Editing Theta function (section)

=== Transformation at the fifth root of the nome ===
The [[Rogers-Ramanujan continued fraction]] can be defined in terms of the '''Jacobi theta function''' in the following way:

: <math>R(q) = \tan\biggl\{\frac{1}{2}\arctan\biggl[\frac{1}{2} - \frac{\theta _{4}(q)^2}{2\,\theta_{4}(q^5)^2}\biggr]\biggr\}^{1/5} \tan\biggl\{\frac{1}{2}\arccot\biggl[\frac{1}{2} - \frac{\theta_{4}(q)^2}{2\,\theta_{4}(q^5)^2}\biggr]\biggr\}^{2/5} </math>

: <math>R(q^2) = \tan\biggl\{\frac{1}{2}\arctan\biggl[\frac{1}{2} - \frac{\theta_{4}(q)^2}{2\,\theta_{4}(q^5)^2}\biggr]\biggr\}^{2/5} \cot\biggl\{\frac{1}{2}\arccot\biggl[\frac{1}{2} - \frac{\theta_{4}(q)^2}{2\,\theta_{4}(q^5)^2}\biggr]\biggr\}^{1/5} </math>

: <math>R(q^2) = \tan\biggl\{\frac{1}{2}\arctan\biggl[\frac{\theta_{3}(q)^2}{2\,\theta_{3}(q^5)^2} - \frac{1}{2}\biggr]\biggr\}^{2/5} \tan\biggl\{\frac{1}{2}\arccot\biggl[\frac{\theta_{3}(q)^2}{2\,\theta_{3}(q^5)^2} - \frac{1}{2}\biggr]\biggr\}^{1/5} </math>

The alternating Rogers-Ramanujan continued fraction function S(q) has the following two identities:

: <math>S(q) = \frac{R(q^4)}{R(q^2)R(q)} = \tan\biggl\{\frac{1}{2}\arctan\biggl [\frac{\theta_{3}(q)^2}{2\,\theta_{3}(q^5)^2} - \frac{1}{2}\biggr]\biggr\}^{1/5} \cot\biggl\{\frac{1}{2}\arccot\biggl[\frac{\theta_{3}(q)^2}{2\,\theta_{3}(q^5)^2} - \frac{1}{2}\biggr]\biggr\}^{2/5}</math>

The theta function values from the fifth root of the nome can be represented as a rational combination of the continued fractions R and S and the theta function values from the fifth power of the nome and the nome itself. The following four equations are valid for all values q between 0 and 1:

: <math>\frac{\theta_{3}(q^{1/5})}{\theta_{3}(q^5)} - 1 = \frac{1}{S(q)}\bigl[S(q)^2 + R(q^2)\bigr]\bigl[1 + R(q^2)S(q)\bigr] </math>
: <math>1 - \frac{\theta_{4}(q^{1/5})}{\theta_{4}(q^5)} = \frac{1}{R(q)}\bigl[R(q^2) + R(q)^2\bigr]\bigl[1 - R(q^2)R(q)\bigr] </math>
: <math>\theta_{3}(q^{1/5})^2 - \theta_{3}(q)^2 = \bigl[\theta_{3}(q)^2 - \theta_{3}(q^5)^2\bigr]\biggl[1+\frac{1}{R(q^2)S(q)}+R(q^2)S(q)+\frac{1}{R(q^2)^2}+R(q^2)^2+\frac{1}{S(q)}-S(q)\biggr] </math>
: <math>\theta_{4}(q)^2 - \theta_{4}(q^{1/5})^2 = \bigl[\theta_{4}(q^5)^2 - \theta_{4}(q)^2\bigr]\biggl[1-\frac{1}{R(q^2)R(q)}-R(q^2)R(q)+\frac{1}{R(q^2)^2}+R(q^2)^2-\frac{1}{R(q)}+R(q)\biggr] </math>