REMARKS FOR BASIC APPELL SERIES

• Journal title : Honam Mathematical Journal
• Volume 31, Issue 4,  2009, pp.463-478
• Publisher : The Honam Mathematical Society
• DOI : 10.5831/HMJ.2009.31.4.463
Title & Authors
REMARKS FOR BASIC APPELL SERIES
Seo, Gyeong-Sig; Park, Joong-Soo;

Abstract
Let k be an imaginary quadratic field, ℌ the complex upper half plane, and let $\small{{\tau}{\in}k{\cap}}$ℌ, q = exp($\small{{\pi}i{\tau}}$). And let n, t be positive integers with $\small{1{\leq}t{\leq}n-1}$. Then $\small{q^{{\frac{n}{12}}-{\frac{t}{2}}+{\frac{t^2}{2n}}}{\prod}^{\infty}_{m=1}(1-q^{nm-t})(1-q^{nm-(n-t)})}$ is an algebraic number [10]. As a generalization of this result, we find several infinite series and products giving algebraic numbers using Ramanujan's $\small{_{1{\psi}1}}$ summation. These are also related to Rogers-Ramanujan continued fractions.
Keywords
Ramanujan theta function;quadratic field;algebraic number;Rogers-Ramanujan continued fraction;Appell series;
Language
English
Cited by
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