## Abstract

This is the second part of our study of the solutions of a *q*-discrete second Painlevé equation (*q*-P_{II}) of type (*A*_{2}+*A*_{1})^{(1)} via its iso-monodromy deformation problem. In part I, we showed how to use the *q*-discrete linear problem associated with *q*-P_{II} to find an infinite sequence of exact rational solutions. In this paper, we study the case giving rise to an infinite sequence of *q*-hypergeometric-type solutions. We find a new determinantal representation of all such solutions and solve the iso-monodromy deformation problem in closed form.

## 1. Introduction

The *q*-P_{II} equation with affine Weyl group symmetry of type (*A*_{2}+*A*_{1})^{(1)} (Ramani & Grammaticos 1996) is
1.1
where *g* is a function of *x*=*x*_{0}*q*^{−n}, , while *x*_{0}, *α*, *q*≠1 are constant parameters. In the continuum limit , *t*=*nϵ*, (with *q*^{2}=(1+*ϵ*^{3}/2)^{−1}, and *α*=(1+*aϵ*^{3})), this equation reduces to the classical second Painlevé equation P_{II}: *y*_{tt}=2*y*^{3}+*ty*−*a*. Later we let *α*=*α*_{k}:=1/*q*^{4k}, where *k* is a *half*-integer, and denote the corresponding solutions of *q*-P_{II} as *g*_{k}(*x*).

In part I, we presented a new approach for finding explicit solutions based on the connection between the *q*-Painlevé equation and its iso-monodromy deformation problem, also known as the associated linear system or ‘Lax pair’ (Joshi & Shi 2011). In particular, we discovered a Schlesinger transformation of the linear system that maps a solution of the linear problem corresponding to *k* to another solution for the case *k*+1. Starting with an explicit solution for the case *k*=0, this approach enabled us to find an infinite hierarchy of rational special solutions of equation (1.1) for integer *k* and provides a new determinantal representation for such solutions.

In the present paper, we show how to proceed in the case when *k* is *half*-integer. In this case, the coefficient matrices of the iso-monodromy deformation problem (2.1) are no longer diagonalizable as in the rational case of part I. In particular, several non-trivial transformations are needed to solve the linear system exactly. We overcome this difficulty to obtain an infinite sequence of *q*-hypergeometric-type solutions of *q*-P_{II}. (Such functions are also called basic hypergeometric functions or series (Gasper & Rahman 1990).) In this case, our approach also provides a new determinantal representation of these solutions, which appears to differ from the case found by Kajiwara & Kimura (2003) and Kajiwara *et al.* (2011) through an algebraic bilinear method.

The paper is organized as follows. In §2, we recall the iso-monodromy deformation problem, i.e. equation (2.1a), and its properties, and solve the case *k*=1/2 explicitly (see proposition 2.3). The solutions of the linear problem in this special case are *q*-hypergeometric functions that can be interpreted as discrete versions of the Airy function. In §3, we derive the determinant formula for hypergeometric type special solutions of equation (1.1) for *α*=1/*q*^{4k}, when *k* are half integers. The main result of this paper is provided here as theorem 3.3. The paper concludes with a summary and discussion in §4.

## 2. Iso-monodromy deformation problem

In this section, we recall the iso-monodromy deformation problem associated with equation (1.1) and some facts about its solutions introduced in part I (Joshi & Shi 2011). In particular, we rely on expansions of the solutions of the linear problem around the origin and infinity, which are known to be convergent (Carmichael 1912). We recall such salient properties here for completeness.

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
and
Note that *ν* is the *q*-discrete monodromy variable (sometimes called spectral variable due 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 the *q*-linear systems (2.1a,*b*)
forces *g*(*x*) to satisfy a second-order nonlinear *q*-discrete equation, namely the *q*-P_{II} equation (1.1). We have chosen to concentrate analysis on equation (2.1a), the first half of the Lax pair, which defines the evolution of *Ψ*(*ν*,*x*) in the *ν*-plane, because 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*(*x*) the solution of *q*-P_{II} equation.

