# §32.10 Special Function Solutions

## §32.10(i) Introduction

For certain combinations of the parameters, $\mbox{P}_{\mbox{\scriptsize II}}$$\mbox{P}_{\mbox{\scriptsize VI}}$ have particular solutions expressible in terms of the solution of a Riccati differential equation, which can be solved in terms of special functions defined in other chapters. All solutions of $\mbox{P}_{\mbox{\scriptsize II}}$$\mbox{P}_{\mbox{\scriptsize VI}}$ that are expressible in terms of special functions satisfy a first-order equation of the form

 32.10.1 $(w^{\prime})^{n}+\sum_{j=0}^{n-1}F_{j}(w,z)(w^{\prime})^{j}=0,$ ⓘ Symbols: $n$: integer and $z$: real Permalink: http://dlmf.nist.gov/32.10.E1 Encodings: TeX, pMML, png See also: Annotations for §32.10(i), §32.10 and Ch.32

where $F_{j}(w,z)$ is polynomial in $w$ with coefficients that are rational functions of $z$.

## §32.10(ii) Second Painlevé Equation

$\mbox{P}_{\mbox{\scriptsize II}}$ has solutions expressible in terms of Airy functions (§9.2) iff

 32.10.2 $\alpha=n+\tfrac{1}{2},$ ⓘ Symbols: $n$: integer and $\alpha$: arbitrary constant Permalink: http://dlmf.nist.gov/32.10.E2 Encodings: TeX, pMML, png See also: Annotations for §32.10(ii), §32.10 and Ch.32

with $n\in\mathbb{Z}$. For example, if $\alpha=\tfrac{1}{2}\varepsilon$, with $\varepsilon=\pm 1$, then the Riccati equation is

 32.10.3 $\varepsilon w^{\prime}=w^{2}+\tfrac{1}{2}z,$ ⓘ Symbols: $z$: real and $\varepsilon=\pm 1$ Permalink: http://dlmf.nist.gov/32.10.E3 Encodings: TeX, pMML, png See also: Annotations for §32.10(ii), §32.10 and Ch.32

with solution

 32.10.4 $w(z;\tfrac{1}{2}\varepsilon)=-\varepsilon\phi^{\prime}(z)/\phi(z),$ ⓘ Symbols: $z$: real, $\varepsilon=\pm 1$ and $\phi(z)$: function Permalink: http://dlmf.nist.gov/32.10.E4 Encodings: TeX, pMML, png See also: Annotations for §32.10(ii), §32.10 and Ch.32

where

 32.10.5 $\phi(z)=C_{1}\mathrm{Ai}\left(-2^{-1/3}z\right)+C_{2}\mathrm{Bi}\left(-2^{-1/3% }z\right),$ ⓘ Symbols: $\mathrm{Ai}\left(\NVar{z}\right)$: Airy function, $\mathrm{Bi}\left(\NVar{z}\right)$: Airy function, $z$: real, $\phi(z)$: function and $C_{j}$: constants Referenced by: §32.10(ii) Permalink: http://dlmf.nist.gov/32.10.E5 Encodings: TeX, pMML, png See also: Annotations for §32.10(ii), §32.10 and Ch.32

with $C_{1}$, $C_{2}$ arbitrary constants.

Solutions for other values of $\alpha$ are derived from $w(z;\pm\tfrac{1}{2})$ by application of the Bäcklund transformations (32.7.1) and (32.7.2). For example,

 32.10.6 $w(z;\tfrac{3}{2})=\Phi-\dfrac{1}{2\Phi^{2}+z},$ ⓘ Symbols: $z$: real and $\Phi$ Permalink: http://dlmf.nist.gov/32.10.E6 Encodings: TeX, pMML, png See also: Annotations for §32.10(ii), §32.10 and Ch.32
 32.10.7 $w(z;\tfrac{5}{2})=\dfrac{1}{2\Phi^{2}+z}+\dfrac{2z\Phi^{2}+\Phi+z^{2}}{4\Phi^{% 3}+2z\Phi-1},$ ⓘ Symbols: $z$: real and $\Phi$ Permalink: http://dlmf.nist.gov/32.10.E7 Encodings: TeX, pMML, png See also: Annotations for §32.10(ii), §32.10 and Ch.32

where $\Phi=\phi^{\prime}(z)/\phi(z)$, with $\phi(z)$ given by (32.10.5).

