## Abstract

In this paper, we present a new method of deducing infinite sequences of exact solutions of *q*-discrete Painlevé equations by using their associated linear problems. The specific equation we consider in this paper is a *q*-discrete version of the second Painlevé equation (*q*-P_{II}) with affine Weyl group symmetry of type (*A*_{2}+*A*_{1})^{(1)}. We show, for the first time, how to use the *q*-discrete linear problem associated with *q*-P_{II} to find an infinite sequence of exact rational solutions and also show how to find their representation as determinants by using the linear problem. The method, while demonstrated for *q*-P_{II} here, is also applicable to other discrete Painlevé equations.

## 1. Introduction

We consider the *q*-discrete second Painlevé equation (Ramani & Grammaticos 1996),
1.1
where *g* is a function of *x*=*x*_{0}*q*^{−n}, and *α*, *q*≠1 are constant parameters. In the continuum limit, *g*(*x*)=1/2(1−*y*(*t*)*ϵ*), *t*=*nϵ*, (with *q*^{2}=(1+*ϵ*^{3}/2)^{−1}, , and *α*=(1+*aϵ*^{3})), this equation becomes the second Painlevé equation P_{II}: *y*_{tt}=2*y*^{3}+*ty*−*a*.

The six classical Painlevé equations (denoted *P*_{I}—*P*_{VI}) were discovered by Painlevé and co-workers in their search for second-order ordinary differential equations (ODEs) with solutions that can be globally continued in the complex plane (Painlevé 1902; Fuchs 1905; Gambier 1910). There has been significant modern interest in these equations owing to their role as physically relevant models in a wide range of applications. They are regarded as modern nonlinear special functions (Deift *et al.* 2010) and have attracted a great deal of attention as universal models in random matrix theory (Deift 2007).

In recent times, there has been enormous interest in discrete versions of the Painlevé equations. The first of these discrete versions were discovered in orthogonal polynomial theory (Shohat 1939) and, more recently, in the study of the partition function in a two-dimensional model of quantum gravity (Its *et al.* 1990). Many of the remaining discrete Painlevé equations were discovered by using the singularity confinement property (Ramani *et al.* 1991), which was regarded as a discrete manifestation of the property used by Painlevé and his colleagues in the context of ODEs. For each continuous Painlevé equation, there are now known to be many integrable second-order discrete versions.

The plethora of such discrete equations was resolved by Sakai’s classification (Sakai 2001) of rational surfaces obtained from a nine-point blow-up of the complex projective plane. For the continuous Painlevé equations, it was Okamoto (1979) who first showed that such surfaces arise from the compactification and regularization of their initial-value spaces. Sakai extended this to discrete equations and showed that the discrete Painlevé equations arise as the Cremona transformations of rational surfaces. In his classification, Sakai found all the known equations as well as a new class of discrete equations in which the iterations occur on an elliptic curve. Moreover, he showed that the classical Painlevé equations arise as degenerate limits of the discrete ones in this geometrical scheme. The particular equation *q*-P_{II} described above is identified in Sakai’s classification as an equation with affine Weyl group symmetry of type (*A*_{2}+*A*_{1})^{(1)}.

The general solutions of the Painlevé equations are highly transcendental functions, called the Painlevé transcendents, which cannot be expressed in terms of earlier known classical special functions, such as hypergeometric functions or elliptic functions. This is also believed to be the case for general solutions of discrete Painlevé equations. However, there exist countable sets of parameter values for which *P*_{II}—*P*_{VI} possess special solutions that can be expressed in terms of earlier known special functions. *P*_{I} does not have any special solutions as there is no parameter in its equation.

*P*_{II}—*P*_{VI} admit hypergeometric-type special solutions in terms of Airy (Airault 1979), Bessel (Lukaševič 1967), parabolic cylinder (Gromak 1987), Whittaker (Gromak & Tsegel’nik 1988) and Gauss hypergeometric (Lukaševič & Yablonskiĭ 1967) functions, respectively. There also exist algebraic special solutions, which are expressed in terms of certain distinguished polynomials. *P*_{II} is associated with Yablonskii–Vorobiev polynomials (Yablonskii 1959; Vorobiev 1965), *P*_{IV} with Okamoto polynomials (Okamoto 1986) and *P*_{III}, *P*_{V} and *P*_{VI} with Umemura polynomials (Umemura 1996).

