3 0 N ov 2 00 9 A study of the associated linear problem for q

We consider the associated linear problem for a q-analogue of the fifth Painlevé equation (q-PV). We identify a lattice of connection preserving deformations in the space of the connection data for the linear problem with the lattice of translational Bäcklund transformations for q-PV, hence, show all translational Bäcklund transformations possess a Lax pair. We shall show that the big q-Laguerre polynomials, and a suitable generalization, solve a special case of the linear problem, hence, find solutions to q-PV in terms of determinants of Hankel matrices with entries consisting of rational or hypergeometric functions. Introduction The Painlevé equations were isolated in the classification of second order non-autonomous differential equations with no movable singularities except for poles [10]. For each Painlevé equation, there exists a system of linear ordinary differential equations with rational coefficients of the form (0.1) d dx Y (x) = A(x)Y (x), such that the Painlevé equation arises as necessary conditions so that the associated monodromy representation is preserved [11, 15, 21, 22, 23]. In addition to the unique continuous isomonodromic deformation of (0.1), Jimbo and Miwa [21] identify a class of discrete isomonodromic deformations that may be interpreted as translations of the monodromy data. These transformations of the linear problem, Y (x) → Ỹ (x), are induced by left multiplication with a rational matrix, R(x), via (0.2) Ỹ (x) = R(x)Y (x). The transformed matrix, Ỹ (x), solves (0.3) d dx Ỹ (x) = Ã(x)Ỹ (x), where (0.4) Ã(x) = ( dR(x) dx R(x) + R(x)A(x)R(x) ) . If Ã(x) is of the same fundamental form as A(x), then the unique continuous isomonodromic deformation of (0.3) specifies that the entries of Ã(x) are also solutions to the same Painlevé equation, with a possible change in parameters governed by Ã(x)’s monodromy data [21]. Therefore, we consider the transformation, (0.2), as inducing a Bäcklund transformation. We may consider 1 2 CHRISTOPHER M. ORMEROD this transformation as a system of difference equations in the “tilde” direction, hence, the resulting Bäcklund transformation may be thought of as the compatibility of a differential-difference system. Jimbo and Miwa state that all the Bäcklund transformations for the Painlevé equations arise in this manner [21]. The discrete Painlevé equations are second order non-autonomous difference equations that admit the Painlevé equations in a continuum limit [32]. There are three types of discrete Painlevé equation; additive, multiplicative and elliptic, which are classified according to how the parameters evolve [33]. The discrete Painlevé equations are integrable in the sense that they possess a discrete analogue of the Painlevé property, known as singularity confinement [17], and many may be solved via associated linear problems [18, 24, 29, 31]. Some of the additive discrete Painlevé equations arise as translational components of the continuous Painlevé equations, hence, (0.1) and (0.2) give rise to differential difference Lax-pairs for known Lax pairs for discrete Painlevé equations [21, 22, 23]. For discrete Painlevé equations of the multiplicative type, where the parameters change by some multiplicative constant, q ∈ C \ {0} and |q| 6 = 1, the associated linear problems are systems of linear q-difference equations [18, 24, 29, 31]. For systems of linear q-difference equations, represented by (0.5) Y (qx) = A(x)Y (x), where A(x) is a rational m×m matrix, we may associate a connection matrix [1, 3, 6, 35, 40]. A q-analogue of the sixth Painlevé equation was found to be equivalent to the conditions necessary for a certain deformation of a system, of the form (0.5), to preserve the associated connection matrix [24]. This condition is that deformed system is related via (0.2), where Ỹ (x) satisfies (0.6) Ỹ (qx) = Ã(x)Ỹ (x), and A(x) 7→ R(qx)A(x)R(x) . In light of Sakai’s framework [33], the discrete Painlevé equations represent just one translational component of the group of Bäcklund transformations. One should expect that other translational Bäcklund transformations are expressible as connection preserving deformations. The aim of this article is to develop this idea in the context of the associated linear problem for q-PV found by Murata [29]. The version of q-PV chosen is a translational component of a birational representation of an extended affine Weyl group of type A (1) 4 , which we denote T4, and is given by (0.7) T4 : { b0, b1, b2 b3, b4 ; f, g } → { b0 q , b1, b2 b3, qb4 ; f̃ , g̃ } , where q = 1/b0b1b2b3b4 and f̃ and g̃ are related to f and g via f̃f = b1b2 b3 1 − g (1 + b2g)(1 − b1b2g) , (0.8a) g̃g = b4b0 b2 (1 − b3f̃)(1 − f̃) f̃(b4 − qf̃) . (0.8b) We intend to show that this system is just one example of the group of connection preserving deformations. In doing so, we shall provide a basis for the lattice of connection preserving deformations, and hence, factorize the connection preserving deformation that defines the evolution of q-PV into more elementary connection preserving deformations. A STUDY OF THE ASSOCIATED LINEAR PROBLEM FOR q-PV 3 Just as one may consider special solutions of the q-Painlevé equations, we shall consider a special solution of the associated linear problem. It is the second aim of this paper to use techniques recently applied to q-PVI [12] to show that the big q-Laguerre polynomials satisfy form a vector solution to the associated linear problem for q-PV. Finding this particular solution of the associated linear problem gives rise to a family of special solutions of q-PV, which may be expressed in terms of variables associated with the orthogonal polynomial framework [12]. These results bear a remarkable similarity to the work of Masuda, who, for a different version of q-PV, forms determinental type solutions where the entries of the relevant matrix are continuous q-Laguerre polynomials [28]. However, this work adopts a significantly different approach to [28], as this approach is entirely based on the associated linear problem alone. 1. q-calculus and q-special functions We first introduce some preliminary material before proceeding. The first set of theory we require is the notion of a q-calculus, which is an difference analogue of regular calculus. For an extensive reference on the q-calculus, see [25]. Just as in the introduction, we fix a q ∈ C\{0}, where we will make the additional assumption that |q| < 1. This assumption holds for the remainder of the paper. We consider the q-difference operator (1.1) Dq,xf(x) = f(x) − f(qx) x(1 − q) . The q-differential analogues of the multiplication and quotient rule are Dq,xf(x)g(x) = f(x)Dq,xg(x) + g(qx)Dq,xf(x), (1.2a) = g(x)Dq,xf(x) − f(qx)Dq,xg(x), Dq,x f(x) g(x) = g(x)Dq,xf(x) − f(x)Dq,xg(x) g(x)g(qx) . (1.2b) If f is continuously differentiable, then lim q→1 Dq,xf(x) = d dx f(x) Associated with the q-derivative is the Jackson q-integral [38], written as (1.3) ∫ z 0 f(t)dqt := z(1 − q) ∞ ∑ n=0 f(zq)q. More generally, we write ∫ b a f(t)dqt = ∫ a 0 f(t)dqt − ∫ b 0 f(t)dqt. We define the q-analogue of the Pochhammer symbol as (a; q)0 = 1, (1.4a) (a; q)k = k−1 ∏ n=0 (1 − aq), (1.4b)