A large family of classical special functions are associated with a linear ordinary differential equation with polynomial or rational coefficients. Typical examples are Gauss' hypergeometric functions, Kummer's confluent hypergeometric functions, and various special functions with the name of Airy, Bessel, Hermite, etc. These special functions satisfy a second order linear ordinary differential equations with at most two or three poles on the Riemann sphere.

Extending this family is obviously rather straightforward: Just increase the number and/or order of poles of the coefficients, the order of the equation itself, and even the number of variables. As the general theory of linear ordinary equations in the complex domain was refined by B. Riemann and L. Fuchs, this line of approach to special functions flourished throughout the second half of the 19th century.

Another family of special functions, also
discovered in the 19th century, are elliptic
functions and their relatives, i.e., theta
functions and modular functions
click **here**
for the history of elliptic functions).
Extending this family of special functions
was a central issue of mathematics in the 19th
century. Riemann studied this issue, too,
and established the theory of Abelian
functions in his his geometric framework.
.

Painlevé's work may be thought of as a combination of these two lines of approaches to special functions. A clue is the fact that elliptic functions are also characterized by a differential equation. For instance, the Weierstrass p-function is well known to satisfy the first order differential equation

Jacobi's three elliptic functions, too, satisfy
a first order system of differential equations.
that single out the solution. (These equations
are essentially the same as the equations of
motion of the Euler-Poinsot rigid body).
Unlike the hypergeometric functions etc.,
however, these differential equations are
*nonlinear*. Just like those elliptic
functions, Painlevé aimed to define
new special functions as solutions of a
*suitable* nonlinear differential equation.

A question emerges here: What kind of nonlinear differential equations are indeed ``suitable''? A guiding principle can be found in the differential equations that the elliptic functions satisfy. One of the difficulties in analyzing a nonlinear differential equation lies in the fact that the position of singularities of solutions can depend on the solution under consideration (or, equivalently, on the initial values). Such singularities are referred to as ``movable singularities''. Furthermore, solutions of a nonlinear differential equation, in general, can have very bad singularities, such as natural boundaries. If movable singularities are inevitable, the best possible option will be to require that those singularities be as simple as possible. This is indeed the case for the differential equations of elliptic functions: All singularities of elliptic functions are poles, though these poles are ``movable'' in the aforementioned meaning.

Bearing these facts in mind, Panlevé proposed to classify all second order differential equation of the form

d^{2}ydx ^{2} |
= R(x,y, | dy dx |
) |

Actually, the irreducibility of the Painlevé
equations was controversial in the days of
Painlevé. R. Liouville argued, without
showing any proof, that the first Painlevé
was reducible to a forth order linear ordinary
differential equation. Painlevé counterattacked
and entered the controversy. This issue remained
open for many years, and was finally settled
in the last decade by Japanese mathematicians
(**K. Nishioka, H. Umemura, M. Noumi, and Y. Murata**).

The concept of Painlevé's work is also
very similar to S. Kovalevskaya's classification
of ``integrable'' rigid bodies
(click **here**
for the history of integrable systems including
Kovalevskaya's work).
Kovalevskaya's method was to determine
the cases where all solutions of the equations
of motion have no singularity other than poles.
This led to the discovery of the third (and final)
example of integrable rigid bodies, which is
now called the Kovalevskaya top.

P. Painlevé |
Memoire sur les équations différentielles
dont l'intégrale générale est uniforme,
Bull. Soc. Math. Phys. France
28 (1900), 201-261. |

Sur les équations différentielles
du second ordre et d'ordre supérieur dont
l'intégrale générale est uniforme,
Acta Math. 21 (1902), 1-85. | |

Painlevé reported his results in these papers. The first three of the Painlevé equations were presented here, but the rest were overlooked. |

B. Gambier |
Sur les équations différentielles
du second ordre et du premier degré dont
l'intégrale générale est à
points critique fixés,
Acta. Math. 33 (1910), 1-55. |

Gambier reexamined Painlevé's previous calculations, and added the remaining three equations. Painlevé's classification program was thus completed. |

Chazy attempted to generalize the work of Painlevé and Gambier to third order differential equations. Chazy's work is closely related to the theory of modular functions. Modular functions are an important family of special functions that satisfy a third order differential quation. Actually they do not have the Painlevé property, because modular functions have movable natural boundaries. Chazy discovered a third order differential equation with the Painlevé, solutions of which are of course not modular, but obeys nearly modular transformations.