Discrete Painlevé equations are also known to have exact solutions of either algebraic or hypergeometric type. Interestingly, both rational and hypergeometric types of special solutions of continuous and discrete Painlevé equations are expressible in terms of determinants. There are deep similarities between these expressions for the continuous and discrete cases. However, the continuous case and the discrete case differ in one important aspect. There are several methods known for finding determinantal expressions for exact solutions of the continuous Painlevé equations. But only one method appears to be known for finding determinantal expressions of exact solutions of discrete Painlevé equations. In the latter case, the single known method is based on the bilinear formalism.

To take one example from the theory of continuous Painlevé equations, Flaschka & Newell (1980) provided a comprehensive study of the solutions of *P*_{II} by using its 2×2 linear iso-monodromy problem (obtained from a similarity reduction of the modified Korteweg-de Vries equation). They provided the complete family of exact rational and Airy-type special solutions (for special cases for the parameter *a*) that were already known earlier from Airault’s work (Airault 1979) as well as the determinantal structure of these solutions. In contrast, the determinantal expressions of special solutions of a *q*-discrete Painlevé equation that has the same affine Weyl group structure as *q*-P_{II} equation (1.1) has only been found through the usage of the bilinear framework (Nakao *et al.* 1998; Kajiwara 2003; Kajiwara & Kimura 2003; Kajiwara *et al.* 2011).

What is appealing about Flaschka and Newell’s technique is that by exploring the connection between the Painlevé equation and its associated linear system, the determinantal structure of the hierarchies of special solutions of the Painlevé equation emerges naturally. This is true for both rational and hypergeometric-type special solutions. The approach of studying the Painlevé equation via its associated linear system was also employed by Dubrovin & Mazzocco (2000), who found new special solutions of *P*_{VI}. The formulation of this method was particularly important for *P*_{VI}, because, at that time, completeness of special solutions was not known. For discrete Painlevé equations, completeness of special solutions is also not known.

We were motivated by such studies to gain information about the solutions of *q*-P_{II} equation (1.1) by starting from its linear problem. In this paper, we use a 2×2 *q*-discrete iso-monodromy deformation problem (equation 2.1) derived from a similarity reduction of the lattice modified Korteweg-de Vries equation (Hay *et al.* 2007) and knowledge of the Bäcklund transformation of *q*-P_{II}, which relates solutions corresponding to different values of *α* (Joshi *et al.* 1998). Our starting point is the asymptotic analysis of the linear problem in the complex plane of the auxiliary variable *ν* (often called the spectral or monodromy variable), which can be used to relate the solutions of the linear problem to solutions of *q*-P_{II}. For a special case, we show how to solve the linear problem exactly. We also find the Schlesinger transformation which allows us to iterate this special case to obtain an infinite sequence of solutions of the linear problem. This sequence naturally leads to the determinantal representation of associated solutions of *q*-P_{II} proved in theorem 4.1. This is an other determinantal expression for the same hierarchy of rational special solutions of a *q*-discrete equation of the same affine Weyl type (*A*_{2}+*A*_{1})^{(1)} found earlier by the bilinear method in Kajiwara (2003) and Kajiwara *et al.* (2011). The correspondence between the two different determinantal forms is interesting and non-trivial. The limitation of space prevents us from providing explicit details about it here. A similar sequence of steps can be pursued for hypergeometric-type special solutions. The latter case will be presented in a separate paper.

The paper is organized as follows. In §2, we carry out the asymptotic analysis of the linear problem (2.1a) in the complex *ν*-plane and study the relationship between the resulting formal asymptotic solutions and the solutions of the *q*-P_{II} equation. We also solve the problem explicitly for a special case, labelled as *k*=0. In §3, we find the Schlesinger transformation that relates solutions of the iso-monodromy deformation problem for successive integer values of *k*. In §4, making use of the Schlesinger transformation, we obtain solutions of the iso-monodromy deformation problem for integer *k* in a closed form and derive the determinantal representation of the solutions of *q*-P_{II} equation corresponding to this case. Finally in §5, we discuss our results.

