§31.7(i) Reductions to the Gauss Hypergeometric Function

 31.7.1 $\mathop{{{}_{2}F_{1}}\/}\nolimits\!\left(\alpha,\beta;\gamma;z\right)=\mathop{% \mathit{H\!\ell}\/}\nolimits\!\left(1,\alpha\beta;\alpha,\beta,\gamma,\delta;z% \right)=\mathop{\mathit{H\!\ell}\/}\nolimits\!\left(0,0;\alpha,\beta,\gamma,% \alpha+\beta+1-\gamma;z\right)=\mathop{\mathit{H\!\ell}\/}\nolimits\!\left(a,a% \alpha\beta;\alpha,\beta,\gamma,\alpha+\beta+1-\gamma;z\right).$

Other reductions of $\mathop{\mathit{H\!\ell}\/}\nolimits$ to a $\mathop{{{}_{2}F_{1}}\/}\nolimits$, with at least one free parameter, exist iff the pair $(a,p)$ takes one of a finite number of values, where $q=\alpha\beta p$. Below are three such reductions with three and two parameters. They are analogous to quadratic and cubic hypergeometric transformations (§§15.8(iii)15.8(v)).

 31.7.2 $\mathop{\mathit{H\!\ell}\/}\nolimits\!\left(2,\alpha\beta;\alpha,\beta,\gamma,% \alpha+\beta-2\gamma+1;z\right)=\mathop{{{}_{2}F_{1}}\/}\nolimits\!\left(% \tfrac{1}{2}\alpha,\tfrac{1}{2}\beta;\gamma;1-(1-z)^{2}\right),$
 31.7.3 $\mathop{\mathit{H\!\ell}\/}\nolimits\!\left(4,\alpha\beta;\alpha,\beta,\tfrac{% 1}{2},\tfrac{2}{3}(\alpha+\beta);z\right)=\mathop{{{}_{2}F_{1}}\/}\nolimits\!% \left(\tfrac{1}{3}\alpha,\tfrac{1}{3}\beta;\tfrac{1}{2};1-(1-z)^{2}(1-\tfrac{1% }{4}z)\right),$
 31.7.4 $\mathop{\mathit{H\!\ell}\/}\nolimits\!\left(\tfrac{1}{2}+i\tfrac{\sqrt{3}}{2},% \alpha\beta(\tfrac{1}{2}+i\tfrac{\sqrt{3}}{6});\alpha,\beta,\tfrac{1}{3}(% \alpha+\beta+1),\tfrac{1}{3}(\alpha+\beta+1);z\right)=\mathop{{{}_{2}F_{1}}\/}% \nolimits\!\left(\tfrac{1}{3}\alpha,\tfrac{1}{3}\beta;\tfrac{1}{3}(\alpha+% \beta+1);1-\left(1-\left(\tfrac{3}{2}-i\tfrac{\sqrt{3}}{2}\right)z\right)^{3}% \right).$

For additional reductions, see Maier (2005). Joyce (1994) gives a reduction in which the independent variable is transformed not polynomially or rationally, but algebraically.

§31.7(ii) Relations to Lamé Functions

With $z={\mathop{\mathrm{sn}\/}\nolimits^{2}}\left(\zeta,k\right)$ and

 31.7.5 $\displaystyle a$ $\displaystyle=k^{-2},$ $\displaystyle q$ $\displaystyle=-\tfrac{1}{4}ah,$ $\displaystyle\alpha$ $\displaystyle=-\tfrac{1}{2}\nu,$ $\displaystyle\beta$ $\displaystyle=\tfrac{1}{2}(\nu+1),$ $\displaystyle\gamma$ $\displaystyle=\delta=\epsilon=\tfrac{1}{2},$

equation (31.2.1) becomes Lamé’s equation with independent variable $\zeta$; compare (29.2.1) and (31.2.8). The solutions (31.3.1) and (31.3.5) transform into even and odd solutions of Lamé’s equation, respectively. Similar specializations of formulas in §31.3(ii) yield solutions in the neighborhoods of the singularities $\zeta=\mathop{K\/}\nolimits$, $\mathop{K\/}\nolimits+i\mathop{{K^{\prime}}\/}\nolimits$, and $i\mathop{{K^{\prime}}\/}\nolimits$, where $\mathop{K\/}\nolimits$ and $\mathop{{K^{\prime}}\/}\nolimits$ are related to $k$ as in §19.2(ii).