§32.10 Special Function Solutions

For certain combinations of the parameters, ${\text{P}}_{\text{II}}$–${\text{P}}_{\text{VI}}$ 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 ${\text{P}}_{\text{II}}$–${\text{P}}_{\text{VI}}$ that are expressible in terms of special functions satisfy a first-order equation of the form

where $F_j(w,z)$ is polynomial in $w$ with coefficients that are rational functions of $z$.

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

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

with solution

where

with $C_1$, $C_2$ arbitrary constants.

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

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

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

where ${\tau }_n(z)$ is the $n\times n$ Wronskian determinant

and

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

with $n\in \mathbb{Z}$, and ${\epsilon }_1=\pm 1$, ${\epsilon }_2=\pm 1$, independently. In the case ${\epsilon }_1\alpha +{\epsilon }_2\beta =2$, the Riccati equation is

If $\alpha \ne {\epsilon }_1$, then (32.10.12) has the solution

where

with $\zeta =\sqrt{{\epsilon }_1{\epsilon }_2}z$, $\nu =\frac{1}{2}\alpha {\epsilon }_1$, 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).

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

or

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

which has the solution

where

with $a=\alpha +\frac{1}{2}\epsilon$, and $C_1$, $C_2$ arbitrary constants. When $a+\frac{1}{2}$ is zero or a negative integer the $U$ parabolic cylinder functions reduce to Hermite polynomials (§18.3) times an exponential function; thus

and

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

where $C$ is an arbitrary constant and $\mathrm{erfc}$ 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).

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

or

where $n\in \mathbb{Z}$, $a={\epsilon }_1\sqrt{2\alpha }$, and $b={\epsilon }_2\sqrt{-2\beta }$, with ${\epsilon }_j=\pm 1$, $j=1,2,3$, independently. In the case when $n=0$ in (32.10.23), the Riccati equation is

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

where

with $\zeta ={\epsilon }_{3}z$, $\kappa =\frac{1}{2}(a-b+1)$, $\mu =\frac{1}{2}(a+b)$, and $C_1$, $C_2$ arbitrary constants.

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

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

where $n\in \mathbb{Z}$, $a={\epsilon }_1\sqrt{2\alpha }$, $b={\epsilon }_2\sqrt{-2\beta }$, $c={\epsilon }_3\sqrt{2\gamma }$, and $d={\epsilon }_4\sqrt{1-2\delta }$, with ${\epsilon }_j=\pm 1$, $j=1,2,3,4$, independently. If $n=1$, then the Riccati equation is

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

where

with $C_1$, $C_2$ arbitrary constants.

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

where the fundamental periods $2{\varphi }_1$ and $2{\varphi }_2$ are linearly independent functions satisfying the hypergeometric equation

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

with $C_1$, $C_2$ arbitrary constants. The solution (32.10.34) is an essentially transcendental function of both constants of integration since ${\text{P}}_{\text{VI}}$ with $\alpha =\beta =\gamma =0$ and $\delta =\frac{1}{2}$ 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).