More generally, if $n=1,2,3,\dots$, then

 32.10.8 $w(z;n+\tfrac{1}{2})=\frac{\mathrm{d}}{\mathrm{d}z}\left(\ln\left(\frac{\tau_{n% }(z)}{\tau_{n+1}(z)}\right)\right),$

where $\tau_{n}(z)$ is the $n\times n$ determinant

 32.10.9 $\tau_{n}(z)=\begin{vmatrix}\phi(z)&\phi^{\prime}(z)&\cdots&\phi^{(n-1)}(z)\\ \phi^{\prime}(z)&\phi^{\prime\prime}(z)&\cdots&\phi^{(n)}(z)\\ \vdots&\vdots&\ddots&\vdots\\ \phi^{(n-1)}(z)&\phi^{(n)}(z)&\cdots&\phi^{(2n-2)}(z)\end{vmatrix},$ ⓘ Symbols: $\det$: determinant, $n$: integer, $z$: real, $\phi(z)$: function and $\tau_{n}(z)$: determinant Permalink: http://dlmf.nist.gov/32.10.E9 Encodings: TeX, pMML, png See also: Annotations for §32.10(ii), §32.10 and Ch.32

and

 32.10.10 $w(z;-n-\tfrac{1}{2})=-w(z;n+\tfrac{1}{2}).$ ⓘ Symbols: $n$: integer and $z$: real Permalink: http://dlmf.nist.gov/32.10.E10 Encodings: TeX, pMML, png See also: Annotations for §32.10(ii), §32.10 and Ch.32

## §32.10(iii) Third Painlevé Equation

If $\gamma\delta\neq 0$, then as in §32.2(ii) we may set $\gamma=1$ and $\delta=-1$. $\mbox{P}_{\mbox{\scriptsize III}}$ then has solutions expressible in terms of Bessel functions (§10.2) iff

 32.10.11 $\varepsilon_{1}\alpha+\varepsilon_{2}\beta=4n+2,$ ⓘ Symbols: $n$: integer, $\varepsilon_{j}=\pm 1$, $\alpha$: arbitrary constant and $\beta$: arbitrary constant Permalink: http://dlmf.nist.gov/32.10.E11 Encodings: TeX, pMML, png See also: Annotations for §32.10(iii), §32.10 and Ch.32

with $n\in\mathbb{Z}$, and $\varepsilon_{1}=\pm 1$, $\varepsilon_{2}=\pm 1$, independently. In the case $\varepsilon_{1}\alpha+\varepsilon_{2}\beta=2$, the Riccati equation is

 32.10.12 $zw^{\prime}=\varepsilon_{1}zw^{2}+(\alpha\varepsilon_{1}-1)w+\varepsilon_{2}z.$ ⓘ Symbols: $z$: real, $\varepsilon_{j}=\pm 1$ and $\alpha$: arbitrary constant Referenced by: §32.10(iii) Permalink: http://dlmf.nist.gov/32.10.E12 Encodings: TeX, pMML, png See also: Annotations for §32.10(iii), §32.10 and Ch.32

If $\alpha\neq\varepsilon_{1}$, then (32.10.12) has the solution

 32.10.13 $w(z)=-\varepsilon_{1}\phi^{\prime}(z)/\phi(z),$ ⓘ Symbols: $z$: real, $\varepsilon_{j}=\pm 1$ and $\phi(z)$: function Permalink: http://dlmf.nist.gov/32.10.E13 Encodings: TeX, pMML, png See also: Annotations for §32.10(iii), §32.10 and Ch.32

where

 32.10.14 $\phi(z)=z^{\nu}\left(C_{1}J_{\nu}\left(\zeta\right)+C_{2}Y_{\nu}\left(\zeta% \right)\right),$

with $\zeta=\sqrt{\varepsilon_{1}\varepsilon_{2}}z$, $\nu=\tfrac{1}{2}\alpha\varepsilon_{1}$, and $C_{1}$, $C_{2}$ arbitrary constants.

