# §32.10 Special Function Solutions

## §32.10(i) Introduction

For certain combinations of the parameters,  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  that are expressible in terms of special functions satisfy a first-order equation of the form

32.10.1

where is polynomial in with coefficients that are rational functions of .

## §32.10(ii) Second Painlevé Equation

has solutions expressible in terms of Airy functions (§9.2) iff

32.10.2

with . For example, if , with , then the Riccati equation is

32.10.3

with solution

32.10.4

where

32.10.5

with , arbitrary constants.

Solutions for other values of are derived from by application of the Bäcklund transformations (32.7.1) and (32.7.2). For example,

32.10.6
32.10.7

where , with given by (32.10.5).

More generally, if , then

where is the determinant

32.10.9

and

32.10.10

## §32.10(iii) Third Painlevé Equation

If , then as in §32.2(ii) we may set and .  then has solutions expressible in terms of Bessel functions (§10.2) iff

with , and , , independently. In the case , the Riccati equation is

32.10.12

If , then (32.10.12) has the solution

32.10.13

where

with , , and , 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

has solutions expressible in terms of parabolic cylinder functions (§12.2) iff either

32.10.15

or

32.10.16

with and . In the case when in (32.10.15), the Riccati equation is

32.10.17

which has the solution

32.10.18

where

with , and , arbitrary constants. When is zero or a negative integer the parabolic cylinder functions reduce to Hermite polynomials (§18.3) times an exponential function; thus

32.10.20,

and

32.10.21.

If , then (32.10.17) has solutions

where is an arbitrary constant and 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 , then as in §32.2(ii) we may set .  then has solutions expressible in terms of Whittaker functions (§13.14(i)), iff

32.10.23

or

32.10.24

where , , and , with , , independently. In the case when in (32.10.23), the Riccati equation is

32.10.25

If , then (32.10.25) has the solution

32.10.26

where

with , , , and , arbitrary constants.

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

## §32.10(vi) Sixth Painlevé Equation

has solutions expressible in terms of hypergeometric functions (§15.2(i)) iff

32.10.28

where , , , , and , with , , independently. If , then the Riccati equation is

32.10.29

If , then (32.10.29) has the solution

where

32.10.31

with , arbitrary constants.

Next, let be the elliptic function (§§22.15(ii), 23.2(iii)) defined by

where the fundamental periods and are linearly independent functions satisfying the hypergeometric equation

Then , with and , has the general solution

32.10.34

with , arbitrary constants. The solution (32.10.34) is an essentially transcendental function of both constants of integration since  with and does not admit an algebraic first integral of the form , with a constant.

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