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*
2.3
*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 integers. Then the system* (2.3) *has fundamental matrix solutions Y* _{0}*(ξ),* *given by
*
2.4
*where* *and (ϵ*_{ij})_{1≤i,j≤n} *and (δ*_{ij})_{1≤i,j≤n} *are n×n matrices of analytic functions that can be expanded as a power series in ξ or 1/ξ around ξ=0 and* *, respectively.*

Applying this theorem on the *q*-discrete linear system (2.1a), we have:

### Proposition 2.2 (Joshi & Shi (2011))

*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,
2.5
*and*
2.6
*where* *and*
2.7a
2.7b
2.7c
and
2.7d
*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*
2.8

### (a) Simplest hypergeometric type solution

Carmichael's theorem no longer applies in the special case that we wish to focus on, namely *α*=1/*q*^{4k}. However, asymptotic series expansions still exist in this case, subject to a consistency condition satisfied by the coefficients of the series. We exploit this idea here to deduce the solution for the simplest case, namely *k*=1/2, *e*_{1}=1/*q*, *e*_{2}=*q*, *e*_{1}/*e*_{2}=*α*=1/*q*^{2}. Denoting the solution of *q*-P_{II} in this case by *g*(*x*)=*g*_{1/2}(*x*), we find the recurrence relation (2.7d) at *j*=1 is
Because *d*_{0}≠0, *m*_{1}(*x*) needs to be zero for the earlier-mentioned equation to be consistent. Recall *m*_{1}(*x*) is defined by equation (2.2a),
Hence *m*_{1}(*x*)=0 implies
2.9
The continuum limit of this result is instructive. In this limit, (2.9) becomes
2.10
where we have let , *t*=*nϵ*, , with *x*=*x*_{0}*q*^{−n}, *q*^{2}=(1+*ϵ*^{3}/2)^{−1}, and *α*=1/*q*^{2}=(1+*ϵ*^{3}/2). Recalling that *α*=(1+*aϵ*^{3}), we get *a*=1/2 for this case. Hence for special parameter *α*=1/*q*^{2}, *q*-P_{II} equation (1.1) reduces to a *q*-discrete analogue of the Riccati equation associated with the Airy-function-type solutions of P_{II}.

Consider now the iso-monodromy deformation problem in this special case. We know from equation (2.8) that
where *a*_{0}(*x*) is the leading coefficient of the vector solution *ϕ*_{1}(*ν*,*x*) of the iso-monodromy deformation problem at *ν*=0. The *q*-Riccati equation (2.9) then implies
which simplifies to
2.11
Equation (2.11) is a *q*-discrete analogue of the Airy equation. Its continuum limit
2.12
is found by taking *a*_{0}(*x*)=*w*(*t*), , 1/*q*^{2}=(1+*ϵ*^{3}/2) as . Thus we have found that when *e*_{1}/*e*_{2}=*α*=1/*q*^{2}, *q*-P_{II} reduces to the *q*-Riccati equation (2.9), and the leading coefficient *a*_{0}(*x*) of a solution of the associated linear system satisfies the *q*-discrete Airy equation with respect to the Painlevé variable *x*.

Using the *q*-Riccati equation (2.9) to replace terms involving *g*_{1/2}(*xq*) in the Lax pair (2.1), and recalling that in this case *m*_{1}(*x*)=0, we obtain
2.13a
and
2.13b
where, for the degenerate case , *e*_{1}=1/*q*, *e*_{2}=*q*, *e*_{1}/*e*_{2}=*α*=1/*q*^{2}, *g*(*x*)=*g*_{1/2}(*x*), we have denoted *A*(*ν*,*x*),*B*(*ν*,*x*) as *A*_{1/2}(*ν*,*x*),*B*_{1/2}(*ν*,*x*):
where

### Proposition 2.3

*A solution of the Lax pair* (2.1) *for the case* *k*=1/2, *e*_{1}=1/*q*, *e*_{2}=*q*, *e*_{1}/*e*_{2}=*α*=1/*q*^{2}, *g*(*x*)=*g*_{1/2}(*x*)=−*i*(*x*/*q*)(*a*_{0}(*x*/*q*)/*a*_{0}(*x*/*q*^{2})), *where* *a*_{0}(*x*) *is a solution of the* *q*-*Airy equation* (2.11), *which* *is given by*
2.14
*where*
2.15a
*and*
2.15b
*in particular*,
*where we have used the notation* .