For examples and plots see Milne et al. (1997). For determinantal representations see Forrester and Witte (2002) and Okamoto (1987c).

## §32.10(iv) Fourth Painlevé Equation

$\mbox{P}_{\mbox{\scriptsize IV}}$ has solutions expressible in terms of parabolic cylinder functions (§12.2) iff either

 32.10.15 $\beta=-2(2n+1+\varepsilon\alpha)^{2},$ ⓘ Symbols: $n$: integer, $\varepsilon=\pm 1$, $\alpha$: arbitrary constant and $\beta$: arbitrary constant Referenced by: §32.10(iv) Permalink: http://dlmf.nist.gov/32.10.E15 Encodings: TeX, pMML, png See also: Annotations for §32.10(iv), §32.10 and Ch.32

or

 32.10.16 $\beta=-2n^{2},$ ⓘ Symbols: $n$: integer and $\beta$: arbitrary constant Permalink: http://dlmf.nist.gov/32.10.E16 Encodings: TeX, pMML, png See also: Annotations for §32.10(iv), §32.10 and Ch.32

with $n\in\mathbb{Z}$ and $\varepsilon=\pm 1$. In the case when $n=0$ in (32.10.15), the Riccati equation is

 32.10.17 $w^{\prime}=\varepsilon(w^{2}+2zw)-2(1+\varepsilon\alpha),$ ⓘ Symbols: $z$: real, $\varepsilon=\pm 1$ and $\alpha$: arbitrary constant Referenced by: §32.10(iv) Permalink: http://dlmf.nist.gov/32.10.E17 Encodings: TeX, pMML, png See also: Annotations for §32.10(iv), §32.10 and Ch.32

which has the solution

 32.10.18 $w(z)=-\varepsilon\phi^{\prime}(z)/\phi(z),$ ⓘ Symbols: $z$: real, $\varepsilon=\pm 1$ and $\phi(z)$: function Permalink: http://dlmf.nist.gov/32.10.E18 Encodings: TeX, pMML, png See also: Annotations for §32.10(iv), §32.10 and Ch.32

where

 32.10.19 $\phi(z)=\left(C_{1}U\left(a,\sqrt{2}z\right)+C_{2}V\left(a,\sqrt{2}z\right)% \right)\exp\left(\tfrac{1}{2}\varepsilon z^{2}\right),$

with $a=\alpha+\tfrac{1}{2}\varepsilon$, and $C_{1}$, $C_{2}$ arbitrary constants. When $a+\tfrac{1}{2}$ is zero or a negative integer the $U$ parabolic cylinder functions reduce to Hermite polynomials (§18.3) times an exponential function; thus

 32.10.20 $w(z;-m,-2(m-1)^{2})=-\frac{H_{m-1}'\left(z\right)}{H_{m-1}\left(z\right)},$ $m=1,2,3,\dots$, ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$: Hermite polynomial, $m$: integer and $z$: real Permalink: http://dlmf.nist.gov/32.10.E20 Encodings: TeX, pMML, png See also: Annotations for §32.10(iv), §32.10 and Ch.32

and

 32.10.21 $w(z;-m,-2(m+1)^{2})=-2z+\frac{H_{m}'\left(z\right)}{H_{m}\left(z\right)},$ $m=0,1,2,\dots$. ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$: Hermite polynomial, $m$: integer and $z$: real Permalink: http://dlmf.nist.gov/32.10.E21 Encodings: TeX, pMML, png See also: Annotations for §32.10(iv), §32.10 and Ch.32

If $1+\varepsilon\alpha=0$, then (32.10.17) has solutions

 32.10.22 $w(z)=\begin{cases}\dfrac{2\exp\left(z^{2}\right)}{\sqrt{\pi}\left(C-i% \operatorname{erfc}\left(iz\right)\right)},&\varepsilon=1,\\ \dfrac{2\exp\left(-z^{2}\right)}{\sqrt{\pi}\left(C-\operatorname{erfc}\left(z% \right)\right)},&\varepsilon=-1,\end{cases}$

where $C$ is an arbitrary constant and $\operatorname{erfc}$ is the complementary error function (§7.2(i)).