## 2. *q*-discrete linear analysis

All discrete Painlevé equations are known to be discrete iso-monodromy conditions for associated linear problems (Murata 2009). For *q*-P_{II}, we use the linear problem (Hay *et al.* 2007):
2.1a
and
2.1b
where , and
Note that *ν* is the *q*-discrete monodromy variable (sometimes called spectral variable owing to its origins in the inverse scattering method of solution for partial difference equations), while *e*_{1}, *e*_{2}, *α*=*e*_{1}/*e*_{2} are constant parameters. The entries of *A* are functions of *g*(*x*) and *x* given by
2.2a
2.2b
2.2c
2.2d
2.2e
and
2.2f
The compatibility condition of *q*-linear systems (2.1a) and (2.1b) is
which forces *g*(*x*) to satisfy a second-order nonlinear *q*-discrete equation, namely the *q*-P_{II} equation (1.1).

Let *α*=*α*_{k}:=1/*q*^{4k} and denote the corresponding solutions of *q*-P_{II} as *g*_{k}(*x*). It is easy to see that equation (1.1) admits a simple rational-type special solution for *k*=0, *α*_{0}=1:
2.3
The Bäcklund transformation of *q*-P_{II} (Joshi *et al.* 1998) is given by
2.4
This transformation produces a solution of equation (1.1) with parameter *α*_{k+1}=1/*q*^{4(k+1)} from a solution corresponding to parameter *α*_{k}=1/*q*^{4k}. Applying the Bäcklund transformation (2.4) on the rational solution (2.3), we obtain a hierarchy of rational-type special solutions of *q*-P_{II} for *α*_{k}=1/*q*^{4k}, where *k* is an integer.

The coefficient matrix *A*(*ν*,*x*) of the equation (2.1a) has polynomial dependence on its variable *ν*, whereas the deformation equation (2.1b) depends transcendentally on its variable *x*, via *g*_{k}(*x*) the solution of *q*-P_{II} equation. For this reason, we concentrate our analysis on the first half of the Lax pair, which defines the evolution of *Ψ*(*ν*,*x*) in the *ν*-plane.

Recall the theorem of Carmichael (1912) on the analysis of *q*-linear systems with polynomial coefficients.

### Theorem 2.1 (Carmichael 1912)

*Consider the n×n q-discrete linear system
*
2.5
*Assume that q*^{θj} *and q*^{ρj}*, j=1,…,n, are eigenvalues of A*_{0} *and A*_{μ}*, respectively, such that for i≠j, none of θ*_{i}*−θ*_{j}*, ρ*_{i}*−ρ*_{j} *are integer. Then the system (2.5) has fundamental matrix solutions Y* _{0}*(ξ),* *given by
*
2.6
*where* *and (ϵ _{ij})*

_{1≤i,j≤n}

*and (δ*

_{ij})_{1≤i,j≤n}

*are n×n matrices of analytical functions which can be expanded as a power series in ξ or 1/ξ around ξ=0 and*

*, respectively.*

We rely on this theorem in the following analysis.

### (a) Expansion in the neighbourhood of *ν*=0

### Proposition 2.2

*For* *e*_{1}/*e*_{2}=*α*_{k}≠1/*q*^{4k}, *k* *is an integer or half integer, there exists a fundamental solution matrix* *Φ*(*ν*,*x*)={*ϕ*_{1}(*ν*,*x*),*ϕ*_{2}(*ν*,*x*)} *of the* *q*-*linear systems (2.1) in the neighbourhood of* *ν*=0,
2.7
*and*
2.8
*where* *and*
2.9a
2.9b
2.9c
*and*
2.9d
*where* *m*_{1},*m*_{2},*n*_{1},*n*_{2},*f*_{1},*f*_{2} *are defined earlier by equations (2.2)*.

