The level 12 analogue of Ramanujan’s function k

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