For examples and plots see Bassom et al. (1995). For determinantal representations see Forrester and Witte (2001) and Okamoto (1986).

## §32.10(v) Fifth Painlevé Equation

If $\delta\neq 0$, then as in §32.2(ii) we may set $\delta=-\tfrac{1}{2}$. $\mbox{P}_{\mbox{\scriptsize V}}$ then has solutions expressible in terms of Whittaker functions (§13.14(i)), iff

 32.10.23 $a+b+\varepsilon_{3}\gamma=2n+1,$ ⓘ Symbols: $n$: integer, $a$, $b$, $\varepsilon_{j}=\pm 1$ and $\gamma$: arbitrary constant Referenced by: §32.10(v) Permalink: http://dlmf.nist.gov/32.10.E23 Encodings: TeX, pMML, png See also: Annotations for §32.10(v), §32.10 and Ch.32

or

 32.10.24 $(a-n)(b-n)=0,$ ⓘ Symbols: $n$: integer, $a$ and $b$ Permalink: http://dlmf.nist.gov/32.10.E24 Encodings: TeX, pMML, png See also: Annotations for §32.10(v), §32.10 and Ch.32

where $n\in\mathbb{Z}$, $a=\varepsilon_{1}\sqrt{2\alpha}$, and $b=\varepsilon_{2}\sqrt{-2\beta}$, with $\varepsilon_{j}=\pm 1$, $j=1,2,3$, independently. In the case when $n=0$ in (32.10.23), the Riccati equation is

 32.10.25 $zw^{\prime}=aw^{2}+(b-a+\varepsilon_{3}z)w-b.$ ⓘ Symbols: $z$: real, $a$, $b$ and $\varepsilon_{j}=\pm 1$ Referenced by: §32.10(v) Permalink: http://dlmf.nist.gov/32.10.E25 Encodings: TeX, pMML, png See also: Annotations for §32.10(v), §32.10 and Ch.32

If $a\neq 0$, then (32.10.25) has the solution

 32.10.26 $w(z)=-z\phi^{\prime}(z)/(a\phi(z)),$ ⓘ Symbols: $z$: real, $a$ and $\phi(z)$: function Permalink: http://dlmf.nist.gov/32.10.E26 Encodings: TeX, pMML, png See also: Annotations for §32.10(v), §32.10 and Ch.32

where

 32.10.27 $\phi(z)=\frac{C_{1}M_{\kappa,\mu}\left(\zeta\right)+C_{2}W_{\kappa,\mu}\left(% \zeta\right)}{\zeta^{(a-b+1)/2}}\exp\left(\tfrac{1}{2}\zeta\right),$

with $\zeta=\varepsilon_{3}z$, $\kappa=\tfrac{1}{2}(a-b+1)$, $\mu=\tfrac{1}{2}(a+b)$, and $C_{1}$, $C_{2}$ arbitrary constants.

For determinantal representations see Forrester and Witte (2002), Masuda (2004), and Okamoto (1987b).

## §32.10(vi) Sixth Painlevé Equation

$\mbox{P}_{\mbox{\scriptsize VI}}$ has solutions expressible in terms of hypergeometric functions (§15.2(i)) iff

 32.10.28 $a+b+c+d=2n+1,$ ⓘ Symbols: $n$: integer, $a$, $b$, $c$ and $d$ Permalink: http://dlmf.nist.gov/32.10.E28 Encodings: TeX, pMML, png See also: Annotations for §32.10(vi), §32.10 and Ch.32

where $n\in\mathbb{Z}$, $a=\varepsilon_{1}\sqrt{2\alpha}$, $b=\varepsilon_{2}\sqrt{-2\beta}$, $c=\varepsilon_{3}\sqrt{2\gamma}$, and $d=\varepsilon_{4}\sqrt{1-2\delta}$, with $\varepsilon_{j}=\pm 1$, $j=1,2,3,4$, independently. If $n=1$, then the Riccati equation is

