Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-23T17:40:36.118Z Has data issue: false hasContentIssue false
  • English
  • Français

THE LEVEL 12 ANALOGUE OF RAMANUJAN’S FUNCTION $k$

Part of: Other special functions Combinatorics Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  22 January 2016

SHAUN COOPER  and
DONGXI YE
SHAUN COOPER*
Affiliation:
Institute of Natural and Mathematical Sciences, Massey University-Albany, Private Bag 102904, North Shore Mail Centre, Auckland, New Zealand email [email protected]
DONGXI YE
Affiliation:
Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, WI 53706, USA email [email protected]
*
✉[email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We provide a comprehensive study of the function $h=h(q)$ defined by

$$\begin{eqnarray}h=q\mathop{\prod }_{j=1}^{\infty }\frac{(1-q^{12j-1})(1-q^{12j-11})}{(1-q^{12j-5})(1-q^{12j-7})}\end{eqnarray}$$
and show that it has many properties that are analogues of corresponding results for Ramanujan’s function $k=k(q)$ defined by
$$\begin{eqnarray}k=q\mathop{\prod }_{j=1}^{\infty }\frac{(1-q^{10j-1})(1-q^{10j-2})(1-q^{10j-8})(1-q^{10j-9})}{(1-q^{10j-3})(1-q^{10j-4})(1-q^{10j-6})(1-q^{10j-7})}.\end{eqnarray}$$

Keywords

Dedekind eta functionDomb numbersEisenstein serieselliptic integralhypergeometric functionJacobi triple product identitymodular equationquintuple product identityRamanujan’s cubic continued fractionRogers–Ramanujan continued fraction

MSC classification

Primary: 11F11: Holomorphic modular forms of integral weight
Secondary: 05A10: Factorials, binomial coefficients, combinatorial functions 33C75: Elliptic integrals as hypergeometric functions 33E05: Elliptic functions and integrals
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 101 , Issue 1 , August 2016 , pp. 29 - 53
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

References

Alaca, A., Alaca, Ş. and Williams, K. S., ‘On the two-dimensional theta functions of the Borweins’, Acta Arith. 124 (2006), 177–195.Google Scholar
Alaca, A., Alaca, Ş. and Williams, K. S., ‘Some infinite products of Ramanujan type’, Canad. Math. Bull. 52 (2009), 481–492.Google Scholar
Almkvist, G., van Straten, D. and Zudilin, W., ‘Generalizations of Clausen’s formula and algebraic transformations of Calabi–Yau differential equations’, Proc. Edinb. Math. Soc. (2) 54 (2011), 273–295.Google Scholar
Andrews, G. E., Askey, R. and Roy, R., Special Functions (Cambridge University Press, Cambridge, 1999).Google Scholar
Bailey, D., Borwein, J. M., Broadhurst, D. and Glasser, M. L., ‘Elliptic integral evaluations of Bessel moments and applications’, J. Phys. A 41(20) (2008), 205203.Google Scholar
Berndt, B. C., Ramanujan’s Notebooks, Part III (Springer, New York, 1991).Google Scholar
Berndt, B. C., Number Theory in the Spirit of Ramanujan (American Mathematical Society, Providence, RI, 2006).Google Scholar
Borwein, J. M., Straub, A., Wan, J. and Zudilin, W., ‘Densities of short uniform random walks’, Canad. J. Math. 64 (2012), 961–990.Google Scholar
Chan, H. H., Chan, S. H. and Liu, Z.-G., ‘Domb’s numbers and Ramanujan-Sato type series for 1/𝜋’, Adv. Math. 186 (2004), 396–410.Google Scholar
Chan, H. H., Cooper, S. and Sica, F., ‘Congruences satisfied by Apéry-like numbers’, Int. J. Number Theory 6 (2010), 89–97.Google Scholar
Chan, H. H. and Zudilin, W., ‘New representations for Apéry-like sequences’, Mathematika 56 (2010), 107–117.Google Scholar
Cooper, S., ‘The quintuple product identity’, Int. J. Number Theory 2 (2006), 115–161.CrossRefGoogle Scholar
Cooper, S., ‘Inversion formulas for elliptic functions’, Proc. Lond. Math. Soc. (3) 99 (2009), 461–483.Google Scholar
Cooper, S., ‘On the number of representations of integers by certain quadratic forms, II’, J. Comb. Number Theory 1 (2009), 53–82.Google Scholar
Cooper, S., ‘On Ramanujan’s function k (q) = r (q)r 2(q 2)’, Ramanujan J. 20 (2009), 311–328.Google Scholar
Cooper, S., ‘Level 10 analogues of Ramanujan’s series for 1/𝜋’, J. Ramanujan Math. Soc. 27 (2012), 59–76.Google Scholar
Cooper, S. and Hirschhorn, M., ‘Factorizations that involve Ramanujan’s function k (q) = r (q)r 2(q 2)’, Acta Math. Sin. (Engl. Ser.) 27 (2011), 2301–2308.Google Scholar
Dharmendra, B. N., Rajesh Kanna, M. R. and Jagadeesh, R., ‘On continued fraction of order twelve’, Pure Math. Sci. 1 (2012), 197–205.Google Scholar
Domb, C., ‘On the theory of cooperative phenomena in crystals’, Adv. Phys. 9 (1960), 149–361.Google Scholar
Lin, B. L. S., ‘On the expansion of a continued fraction of order 12’, Int. J. Number Theory 9 (2013), 2019–2031.Google Scholar
Mahadeva Naika, M. S., Chandankumar, S. and Bairy, K. S., ‘Some new identities for a continued fraction of order 12’, South East Asian J. Math. Math. Sci. 10 (2012), 129–140.Google Scholar
Mahadeva Naika, M. S., Dharmendra, B. N. and Shivashankara, K., ‘A continued fraction of order twelve’, Cent. Eur. J. Math. 6 (2008), 393–404.Google Scholar
Ramanujan, S., ‘Modular equations and approximations to 𝜋’, Q. J. Math. 45 (1914), 350–372; Reprinted in: Collected Papers of Srinivasa Ramanujan, Cambridge University Press, 1927; reprinted by Chelsea, New York, 1962; reprinted by the American Mathematical Society, Providence, RI, 2000.Google Scholar
Richmond, L. B. and Shallit, J., ‘Counting abelian squares’, Electron. J. Combin. 16 (2009), Research Paper 72.Google Scholar
Rogers, M. D., ‘New 5F 4 hypergeometric transformations, three-variable Mahler measures, and formulas for 1/𝜋’, Ramanujan J. 18 (2009), 327–340.CrossRefGoogle Scholar
Vasuki, K. R., Kahtan, A. A. A., Sharath, G. and Sathish Kumar, C., ‘On a continued fraction of order 12’, Ukrainian Math. J. 62 (2011), 1866–1878.CrossRefGoogle Scholar
Venkatachaliengar, K. and Cooper, S., Development of Elliptic Functions According to Ramanujan (World Scientific, Singapore, 2012).Google Scholar