Noumi–Shiraishi Function in Integrable Systems

• The Noumi–Shiraishi function is an explicit multivariate power series that interpolates between symmetric Macdonald polynomials and Baker–Akhiezer functions in integrable systems.
• It acts as a kernel for Macdonald–Ruijsenaars operators, satisfying difference equations with clear spectral eigenvalue reductions and bispectral properties.
• Generalizations, including twisted and elliptic deformations, extend its application in advanced symmetric function theory and quantum integrable systems.

The Noumi–Shiraishi function forms a central object in Macdonald–Ruijsenaars integrable systems, interpolating between symmetric Macdonald polynomials and nonsymmetric Baker–Akhiezer functions within the so-called triad of bispectrally related wavefunctions. It is defined as an explicit multivariate formal power series in variables $x = (x_1, ..., x_N)$ with deformation parameters $q$, $t$ and spectral parameters $y = (y_1, ..., y_N)$. The series specializes to Macdonald polynomials for $y_i$ at the Young diagram locus and to Baker–Akhiezer functions for $t = q^{-m}$ and arbitrary $y_i$. It admits generalizations along “twisted” rays in the Ding–Iohara–Miki algebra, as well as elliptic deformations underpinning advanced symmetric function theory, Seiberg–Witten theory, and the elliptic Ruijsenaars–Schneider hierarchy.

1. Formal Power-Series Definition

The original Noumi–Shiraishi (NS) function is given for $N$ variables by

$$P^{(q,t)}(z;\,\lambda) = q^{\sum_i z_i\lambda_i} \cdot t^{\rho\cdot z} \sum_{k_{ij}\ge0} \psi\bigl(\lambda,\{k_{ij}\};q,t\bigr) \prod_{1\le i<j\le N} q^{k_{ij}(z_j-z_i)}$$

where $z = (z_1, ..., z_N)$, $\lambda = (\lambda_1, ..., \lambda_N)$, and $\rho_i = \frac{N-2i+1}{2}$ is the Weyl vector. The coefficient $\psi$ is a product of $q$-Pochhammer-type factors,

$$\psi(\lambda, \{k_{ij}\};q,t) = \prod_{n=2}^N \prod_{i<n} \frac{(q^{1-k_{in}};q)_{k_{in}} (t q^{\lambda_n-\lambda_i+1};q)_{k_{in}}} {(q;q)_{k_{in}} (tq;q)_{k_{in}}} \times \prod_{n=2}^N \prod_{i<j<n} \frac{(q^{\lambda_j-\lambda_i+1};q)_{k_{ij}} (t^{-1}q^{\lambda_j-\lambda_i+1};q)_{k_{ij}}} {(q;q)_{k_{ij}}^2}$$

with $(a;q)_n := \prod_{s=0}^{n-1}(1-aq^s)$ (Mironov et al., 2024).

The series is a formal expansion in the ratios $x_j/x_i=q^{z_j-z_i}$ with generic spectral parameters $\lambda$ not necessarily corresponding to Young diagrams; symmetry in $x$ is absent unless $\lambda$ is specialized.

2. Role in the Macdonald–Ruijsenaars Triad

The NS function encapsulates the “triad” linking Macdonald polynomials, Baker–Akhiezer functions, and the kernel of Macdonald–Ruijsenaars operators (Mironov et al., 2024, Mironov et al., 10 Mar 2025):

• Symmetric Macdonald polynomials: At $\lambda = \mu - \rho \log_q t$ for a partition $\mu$, $P^{(q,t)}(z;\lambda)$ truncates to $M_{\mu}(x;q,t)$ up to monomial factor.
• Baker–Akhiezer functions: For $t=q^{-m}$, the NS series in $x$ truncates to a polynomial of total degree~$m$ in each ratio, yielding the multivariable BA function of Chalykh–Feigin–Veselov type.
• NS kernel function: In general, $P^{(q,t)}$ acts as an explicit kernel intertwining Macdonald and BA eigenbases.

This structure is central to representations of the Ding–Iohara–Miki algebra (Mironov et al., 10 Mar 2025, Mironov et al., 2024).