*In particular, the solution of* *q*-P_{II} *is related to the leading coefficient of the formal solution of the associated linear system by*
2.10

### Proof.

Noting that *e*_{1}/*e*_{2} is not a multiple of 1/*q*^{2}, we have by applying Carmichael’s theorem (Carmichael 1912) in the neighbourhood of *ν*=0, that the solution matrix of spectral system (2.1a) has the form:
2.11
To obtain the recurrence relations (2.9), substitute solution (2.11) into equation (2.1a) and equate powers of *ν* near 0. We see that for *e*_{1}≠*e*_{2}, *b*_{0}=*c*_{0}=0, while *a*_{0},*d*_{0} are arbitrary. The fact that *b*_{0}=*c*_{0}=0 further implies
2.12
Let *Φ*={*ϕ*_{1},*ϕ*_{2}}, then we have
and
Note that *a*_{j},*b*_{j},*c*_{j} and *d*_{j} are in general functions of *x* as *m*_{1},*m*_{2},*n*_{1},*n*_{2},*f*_{1} and *f*_{2} are functions of *x*. To find the *x* dependence of *a*_{j}(*x*) and *c*_{j}(*x*) substitute *ϕ*_{1}(*ν*,*x*) into deformation equation (2.1b) equating powers in *ν*,
From terms we have
2.13
Therefore, the solution of *q*-P_{II} is given by the ratio of the leading coefficient of the formal solution of its associated linear system around *ν*=0, with different shifts in the *x*-direction. This difference in shift corresponds exactly to differentiation in a similar formula, which relates the solution of *P*_{II} to the solution of its associated linear system in the continuous case (Flaschka & Newell 1980). The remaining coefficients in the expansion (2.11) can be found similarly. ■

### (b) Expansion in a neighbourhood of

We first recall the definition of the *q*-discrete analogue of the Gamma function, *Γ*_{q}(1−*z*). It is the solution of the *q*-Gamma equation:
2.14
which has the infinite product expression
2.15
where .

### Proposition 2.3

*There exists a fundamental solution matrix* *Ψ*={*ψ*_{1},*ψ*_{2}} *of the* *q*-*linear systems (2.1a) and (2.1b) in the neighbourhood of* *given by*
2.16
*and*
2.17
*where*

### Proof.

First, for the leading behaviour of the solution of spectral system (2.1a) near , we need to diagonalize its coefficient matrix near ,
Since *f*_{1}(*x*)/*f*_{2}(*x*)=*q*^{2}, this can be done by conjugation with the constant matrix . Let *Ψ*(*ν*,*x*)=*CΨ*_{1}(*ν*,*x*), then
2.18a
and
2.18b
where
and
The matrix solution of the *q*-linear system (2.18a) at , therefore has the form:
2.19
Substitute into spectral linear system (2.18a), we have conditions:
2.20
2.21
and
2.22
Solution (2.19) also needs to satisfy deformation equation (2.18b). This gives us the following conditions with respect to the Painlevé variable *x*:
2.23
2.24
and
2.25
It is easy to check by substitution that equations (2.20) and (2.23) for *I*(*ν*) and *J*(*x*) are solved by
and
whereas
and
solve equations (2.21) and (2.24) for *u*(*ν*,*x*) and equations (2.22) and (2.25) for *v*(*ν*,*x*). The recurrence relations for the coefficients *α*_{j}, *β*_{j}, *γ*_{j}, *δ*_{j} can be found by substituting solution (2.19) into (2.18a) and equating powers of *ν* at . ■

### (c) Special solutions

Carmichael’s theorem no longer applies when *e*_{1}/*e*_{2} is an integer power of 1/*q*^{2}. However, formal series expansions may still exist in the limits *ν*→0 or . In this subsection, we consider such cases in further detail. Consider the case when *α*_{k}=1/*q*^{4k}. Since *q*-P_{II} has the symmetry
we only need to consider *k*>0.

The case *e*_{1}/*e*_{2}=*α*_{k}=1/*q*^{4k} separates into two types: (I) *k* is an integer, (II) *k* is a half integer.

I. The solutions near

*ν*=0 of the forms (2.7) and (2.8) are still valid in this case. This can be easily checked as follows, when*e*_{1}/*e*_{2}=1/*q*^{4k}, the*j*in equation (2.9d) is even: Since we found in the expansion of solution that*d*_{odd}=*b*_{even}=0 (equation 2.12), the right side here is also 0. Hence, no inconsistency arises in this case.II. In this case,

*k*is a positive half integer and inconsistency can occur for the recurrence relations. For example when*e*_{1}/*e*_{2}=1/*q*^{4k}, the*j*in equation (2.9d) is odd: but*b*_{odd},*d*_{even}are not necessarily 0, so the right side is not 0 in general. However, it is possible to avoid inconsistency if special conditions are imposed on the coefficients*m*_{1}(*x*),*n*_{1}(*x*) and*f*_{1}(*x*).

