 107–124, 253–254 (English, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 137 (2006), no. 2, 4673–4683. MR 2138454, DOI 10.1007/s10958-006-0263-y