Multiple Skew-Orthogonal Polynomials and 2-Component Pfaff Lattice Hierarchy

Abstract

In this paper, we introduce multiple skew-orthogonal polynomials and investigate their connections with classical integrable systems. By using Pfaffian techniques, we show that multiple skew-orthogonal polynomials can be expressed by multi-component Pfaffian tau-functions upon appropriate deformations. Moreover, a two-component Pfaff lattice hierarchy, which is equivalent to the Pfaff–Toda hierarchy studied by Takasaki, is obtained by considering the recurrence relations and Cauchy transforms of multiple skew-orthogonal polynomials.

Appendices

Appendix A. Pfaffian identities

There are two different kinds of Pfaffian identities, c.f. [31, eq. 2.95’ & 2.96’]

$$\begin{aligned}&\text {pf}(*,a_1,a_2,a_3,a_4)\text {pf}(*)=\text {pf}(*,a_1,a_2)\text {pf}(*,a_3,a_4)\nonumber \\&\quad -\text {pf}(*,a_1,a_3)\text {pf}(*,a_2,a_4)+\text {pf}(*,a_1,a_4)\text {pf}(*,a_2,a_3), \end{aligned}$$

(A.1a)

$$\begin{aligned}&\text {pf}(\star ,a_1,a_2,a_3)\text {pf}(\star ,a_4)=\text {pf}(\star ,a_2,a_3,a_4)\text {pf}(\star ,a_1)\nonumber \\&\quad -\text {pf}(\star ,a_1,a_3,a_4)\text {pf}(\star ,a_2)+\text {pf}(\star ,a_1,a_2,a_4)\text {pf}(\star ,a_3), \end{aligned}$$

(A.1b)

where $*$ and $\star$ are sets of even and odd number symbols, respectively.

Appendix B. Derivative Formulas for Wronskian-Type Pfaffians

Wronskian-type Pfaffians are well investigated in soliton theory due to its wide applications in coupled KP theory. In [31, Sec. 3.4], Pfaffian element $\text {pf}(i,j)$ satisfying the differential rules with respect to the variables $\textbf{t}=(t_1,t_2,\cdots )$ by

$$\begin{aligned} \partial _{t_n}\text {pf}(i,j)=\text {pf}(i+n,j)+\text {pf}(i,j+n) \end{aligned}$$

(B.1)

was called Wronskian-type Pfaffians. For more details about Wronskian-type Pfaffian and its discrete counterparts, please refer to [48]. It was shown that if Pfaffian elements satisfy (B.1), then

$$\begin{aligned} \partial _{t_n}\text {Pf}(i_0,i_1,\cdots ,i_{2N-1})=\sum _{k=0}^{2N-1}\text {Pf}(i_0,\cdots ,i_k+n,\cdots ,i_{2N-1}). \end{aligned}$$

This was proved by induction.

In this paper, we need to introduce 2-component Pfaffian $\tau$-functions, indexed by ${\mathcal {I}}=\{i_0,\cdots ,i_n\}$ and ${\mathcal {J}}=\{j_0,\cdots ,j_m\}$ with $n,m\in {\mathbb {N}}$ and $n+m\in 2{\mathbb {N}}$. Pfaffian elements in this case should satisfy a 2-component Wronskian-type generalization (c.f. Prop 3.3)

$$\begin{aligned}&\partial _{t_k}\text {pf}(i_\alpha ,i_\beta )=\text {pf}(i_\alpha +k,i_\beta )+\text {pf}(i_\alpha ,i_\beta +k),\\&\partial _{t_k}\text {pf}(i_\alpha ,j_\beta )=\text {pf}(i_\alpha +k,j_\beta ),\\&\partial _{t_k}\text {pf}(j_\alpha ,j_\beta )=0,\\&\partial _{s_k}\text {pf}(j_\alpha ,j_\beta )=\text {pf}(j_\alpha +k,j_\beta )+\text {pf}(j_\alpha ,j_\beta +k),\\&\partial _{s_k}\text {pf}(i_\alpha ,j_\beta )=\text {pf}(i_\alpha ,j_\beta +k),\\&\partial _{s_k}\text {pf}(i_\alpha ,i_\beta )=0, \end{aligned}$$

and then, we have the following proposition.

