Combinatorial expressions for the tau functions of $q$-Painlev\'e V and III equations

We derive series representations for the tau functions of the $q$-Painlev\'e V, $\mathrm{III_1}$, $\mathrm{III_2}$, and $\mathrm{III_3}$ equations, as degenerations of the tau functions of the $q$-Painlev\'e VI equation in \cite{JNS}. Our tau functions are expressed in terms of $q$-Nekrasov functions. Thus, our series representations for the tau functions have explicit combinatorial structures. We show that general solutions to the $q$-Painlev\'e V, $\mathrm{III_1}$, $\mathrm{III_2}$, and $\mathrm{III_3}$ equations are written by our tau functions. We also prove that our tau functions for the $q$-Painlev\'e $\mathrm{III_1}$, $\mathrm{III_2}$, and $\mathrm{III_3}$ equations satisfy the three-term bilinear equations for them.


INTRODUCTION
The q-Painlevé equations [RGH], [KNY] are q-difference analogs of the Painlevé equations, which were introduced as new special functions beyond elliptic functions and the hypergeometric functions more than one hundred years ago [P], [G], and are now considered as important special functions with many applications both in mathematics and physics.
Similarly as for other integrable systems, tau functions play a crucial role of the studies of the Painelvé equations. Recent discovery by [GIL] states that the tau function of the sixth Painlevé equation is a Fourier transform of conformal blocks at c = 1, which admit explicit combinatorial formulas by AGT correspondence [AGT]. Series representations of the tau functions of other types are also studied in [GIL1], [N], [N1], [BLMST] for differential cases, [BS1], [JNS] for q-difference cases.
In [JNS], a general solution (y, z) to the q-Painlevé VI equation [JS] was expressed by the tau functions having q-Nekrasov type expressions, and it was conjectured that the tau functions satisfy the bilinear equations for the q-Painlevé VI equation. In this paper, we give explicit expressions for general solutions to the q-Painlevé V, III 1 , III 2 , and III 3 equations using degenerations of the tau functions of the q-Painlevé VI equation. We also give conjectures on the bilinear equations satisfied by the tau functions of the q-Painlevé V equation and prove that the tau functions of the q-Painlevé III 1 , III 2 , and III 3 equations satisfy the bilinear equations.
Our q-difference equations are as follows. (i) the q-Painlevé VI equation: a 4 (y − a 3 ) .
From the point of view of Sakai's classification for the discrete Painlevé equations [S1], the q-Painlevé VI, V, III 1 , III 2 and III 3 equations are derived from the symmetries/surfaces of type respectively. The degeneration scheme of Painlevé equations is as follows.
The degeneration pattern of the q-Painlevé equations we use is similar to the one in [Mu] but not exactly same. Rather, our limiting procedure is a q-version for the one used in [GIL1] in order to derive combinatorial expressions of tau functions of P V , P III 1 , P III 2 , and P III 3 from the Nekrasov type expression of the tau function of P VI [GIL]. For the case of the q-Panlevé III 3 equation, series representations for the tau functions was proposed in [BS1], which are expressed by q-Virasoro Whittaker conformal blocks which equal Nekrasov partition functions for pure SU (2) 5d theory [AY], [Y]. Our tau functions for the q-Panlevé III 3 equation obtained by the degeneration are equivalent to them.
Our plan is as follows. In Section 2, we recall the result on q-Painlevé VI equation in [JNS]. In Section 3-6, we compute limits of tau functions and derive combinatorial expressions of general solutions and bilinear equations for q-Painlevé V, III 1 , III 2 and III 3 equations.
We use the q Gamma function, q Barnes function and the theta function defined by which satisfy Γ q (1) = G q (1) = 1 and A partition is a finite sequence of positive integers λ = (λ 1 , . . . , λ l ) such that λ 1 ≥ . . . ≥ λ l > 0. Denote the length of the partition by ℓ(λ) = l . The conjugate partition We regard a partition as a Young diagram. Namely, we regard a partition λ also as the subset {(i , j ) ∈ Z 2 | 1 ≤ j ≤ λ i , i ≥ 1} of Z 2 , and denote its cardinality by |λ|. We denote the set of all partitions by Y. For = (i , j ) ∈ Z 2 >0 we set a λ ( ) = λ i − j (the arm length of ) and ℓ λ ( ) = λ ′ j −i (the leg length of ) . In the last formulas we set . For a pair of partitions (λ, µ) and u ∈ C we set which we call a Nekrasov factor.
2. RESULTS ON q -P VI FROM [JNS] In this section, we recall the results of [JNS] on the q-Painlevé VI equation. Define the tau function by with the definition Here and after we write Theorem 2.1. [JNS] The functions y and z defined by are solutions to the q-Painlevé VI equation Theorem 2.1 was obtained by constructing the fundamental solution of the Lax-pair for q-P VI in [JS], in terms of q-conformal blocks in [AFS]. The method of construction of the fundamental solution is a q analogue of the CFT approach used in [ILT]. In the derivation of Theorem 2.1 convergence of the fundamental solution was assumed and it has not been proved, Recently, analyticity of K-theoretic Nekrasov functions in certain domains was discussed in [FM].
Then, the function y, z The function y in Conjecture 2.2 is expressed as the same form in Theorem 2.1, while the function z in Conjecture 2.2 is not. By the bilinear equations (2.7) and (2.8), we obtain the expression of z in (2.11) from the expression of z in (2.1).
We note that in [JNS] we have a Lax pair with respect to the shift t → q t , namely, a fundamental solution of the linear q-difference equations for certain 2 by 2 matrices A(x, t ) and B(x, t ) was constructed in terms of q-Nekarasov functions. From (2.12) we obtain the four-term bilinear equation in Remark 3.5 of [JNS]: 3. FROM q -P VI TO q -P V In this section, we take a limit of the tau functions of q-P VI to q-P V . For θ = (θ * , θ t , θ 0 ) and α = (α 1 , α 2 , α 3 , α 4 , α 5 ) we set We set We define tau functions for q-P V by Proposition 3.1. Set for i = 1, . . ., 8. Here, we denote by τ VI i ( θ, s, σ, t ) the tau functions of q-P VI presented in the previous section.
Proof. First, we verify the limit of the series part. For any partition λ we have Hence, the series Z Second, we examine the limits of the coefficients of Z . By the identities (1.1) on q Gamma function and q Barnes function, for n ∈ Z we have Using the identity above, we compute the coefficient of Z in τ VI 1 multiplied by C (a 1 ) as follows.
Similarly, we can compute the coefficients of Z in the other tau functions and obtain the desired results.
Hence, by (3.1) the solution (y, z) of the q-Painlevé VI equation has the following limit It is easy to see that the q-Painlevé VI equation (2.2) degenerates the q-Painlevé V equation (3.3) for y = y 1 and z = z 1 as Λ → ∞.
Let us define the tau functions for q-P III 1 by Proposition 4.1. Set Proof. For any partition λ we have Hence, the series Z V [(θ * , θ t , θ 0 ), σ, t ] goes to Z III 1 [(θ * , θ ⋆ ), σ, t 1 ] as Λ → ∞. The coefficients of Z V are computed in the same way as in the proof of Proposition 3.1 using (3.2) and we obtain the desired results.
Theorem 5.2. The functions solves the q Painlevé III 2 equation with the parameters Furthermore, the tau functions τ i (i = 1, . . . , 8) satisfy the following bilinear equations.
Then we have C (a i )τ In what follows, we abbreviate τ III 3 i (s, σ, t ) to τ i .