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The level 12 analogue of Ramanujan’s function \(k\). (English) Zbl 1404.11040

Summary: We provide a comprehensive study of the function \(h=h(q)\) defined by \[ h=q\mathop{\prod }_{j=1}^{\infty }\frac{(1-q^{12j-1})(1-q^{12j-11})}{(1-q^{12j-5})(1-q^{12j-7})} \] and show that it has many properties that are analogues of corresponding results for Ramanujan’s function \(k=k(q)\) defined by \[ k=q\mathop{\prod }_{j=1}^{\infty }\frac{(1-q^{10j-1})(1-q^{10j-2})(1-q^{10j-8})(1-q^{10j-9})}{(1-q^{10j-3})(1-q^{10j-4})(1-q^{10j-6})(1-q^{10j-7})}. \]

MSC:

11F11 Holomorphic modular forms of integral weight
05A10 Factorials, binomial coefficients, combinatorial functions
33C75 Elliptic integrals as hypergeometric functions
33E05 Elliptic functions and integrals
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