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We prove three new theta function identities for the continued fraction H ( q ) defined by $$\begin{aligned} H(q):=q^{1/8}-\frac{q^{7/8}}{1-q}_{+}\frac{q^2}{1+q^2}_{-}\frac{q^3}{1-q^3}_{+}\frac{q^4}{1+q^4}_{- \cdots }, \vert q\vert Click to show full abstract
We prove three new theta function identities for the continued fraction H ( q ) defined by $$\begin{aligned} H(q):=q^{1/8}-\frac{q^{7/8}}{1-q}_{+}\frac{q^2}{1+q^2}_{-}\frac{q^3}{1-q^3}_{+}\frac{q^4}{1+q^4}_{- \cdots }, \vert q\vert <1. \end{aligned}$$ H ( q ) : = q 1 / 8 - q 7 / 8 1 - q + q 2 1 + q 2 - q 3 1 - q 3 + q 4 1 + q 4 - ⋯ , | q | < 1 . The theta-function identities are then used to prove integral representations for the continued fraction H ( q ). We also prove general theorems and reciprocity formulas for the explicit evaluation of the continued fraction H ( q ). The results are analogous to those of the famous Rogers-Ramanujan continued fraction.
Journal Title: Afrika Matematika
Year Published: 2020
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