We will see that type (I) solutions of the iso-monodromy deformation problem correspond to equation (1.1) admitting rational-type special solutions, while type (II) solutions of the iso-monodromy deformation problem correspond to *q*-hypergeometric-type special solutions. In this paper, we consider type (I) only. The problem of type (II) will be considered in a separate paper.

#### (i) Simplest rational type solution

We start by considering the case *k*=0, *e*_{1}/*e*_{2}=1, which is the simplest case of type (I). Note that *q*-linear problem (2.1a) is not easy to solve in general, being a 2×2 system with polynomial coefficient of degree 3. However, the problem simplifies when *M*_{0}(*x*), *M*_{1}(*x*), *M*_{2}(*x*) and *M*_{3}(*x*) commute, which implies *A*(*ν*,*x*) can be diagonalized by conjugation with a constant matrix, and the second-order linear problem then reduces to two first-order ones. On demanding that the four coefficient matrices *M*_{0}(*x*), *M*_{1}(*x*), *M*_{2}(*x*) and *M*_{3}(*x*) in equation (2.1a) commute with each other, we arrive at the conditions
2.26
and
2.27
where *g*_{0}(*x*) is a special solution of *q*-P_{II} with parameter *α*_{0}=*e*_{1}/*e*_{2}=1. In this case, we label the corresponding matrices of the linear problem *A*(*ν*,*x*),*B*(*ν*,*x*) as *A*_{0}(*ν*,*x*),*B*_{0}(*ν*,*x*) and the solution matrix as .

### Proposition 2.4

*A solution of the Lax pair* (2.1a) *and* (2.1b) *when* *α*_{k}=*e*_{1}/*e*_{2}=1, i.e. *k*=0, *when the corresponding* *q*-*P*_{II} *equation* (1.1) *admits rational solution* *g*_{0}(*x*)=−*ix*, *is given by*
2.28
*where* *Γ*_{q}(1−*z*) *is the* *q*-*Gamma function defined by equation (2.14), and*
2.29a
*and*
2.29b
*where* *T*_{j}(*x*) *has the generating function*
2.30

### Proof.

Substituting the special solution (2.27) into the linear problems (2.1a) and (2.1b) gives 2.31a and 2.31b where and

The constant matrix which diagonalizes *A*_{0}(*ν*,*x*) is . Let *Ψ*(*ν*,*x*)=*CΨ*_{1}(*ν*,*x*). Then
2.32a
and
2.32b
where
Let the first column of the solution matrix of equations (2.32) be , that is
where
2.33a
2.33b
2.33c
and
2.33d
We see that equation (2.33a) can be solved in terms three *q*-Gamma functions
2.34
It can be easily checked that the solution (2.34) also satisfies equation (2.33b), using the infinite product expression (2.15) of *Γ*_{q}(1−*z*). Similarly, we have
2.35
which solves equations (2.33c) and (2.33d).

To show the series summation expression of in equation (2.28), let
2.36
and substitute into equations (2.33a) and (2.33b) for *u*(*ν*,*x*), we then have the relations which define *T*_{j}(*x*), *j*=1,2,…
and
Later, we will show that these are the polynomial entries in the determinants of the determinant form of rational-type special solutions of our *q*-P_{II} equation.

Similarly it can be shown and finally a solution of equations (2.31a) and (2.31b) is as required. ■

## 3. Schlesinger transformation

### (a) Schlesinger transformation *L*_{k}(*ν*,*x*)

Let the Lax pair (2.1) for the case *e*_{1}=1/*q*^{2k}, *e*_{2}=*q*^{2k}, *α*_{k}=*e*_{1}/*e*_{2}=1/*q*^{4k} be denoted by
3.1a
and
3.1b
where
3.2
We denote the matrix solution of the Lax equations (3.1a) and (3.1b) by . It has a series expansion in the neighbourhood of *ν*=0 given by proposition 2.2.

