New Transformations for Painlevé’s Third Transcendent

We present transformations relating the third transcendent of Painlevé with parameter sets located at the corners of the Weyl chamber for the symmetry group of the system, the affine Weyl group of the root system B (1) 2 , to those at the origin. This transformation entails a scaling of the independent variable, and implies additive identities for the canonical Hamiltonians and product identities for the τ -functions with these parameter sets. A curious anomaly has existed in the theory of the Painlevé transcendents for many years, for almost a century in fact. This anomaly is the existence of a particular transformation for the second Painlevé transcendent q(t;α+ /2) which has no known analogue for any of the other transcendents. One recalls the second Painlevé transcendent q = q(t;α+ /2) is by definition a solution of the PII differential equation (1) dq dt2 = 2q + tq + α , with parameter α ∈ C. The transformation, found by Gambier in 1909 [5], relates solutions of the PII transcendent with parameter values at v1 := α + /2 = 0 and v1 = /2 (also between the solutions at v1 = /2 and v1 = 1 but this can be constructed from the previous one and the Bäcklund transformation v1 7→ v1+1). The Bäcklund symmetries of PII are a realisation of the affine Weyl group associated with the root system A 1 , and using v1 7→ v1 + 1, shift v1 7→ −v1, reflection, (2) transformations of this group, solutions for arbitrary v1 can be related to a solution whose parameter v1 ∈ (0, /2] (or in [/2, 1)). This interval defines the Weyl chamber and the above transformation in question relates the solutions at the endpoints of this interval (see Fig. 1). Specifically the transformation takes the form T = −2−1/3t 2Q(T ) = dq(t) dt + q(t) + /2t , (3) where q(t) := q(t; 0), Q(T ) := q(T ; /2) and its inverse is t = −2T q(t) = −2−1/3 1 Q(T ) dQ(T ) dT . (4) Received by the editors January 26, 2002 and, in revised form, XXX XX, 2002. 1991 Mathematics Subject Classification. Primary 34M55, 33E17; Secondary 20F55.