### Proof.

We aim to transform the reduced Lax pair (2.13) to a simpler form. Since
we know from equation (2.5) that the Lax pair (2.13) admits a vector solution
with *a*_{0}(*x*) being a solution of equation (2.11). The first transformation
2.16
gives us
2.17a
2.17b
and
2.17c
Now let
be a vector solution of the Lax pair (2.17), that is
2.18a
and
2.18b
The top entry of equation (2.18a) gives
2.19
The combination on the left is reminiscent of a *q*-derivative:
To transform to a new ‘*v*’ which relates to *u* more simply, we let
2.20
This choice of *v*_{1} is similar to transformations taken in the associated linear system of PII (Flaschka & Newell 1980) for the special case .

Because we now have
2.21
the Lax pair now becomes
2.22a
and
2.22b
where
and
We see that the transformation equation (2.21) simplifies the Lax pair significantly. For instance, equations (2.22) no longer have the solution of *q*-P_{II}, i.e. *g*_{1/2}(*x*), in their coefficient matrices. The top entry of the vector equation (2.22b)
2.23
relates iteration in *x* to the iteration in *ν*, which suggests a further transformation
that is
2.24
The Lax pair is now
2.25a
and
2.25b
Recall *u*(*ν*,*x*) in the neighbourhood of *ν*=0 has the series expansion:
and
So we have now transformed the Lax pair into a form where we have a vector solution expressed in terms of only one set of coefficients of expansion, *a*_{2j}(*x*), *j*=0,1,2,…. The task now is to find the recurrence relations that define *a*_{2j}(*x*). Equation (2.23) gives us one relation (2.15a) for *a*_{2j}:
The bottom entry of equation (2.25b)
gives us the other relation (2.15b) for *a*_{2j}:
Transformations (2.16), (2.21) and (2.24) maps the Lax pair (2.13) to (2.17) to (2.22) and finally to (2.25), where a solution of the last Lax pair (2.25) has the expansion
with *a*_{2j}(*x*) satisfying the recurrence type relations (2.15a,*b*). In particular, *a*_{0}(*x*) solves the *q*-Airy equation (2.11). The solution of the original Lax pair (2.13) can be reconstructed by using the transformations (2.16), (2.21) and (2.24):
as required. ■

## 3. Schlesinger transformation

In this section, we use the Schlesinger transformation given in part I to iterate solutions of the Lax pair as half-integer values of *k* increase by unity. To emphasize this iteration in *k*, we denote (2.1) when *e*_{1}=1/*q*^{2k}, *e*_{2}=*q*^{2k}, *α*=*e*_{1}/*e*_{2}=1/*q*^{4k} by
3.1a
and
3.1b
Note that the Lax pair (3.1) has the matrix solution ,
3.2
and
3.3
in the neighbourhood of *ν*=0 that are consistent extensions of those given by proposition 2.2, as explained in §2.

### Definition 3.1

Define an auxiliary system of vector functions *F*^{(k)}(*ν*,*x*), which are related to by
3.4

### Proposition 3.2 (Joshi & Shi (2011))

*The functions* *F*^{(k)}(*ν*,*x*) *are related by the Schlesinger transformation*
3.5
*where*
3.6
*and*
3.7

### Theorem 3.3

*The equation q-P*_{II} (1.1*) with parameter α=1/q*^{4k}*, where* *n=1,2,… admits a hierarchy of q-hypergeometric-type special solutions g*_{n−1/2}*(x) given by
*
3.8
*where τ*_{n}*(x) is the determinant of a n×n matrix
*
3.9a
*and
*
3.9b
*where
*
3.10

### Proof.

For *k* a half-integer, recursive iteration of equation (3.5) gives
where
with
We can rewrite the product *Λ*_{k}*Λ*_{k−1},…,*Λ*_{1/2} in terms of different powers of *ν*, according to whether *n* is odd or even in . The case of odd integer *n*, *n*=1,3,…, can be expressed as
while for even integer *n*, *n*=2,4,…, we write
Note that *s*_{j} and *t*_{j} act as notation for coefficients; it is not implied that the coefficients are necessarily the same for the odd and even cases. In either case, it follows that *s*_{0} is a constant and that *t*_{1}=−*i*/*xq*^{2}.