Let be the first column of the fundamental matrix solution of the Lax pair (3.1a) and (3.1b) given by proposition 2.2. It has asymptotic behaviour:
and when *α*_{k+1}=*e*_{1}/*e*_{2}=1/*q*^{4(k+1)},
We want to find the Schlesinger transformation *L*_{k}(*ν*,*x*) such that
We first relate *v*^{(k)}(*ν*,*x*) with *u*^{(k+1)}(*ν*,*x*), making use of Bäcklund transformation (2.4) of *g*_{k}(*x*). We are motivated by an observation that *v*^{(k)}(*ν*,*x*) has the same order of leading behaviour in the spectral variable *ν* with *u*^{(k+1)}(*ν*,*x*) in the neighbourhood of *ν*=0, namely,
and

### Proposition 3.1

*There exists a constant* *μ* *such that*

### Proof.

Recall equation (2.13)
3.3
Then proposition 3.1 is true if we can show that
From the recurrence relation (2.9b) for we have
Using the *q*-P_{II} equation (1.1) to eliminate *g*_{k}(*x*/*q*^{2}), and rewrite in terms of *g*_{k}(*x*), we have
Then
The last line is exactly Bäcklund transformation (2.4) of our *q*-P_{II} equation (1.1). That is
hence is proportional to . ■

The above result motivates the following statement.

### Proposition 3.2

### Proof.

Recall spectral equation (3.1a) is of the form:
3.4
where it is useful to note that
3.5
This 2×2 system of coupled first-order *q*-discrete system can be rewritten as two second-order *q*-discrete equations:
3.6
and
3.7

The equation for *u*^{(k+1)} is then
3.8
We observe from equation (3.5) that
3.9
and using *q*-P_{II} equation (1.1) for *g*_{k}(*x*) and its Bäcklund transformation (2.4) we can easily show that *v*^{(k)}(*ν*,*x*) and *u*^{(k+1)}(*ν*/*q*^{2},*x*) satisfy the same equation. The leading behaviours of *v*^{(k)}(*ν*,*x*) and *u*^{(k+1)}(*ν*/*q*^{2},*x*) are
If we choose *μ* to be 1/*q*^{2k+2}, so that
3.10
then we have
■

### Proposition 3.3

*The Schlesinger transformation of linear system (3.1a) is given by*
3.11a
*where*
3.11b

### Proof.

Recall
and
Using proposition 3.2, the problem of writing *u*^{(k+1)}(*ν*,*x*) in terms of *u*^{(k)}(*ν*,*x*) and *v*^{(k)}(*ν*,*x*) is reduced now to writing *v*^{(k)}(*νq*^{2},*x*) in terms of *u*^{(k)}(*ν*,*x*) and *v*^{(k)}(*ν*,*x*). Using equation (3.4), we have
3.12
Note we have made the *x* dependence implicit, since all the operations are on the variable *ν* and *x* does not change. Now it is only left for us to write *v*^{(k+1)} in terms of *u*^{(k)} and *v*^{(k)} to obtain the expression for *L*_{k}(*ν*,*x*). From spectral equation (3.4) for *k*+1, we have for *v*^{(k+1)}(*ν*) in terms of and *u*^{(k+1)} and using equation (3.12) we have
3.13
and we have proved the proposition. ■

We see that as ,
and
so
is a matrix function with infinite series expansion in the spectral variable *ν* in the neighbourhood of *ν*=0. We have found that Schlesinger transformation *L*_{k}(*ν*,*x*) is not particularly helpful for finding the determinant expression of . In what follows, we show that using another Schlesinger-type transformation, the determinant structure of and hence that of *g*_{k}(*x*) can be readily obtained.

### (b) Schlesinger transformation *Λ*_{k}(*ν*,*x*)

### Definition 3.4

Let us define an auxiliary vector function *F*^{(k)}(*ν*,*x*) related to by
3.14

### Proposition 3.5

*Now we show that vector function* *F*^{(k)}(*ν*,*x*) *has a Schlesinger transformation defined by*
3.15a
*where*
3.15b
*and*
3.15c

### Proof.

We know
where we have used equation (3.1a) and proposition 3.3. Now let ,
and
Use the definition of *A*_{k+1}(*ν*,*x*) and *L*_{k}(*ν*,*x*) we have
Using Bäcklund transformation (2.4) of *g*_{k}(*x*), *q*-P_{II} equations (1.1) and (3.9), it can be easily checked that
and
Hence
■

Proposition 3.5 gives a much simpler Schlesinger transformation *Λ*_{k}(*ν*,*x*) compared with *L*_{k}(*ν*,*x*) with respect to the spectral variable *ν*. Instead of being a rational function in *ν* and hence having an infinite series expansion in the neighbourhood of *ν*=0, it has simple polynomial dependence in 1/*ν*. This observation is crucial in allowing us to obtain the determinant form of *g*_{k}(*x*) for special values of *k*.

