§31.7 Relations to Other Functions

Permalink:: http://dlmf.nist.gov/31.7

§31.7(i) Reductions to the Gauss Hypergeometric Function
§31.7(ii) Relations to Lamé Functions

§31.7(i) Reductions to the Gauss Hypergeometric Function

Keywords:: Heun functions, hypergeometric function
Permalink:: http://dlmf.nist.gov/31.7.i

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

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\mathop{\mathit{H\!\ell}\/}\nolimits\!\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$ : Heun functions, $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $a$ : complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/31.7.E1
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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 $\mathop{\mathit{H\!\ell}\/}\nolimits\!\left(2,\alpha\beta;\alpha,\beta,\gamma,\alpha+\beta-2\gamma+1;z\right)=\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left(\tfrac{1}{2}\alpha,\tfrac{1}{2}\beta;\gamma;1-(1-z)^{2}\right),$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\mathop{\mathit{H\!\ell}\/}\nolimits\!\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$ : Heun functions, $z$ : complex variable, $\gamma$ : real or complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/31.7.E2
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31.7.3 $\mathop{\mathit{H\!\ell}\/}\nolimits\!\left(4,\alpha\beta;\alpha,\beta,\tfrac{1}{2},\tfrac{2}{3}(\alpha+\beta);z\right)=\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left(\tfrac{1}{3}\alpha,\tfrac{1}{3}\beta;\tfrac{1}{2};1-(1-z)^{2}(1-\tfrac{1}{4}z)\right),$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\mathop{\mathit{H\!\ell}\/}\nolimits\!\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$ : Heun functions, $z$ : complex variable, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/31.7.E3
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31.7.4 $\mathop{\mathit{H\!\ell}\/}\nolimits\!\left(\tfrac{1}{2}+i\tfrac{\sqrt{3}}{2},\alpha\beta(\tfrac{1}{2}+i\tfrac{\sqrt{3}}{6});\alpha,\beta,\tfrac{1}{3}(\alpha+\beta+1),\tfrac{1}{3}(\alpha+\beta+1);z\right)=\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left(\tfrac{1}{3}\alpha,\tfrac{1}{3}\beta;\tfrac{1}{3}(\alpha+\beta+1);1-\left(1-\left(\tfrac{3}{2}-i\tfrac{\sqrt{3}}{2}\right)z\right)^{3}\right).$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\mathop{\mathit{H\!\ell}\/}\nolimits\!\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$ : Heun functions, $z$ : complex variable, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/31.7.E4
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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

Notes:: See Ronveaux (1995, Part A, p. 10).
Keywords:: Heun functions, Lamé functions
Permalink:: http://dlmf.nist.gov/31.7.ii

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

31.7.5

$a=k^{{-2}},$

$q=-\tfrac{1}{4}ah,$

$\alpha=-\tfrac{1}{2}\nu,$

$\beta=\tfrac{1}{2}(\nu+1),$

$\gamma=\delta=\epsilon=\tfrac{1}{2},$

Symbols:: $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $\nu$ : real or complex parameter, $a$ : complex parameter, $q$ : real or complex parameter, $\alpha$ : real or complex parameter, $\beta$ : real or complex parameter and $k$ : modulus
Permalink:: http://dlmf.nist.gov/31.7.E5
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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).

§31.7 Relations to Other Functions

Contents

§31.7(i) Reductions to the Gauss Hypergeometric Function

§31.7(ii) Relations to Lamé Functions