Proposition B.1

If Pfaffian elements satisfy the above derivative relations, then one has

$$\begin{aligned}&\partial _{t_k}\text {Pf}(i_0,\cdots ,i_n,j_0,\cdots ,j_m)=\sum _{\alpha =0}^n \text {Pf}(i_0,\cdots ,i_\alpha +k,\cdots ,i_n,j_0,\cdots ,j_m),\\&\partial _{s_k}\text {Pf}(i_0,\cdots ,i_n,j_0,\cdots ,j_m)=\sum _{\alpha =0}^m \text {Pf}(i_0,\cdots ,i_n,j_0,\cdots ,j_\alpha +k,\cdots ,j_m). \end{aligned}$$

Proof

Here, we only prove the first equation by using induction, and the second one can be similarly proved. Note that

$$\begin{aligned} \partial _{t_k}\text {Pf}(i_0,\cdots ,i_n,j_0,\cdots ,j_m)&=\partial _{t_k}\left( \sum _{i_l\in {\mathcal {I}}}(-1)^{l-1}\text {pf}(i_0,i_l)\text {Pf}(i_1,\cdots ,{\hat{i}}_l,\cdots ,i_n,j_0,\cdots ,j_m)\right. \\&\qquad \left. +\sum _{j_l\in {\mathcal {J}}}(-1)^{n+l}\text {pf}(i_0,j_l)\text {Pf}(i_1,\cdots ,i_n,j_0,\cdots ,{\hat{j}}_l,\cdots ,j_m)\right) \end{aligned}$$

where the first part is equal to

$$\begin{aligned}&\sum _{i_l\in {\mathcal {I}}}(-1)^{l-1}\text {pf}(i_0+k,i_l)\text {Pf}(i_1,\cdots ,{\hat{i}}_l,\cdots ,i_n,j_0,\cdots ,j_m) \end{aligned}$$

(B.2a)

$$\begin{aligned}&+\sum _{i_l\in {\mathcal {I}}}(-1)^{l-1}\text {pf}(i_0,i_l+k)\text {Pf}(i_1,\cdots ,{\hat{i}}_l,\cdots ,i_n,j_0,\cdots ,j_m) \end{aligned}$$

(B.2b)

$$\begin{aligned}&+\sum _{i_l\in {\mathcal {I}}}(-1)^{l-1}\text {pf}(i_0,i_l)\sum _{\alpha \ne l}\text {Pf}(i_1,\cdots ,i_\alpha +k,\cdots ,{\hat{i}}_l,\cdots ,i_n,j_0,\cdots ,j_m), \end{aligned}$$

(B.2c)

while the derivative of the second part is equal to

$$\begin{aligned}&\sum _{j_l\in {\mathcal {J}}}(-1)^{n+l}\text {pf}(i_0+k,j_l)\text {Pf}(i_1,\cdots ,i_n,j_0,\cdots ,{\hat{j}}_l,\cdots ,j_m) \end{aligned}$$

(B.3a)

$$\begin{aligned}&\quad +\sum _{j_l\in {\mathcal {J}}}(-1)^{n+l}\text {pf}(i_0,j_l)\sum _{\alpha =1}^n\text {Pf}(i_1,\cdots ,i_\alpha +k,\cdots ,i_n,j_0,\cdots ,{\hat{j}}_l,\cdots ,j_m). \end{aligned}$$

(B.3b)

Therefore, by summing (B.2a) and (B.3a) up, one obtains

$$\begin{aligned} \text {Pf}(i_0+k,i_1,\cdots ,i_n,j_0,\cdots ,j_m). \end{aligned}$$

The summation of rest three equations is equal to

$$\begin{aligned} \sum _{\alpha =1}^n \text {Pf}(i_0,\cdots ,i_\alpha +k,\cdots ,i_n,j_0,\cdots ,j_m), \end{aligned}$$

and our proof is complete. $\square$