## 4. Determinant representations of special solutions

### Theorem 4.1

*q-P*_{II} *equation (1.1) with parameter α*_{k}*=1/q*^{4k} *for integer k admits a hierarchy of rational-type special solutions g*_{k}*(x), given by
*
4.1
*and
*
4.2
*T*_{k}*(x) is some q-discrete polynomial of degree k in x
*
4.3a
*and
*
4.3b
*and T*_{j}*(x) has the generating function
*
4.4

### Proof.

We have for integer *k*
4.5
and from definition (3.14) of *F*^{(k)}(*ν*,*x*)
4.6
we have omitted the superscript ^{(0)} on the coefficients *a*_{2j}, *c*_{2j+1} for simplicity. Using proposition 3.5 for *k*, an odd integer
and for *k*, an even integer
We have named the coefficients of *ν*^{−j} in the (1,1) and (2,1) entries in each case to be *s*_{j} and the coefficients of *ν*^{−j−1} in the (1,2) and (2,2) entries to be *t*_{j} (*j*=0,…,*l*). Note the particular form of *Λ*_{k}(*ν*,*x*) implies that the *s*_{0} and *t*_{−1} in each case are constants whereas *s*_{j} and *t*_{j}, in general, are functions of *g*_{k}(*x*) and *x*. We consider the case when *k* is an odd integer, and show that the case for *k* even can be proved along the way.

For *k* is odd, the Schlesinger transformation relating *F*^{(k+1)} to *F*^{(0)} is,
and again from definition (3.14) of *F*^{(k)} we know
Equating the two expressions of *F*^{(k+1)} in powers of *ν* in the neighbourhood of 0:

— the top entry gives

— the bottom entry gives

We can rewrite the equations from the top entry in a (*k*+1)×(*k*+1) matrix form:
4.7
and the equations from the bottom entry in a (*k*+2)×(*k*+2) matrix form:
4.8

### Definition 4.2

Recall *k* is an odd integer, therefore *k*+1 is even, let *σ*_{even}(*x*) be
and *σ*_{odd}(*x*) be

Apply Cramer’s rule on matrix equation (4.7) to evaluate *s*_{0}, recall *s*_{0} is a constant,
4.9
where *σ*_{k}=*σ*_{odd}, *σ*_{k+1}=*σ*_{even} are defined by definition 4.2. We now have in terms of determinants.

Apply Cramer’s rule on matrix equation (4.8) for *t*_{−1}, recall *t*_{−1} is also a constant,
however from equation (3.10), we know that , hence
4.10
Now we also have in terms of determinants. Recall proposition 2.4
that is
Then for integer *k*, let
4.11
The *τ*_{k} function defined by equation (4.11) is proportional to the *σ*_{k} function defined in definition 4.2 and
where *μ*_{k} is a constant. Finally, recall equation (3.3) we have
A few examples of *τ*_{k}(*x*). For *k*=0,1,2,3, respectively, we have
4.12
4.13
4.14
where
4.15
4.16
4.17
4.18
4.19 ■

## 5. Discussion

In this paper, a general relation between the solution of the iso-monodromy deformation problem and the solution of *q*-P_{II} equation was established. We have found exact solutions of the iso-monodromy deformation problem for a special case for integer *k*, corresponding to *q*-P_{II} admitting rational-type special solutions. This fact was then used to find the determinant form of the rational-type special solutions of *q*-P_{II}. The calculation has revealed that the generating function for the entries of the determinant of a type of special solutions (integer *k*) is exactly the solution of the iso-monodromy deformation problem (proposition 2.4), when *q*-P_{II} equation is solved by the seed solution of that type, for *k*=0.

- Received March 15, 2011.
- Accepted June 20, 2011.

- This journal is © 2011 The Royal Society