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New properties for the Ramanujan-Göllnitz-Gordon continued fraction. (English) Zbl 1254.11008
The Ramanujan-Göllnitz-Gordon continued fraction \(v(q)\) is defined by \[ v(q)=\frac{q^{1/2}}{1+q}\thickfrac\thickness{0}{}{+}\frac{q^2}{1+q^3}\thickfrac\thickness{0}{}{+}\frac{q^4}{1+q^5} \thickfrac\thickness{0}{}{+}\frac{q^6}{1+q^7}\cdots,\quad |q|<1. \] In is second notebook, Ramanujan recorded a product representation for \(v(q)\), \[ \;v(q)=q^{1/2}\frac{(q;q^8)_{\infty}(q^7;q^8)_{\infty}}{(q^3;q^8)_{\infty}(q^5;q^8)_{\infty}}, \] where \[ (a;q)_{\infty}=\prod_{n=0}^{\infty}(1-aq^n). \] This identity was later proved independently by H. Göllnitz [J. Reine Angew. Math. 225, 154–190 (1967; Zbl 0166.00803)] and B. Gordon [Duke Math. J. 32, 741–748 (1965; Zbl 0178.33404)]. Motivated by identities due to Ramanujan for the Rogers-Ramanujan continued fraction, and based on the product representation described above, the author establishes several new identities, and properties, for \(v(q)\), namely, formulas for \(1/\sqrt{v}\pm\alpha\sqrt{v},\,\,1/v\pm 2-v,\,\,1/v\pm 2\sqrt{2}+v,\,\,1/v-(\sqrt{2}-1)^2v,\,\,1/v-(\sqrt{2}+1)^2v,\) and \(1/v^2-6+v^2\), where \(\alpha=1,i,\sqrt{2}-1,\sqrt{2}+1\). A new proof of Hirschhorn’s 8-dissection identities for \(v(q)\) and \(1/v(q)\) [M. D. Hirschhorn, “On the expansion of a continued fraction of Gordon”, Ramanujan J. 5, No. 4, 369–375 (2001; Zbl 0993.30003)] is also presented.

MSC:
11A55 Continued fractions
11P83 Partitions; congruences and congruential restrictions
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