We give the details of the even case when *n*=2,4,… here. The results for the odd *n* case are provided along the way. Firstly, we have
3.11
where is defined by equations (2.15a,*b*). From the definition of *F*^{(n+1/2)}, we also know that as
3.12
For *F*^{(n+1/2)}(*ν*,*x*) given by equation (3.11) to have the correct leading behaviour (3.12) in the neighbourhood of *ν*=0, coefficient *s*_{j} and *t*_{j} (*j*=0,…,*n*+1) must in general be functions of *x* and *g*_{k}(*x*) necessarily must satisfy appropriate systems of equations. Equating the two expressions of *F*^{(n+1/2)}(*ν*,*x*), from the top entry we get *n*+1 equations (*ν*^{−n},*ν*^{−n+2},…,*ν*^{n}):
which can be rewritten in the form of an (*n*+1)×(*n*+1) matrix equation
3.13
From the bottom entry, we get *n*+2 equations (*ν*^{−n−1},*ν*^{−n+1},…,*ν*^{n+1}):
which can be rewritten in the form of an (*n*+2)×(*n*+2) matrix equation
3.14
Let *τ*_{n}(*x*) be the determinant of the *n*×*n* matrix
3.15
and
3.16
By Cramer's rule on equation (3.13) for *s*_{0}, recalling *s*_{0} is a constant and that *n* is an even integer, we have
where *μ*_{n} is constant. Applying Cramer's rule on equation (3.14) for *t*_{1}, recalling *t*_{1}=−*i*/*xq*^{2},
where *η*_{n}, *ς*_{n}′ are constants.

Since *n*+1 is an odd integer when *n* is even, we now also have the formula for , where *n* is odd
Recall
for *n* is an even integer is
and when *n* is an odd integer is
where *τ*_{n}(*x*) is defined by equations (3.15) and (3.16). Recall *μ*_{n} and *ς*_{n} are constants, and
Hence for *n* even, we find
while for *n* odd, we get
as desired. ■

## 4. Discussion

In this paper, we derived *q*-hypergeometric function solutions of a *q*-discrete Painlevé equation from its associated linear system. Our first step was to solve the iso-monodromy deformation problem of equation (1.1) for the special case *k*=1/2 in terms of *q*-discrete Airy functions; see equation (2.8). We then applied an approach developed in (Joshi & Shi 2011) to obtain a new determinantal formula for a hierarchy of *q*-hypergeometric type special solutions of equation (1.1), in the case *α*=1/*q*^{4k} when *k* is a half-integer.

Our determinantal expressions differ from those found earlier by the bilinear method for the same hierarchy of hypergeometric special solutions of a *q*-discrete equation of the affine Weyl type (*A*_{2}+*A*_{1})^{(1)} in Kajiwara & Kimura (2003) and Kajiwara *et al.* (2011). It is worth noting that our derivation relies entirely on explicit, elementary methods, such as the usage of Cramer's rule. Some explicit examples of our determinant formula (3.8) are provided here for comparison purposes. For *n*=1,
for *n*=2,
for *n*=3,
where
and satisfies equation (3.10). The problem of relating different determinantal forms is highly non-trivial as demonstrated in Kajiwara *et al.* (2011) and is beyond the scope of this paper.

While we have provided explicit constructions of infinite sequences of special solutions for the case *α*=1/*q*^{4 k}, with *k* being integer in part I (Joshi & Shi 2011) and *k* being half-integer in the current paper, we note that there exist other solutions both for general values of *α* and special values taken here, which are believed not to be expressible in terms of algebraic functions or earlier known *q*-special functions, such as *q*-hypergeometric functions. Such generic solutions of *q*-P_{II} are highly transcendental functions of the variable *x*, in the same sense as the solutions of the second Painlevé equation P_{II} and will form the subject of future studies.

## Acknowledgements

Research supported in part by the Australian Research Council Discovery grant nos. DP0985615 and DP110102001.

- Received April 13, 2012.
- Accepted May 17, 2012.

- This journal is © 2012 The Royal Society