J. Chazy |
Sur les équations différentielles
dont l'intégrale générale possede
un coupure essentielle mobile,
C.R. Acad. Sci. (Paris) 150 (1910), 456-458. |

Sur les équations différentielles
de troisième ordre et d'ordre supérieur
dont l'ntégrale générale a ses
points critiques fixés,
Acta Math. 33 (1911), 317-385. |

Chazy's third order equation has a special solution that is also related to the sixth Painlevé equation. This special solution is related to complete period integrals (thereby to modular functions as well). The so called ``Picard solution'' of the sixth Painlevé equation is also realized by complete elliptic integrals and an elliptic function. These special solutions of the sixth Painlevé equations are linked with the following work of R. Fuchs:

R. Fuchs |
Sur quelques équations différentielles
linéaires du second ordre,
C. R. Acad. Sci. (Paris) 141 (1905), 555-558. |

Über lineare homogene Differentialgleichungen
zweiter Ordnung mit drei im Endlichen gelegene
wesentlich singuläre Stellen,
Math. Ann. 63 (1907), 301-321. |

R. Fuchs (son of L. Fuchs) reported two significant discoveries in this paper.

The first discovery is a new interpretation of the sixth Painlevé equation in the languague of monodromy. He took a second order Fuchsian linear ordinary differential equation on the Riemann sphere, and considered the deformations of this lienar equation that leave invariant the monodromy around regular singular points. Such deformations are nowadays called ``isomonodromic deformations''. Fuchs noticed that those deformations were described by a nonlinear differential equation in suitably selected variables, and pointed out that this nonlinear equation is nothing but the sixth Painlevé equation.

Another discovery is to rewrite the sixth Painlevé equation in terms of an incomplete elliptic integral. In a sense, this is an extension of the Picard-Fuchs equation for complete elliptic integrals. Inspired by this remark, Painlevé proposed yet another expression of his sixth equation using an elliptic function:

P. Painlevé |
Sur les équations différentielles
du second ordre à point critiques fixes,
C.R. Acad. Sci. (Paris) 143 (1906), 1111-1117. |

These expression of Fuchs and Painlevé revived
in **Yu. Manin**'s work more than ninety years later
(Yu. I. Manin,
Sixth Painlevé equation, universal elliptic curve,
and mirror of P^{2}, AMS Transl. (2) 186 (1998),
131-151).

The discovery of isomonodromic deformations indicated a new direction for extending the Painlevé equations. Those extensions were soon attempted by L. Schlesinger and R. Garnier:

L. Schlesinger |
Über eine Klasse von Differentialsystemen
beliebliger Ordnumg mit festen kritischer Punkten,
J. fÜr Math. 141 (1912), 96-145. |

R. Garnier |
Sur des équations différentielles
du troisième ordre dont l'intégrale est
uniform et sur une classe d'équations
nouvelles d'ordre supérieur dont l'intégrale
générale a ses point critiques fixés,
Ann. Sci. de l'ENS 29 (1912), 1-126. |

Etudes de l'intégrale générale
de l'équation VI de M. Painlevé dans le
voisinage de ses singularité transcendentes,
Ann. Sci. Ecole Norm. Sup. (3) 34 (1917), 239-353. |

Garnier also noted that an integrable system, solvable by hyoerelliptic Abelian functions, can be obtained by slightly modifying the setup of Schlesinger's isomonodromic deformations:

R. Garnier |
Sur une classe de systèmes differentiels
abéliens deduits de la théorie
des équations linéaires,
Rend. Circ. Mat. Palermo 43 (1918-19), 155-191. |

This paper of Garnier is one of the earliest papers
that pointed out a hidden relation between integrable
systems and equations of Painlevé type
(click here
for the history of integrable systems). |

The point of view of isomonodromic deformations
experienced a dramatic revival due to
**M. Jimbo, T. Miwa, Y. Môri and M. Sato**
in the late seventies
(M. Sato, T. Miwa and M. Jimbo,
Holonomic quantum fields II,
Publ. RIMS, Kyoto Univ., 15 (1979), 201-27;
M. Jimbo, T. Miwa, Y. Môri and M. Sato,
Density matrix of an impenetrable Bose gas and
the fifth Painlevé transcendent,
Physica 1D (1980), 80-158) soon after
**T.T. Wu, B.M. McCoy, C.A. Tracy and
E. Barouch** encountered the Painlevé
equation in the Ising model of statistical physics
(E. Barouch, B.M. McCoy and T.T. Wu,
Zero-field susceptibility of the two-dimensional
Ising model near T_{c}, Phys. Rev. Lett. 31
(1973), 1409-1411;
T.T. Wu, B.M. McCoy, C.A. Tracy and E. Barouch,
Spin-spin correlation functions for the
two-dimensional Isong model, Phys. Rev. B13
(1976), 316-374).

Painlevé himself pursued his equations in a different directions, in particular, in the language of algebraic geometry of surfaces. The famous Stockholm lectures was intended to present such ideas.

P. Painlevé |
Leçons de Stockholm, Oeuvre t.I
(C.N.R.S., Paris, 1972). |

The algebro-geometric point of view is also a keynote
of recent researches on the Painlevé equations,
which originate in **K. Okamoto**'s work in the late
seventies (K. Okamoto, Sur les feuilletages associés aux
équations du second ordre à points critiques fixes
de P. Painlevé, Japan. J. Math. 5 (1979), 1-79).