 32.10.29 $w^{\prime}=\frac{aw^{2}}{z(z-1)}+\frac{(b+c)z-a-c}{z(z-1)}w-\frac{b}{z-1}.$ ⓘ Symbols: $z$: real, $a$, $b$ and $c$ Referenced by: §32.10(vi) Permalink: http://dlmf.nist.gov/32.10.E29 Encodings: TeX, pMML, png See also: Annotations for §32.10(vi), §32.10 and Ch.32

If $a\neq 0$, then (32.10.29) has the solution

 32.10.30 $\displaystyle w(z)$ $\displaystyle=\frac{\zeta-1}{a\phi(\zeta)}\frac{\mathrm{d}\phi}{\mathrm{d}% \zeta},$ $\displaystyle\zeta$ $\displaystyle=\frac{1}{1-z},$ ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative, $z$: real, $a$ and $\phi(z)$: function Permalink: http://dlmf.nist.gov/32.10.E30 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §32.10(vi), §32.10 and Ch.32

where

 32.10.31 $\phi(\zeta)=C_{1}F\left(b,-a;b+c;\zeta\right)+C_{2}\zeta^{-b+1-c}\*F\left(-a-b% -c+1,-c+1;2-b-c;\zeta\right),$

with $C_{1}$, $C_{2}$ arbitrary constants.

Next, let $\Lambda=\Lambda(u,z)$ be the elliptic function (§§22.15(ii), 23.2(iii)) defined by

 32.10.32 $u=\int_{0}^{\Lambda}\frac{\mathrm{d}t}{\sqrt{t(t-1)(t-z)}},$ ⓘ Symbols: $\mathrm{d}\NVar{x}$: differential, $\int$: integral, $z$: real and $\Lambda(u,z)$: elliptic function Permalink: http://dlmf.nist.gov/32.10.E32 Encodings: TeX, pMML, png See also: Annotations for §32.10(vi), §32.10 and Ch.32

where the fundamental periods $2\phi_{1}$ and $2\phi_{2}$ are linearly independent functions satisfying the hypergeometric equation

 32.10.33 $z(1-z)\frac{{\mathrm{d}}^{2}\phi}{{\mathrm{d}z}^{2}}+(1-2z)\frac{\mathrm{d}% \phi}{\mathrm{d}z}-\tfrac{1}{4}\phi=0.$ ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative, $z$: real and $\phi(z)$: function Permalink: http://dlmf.nist.gov/32.10.E33 Encodings: TeX, pMML, png See also: Annotations for §32.10(vi), §32.10 and Ch.32

Then $\mbox{P}_{\mbox{\scriptsize VI}}$, with $\alpha=\beta=\gamma=0$ and $\delta=\tfrac{1}{2}$, has the general solution

 32.10.34 $w(z;0,0,0,\tfrac{1}{2})=\Lambda(C_{1}\phi_{1}+C_{2}\phi_{2},z),$ ⓘ Symbols: $z$: real, $\Lambda(u,z)$: elliptic function, $\phi_{j}$: periods and $C$: constant Referenced by: §32.10(vi) Permalink: http://dlmf.nist.gov/32.10.E34 Encodings: TeX, pMML, png See also: Annotations for §32.10(vi), §32.10 and Ch.32

with $C_{1}$, $C_{2}$ arbitrary constants. The solution (32.10.34) is an essentially transcendental function of both constants of integration since $\mbox{P}_{\mbox{\scriptsize VI}}$ with $\alpha=\beta=\gamma=0$ and $\delta=\tfrac{1}{2}$ does not admit an algebraic first integral of the form $P(z,w,w^{\prime},C)=0$, with $C$ a constant.

For determinantal representations see Forrester and Witte (2004) and Masuda (2004).