3. Algebraic and Bispectral Properties

The NS function satisfies the Macdonald–Ruijsenaars difference equation in the $z$-variables: $$\widehat{H}_{MR} P^{(q,t)}(z;\lambda) = \left( \sum_{i=1}^N q^{\lambda_i} \right) P^{(q,t)}(z;\lambda)$$ where

$$\widehat{H}_{MR} = \sum_{i=1}^{N} \left( \prod_{j\ne i} \frac{t q^{z_i} - q^{z_j}}{q^{z_i} - q^{z_j}} \right) q^{\partial_{z_i}}$$

with $$q^{\partial_{z_i}} f(z_1,...,z_i,...) = f(z_1,...,z_i+1,...)$$ (Mironov et al., 2024, Mironov et al., 10 Mar 2025). In $x$-variables, the operator becomes

$$\widehat{D} = \sum_{i=1}^{N} \left( \prod_{j\ne i} \frac{t x_i - x_j}{x_i - x_j} \right) T_{q,x_i},$$

with $T_{q,x_i}f(\dots,x_i,\dots) = f(\dots,qx_i,\dots)$.

Eigenvalue reduction matches the spectral parameter $y_i=q^{\lambda_i}$, and the action of the operator is diagonal when the NS function is expanded in the respective basis.

4. Cramer’s Rule and Factorization Structures

A key algebraic property is that the NS function solves a finite system of linear equations by Cramer’s rule, leading to partial factorization of all involved determinants (Mironov et al., 2 Apr 2025). For the $N=2$ case, coefficients $u_k$ in $y(Z) = \sum_{k=0}^{m} u_k Z^k$ arise as

$u_k = (-1)^k \frac{\det M^{(k)}}{\det M^{(0)}}$

where $M^{(k)}$ is obtained by deleting the $k$th column from the matrix defining the system.

• For generic twist parameter $a$, NS functions generalize the BA system to “twisted rays” of the DIM algebra.
• Determinants undergo remarkable factorization, with only boundary terms introducing non-analytic jumps in the parameter space (Mironov et al., 2 Apr 2025).

5. Nested Ansatz and Resolution of Ambiguities

The nested-ansatz approach constructs NS functions recursively in the number of variables, resolving root ambiguities and ensuring full factorization of coefficients (Mironov et al., 24 Jan 2026). The $N$-variable Baker–Akhiezer function can be written as

$$\Psi_m^{[N]}(\underline{x}\mid\underline{\lambda}) = x_1^{\lambda_1+ \frac m2(N-1)} \sum_{k_{12},...,k_{1N}=0}^m \tilde\psi_{m;k_{12},...,k_{1N}}(\underline{\lambda}) x_1^{\sum_{i=2}^{N} k_{1i}} \Psi_m^{[N-1]}(x_2,...,x_N | \lambda_2 + k_{12} - m/2,\ldots)$$

with coefficients fixed layer-by-layer from functional equations. This construction uniquely determines all polynomial coefficients, including those attached to non-simple roots in higher rank (Mironov et al., 24 Jan 2026).

6. Generalizations: Shiraishi Functor and Elliptic Extensions

Awata–Kanno–Mironov–Morozov introduced the Shiraishi functor, which replaces the $q$-Pochhammer symbol $(1-z)$ in NS series with a general meromorphic kernel $\xi(z)$ satisfying $\xi(1)=0$, $\xi(z^{-1})\propto z^{-1}\xi(z)$ (Awata et al., 2020). The construction

$$P_N^\xi(x;p\mid y;s\mid q,t) = \sum_{\lambda^{(1)},\dots,\lambda^{(N)}} \prod_{i,j=1}^{N} {}_{n}\Xi\left(\frac{y_j}{y_i}\mid q,s\right)_{\lambda^{(i)}-\lambda^{(j)}} \prod_{i=1}^N (t p x_i/y_i)^{|\lambda^{(i)}|}$$

produces the generalized Noumi–Shiraishi (GNS) polynomials upon specializing $y$ to the Young diagram locus. These exhibit new non-Kerov triangularity features and admit biorthogonal bases and generalized Littlewood–Richardson coefficients.

Elliptic deformations of NS series are obtained by replacing $q$-Pochhammer factors with elliptic gamma and theta functions, as in the Komori–Noumi–Shiraishi elliptic kernel (Saito, 2013), leading to elliptic Ruijsenaars–Schneider polynomials, duality phenomena, Pieri rules, and novel Cauchy identities (Mironov et al., 2021).

7. Significance and Open Problems

The NS function provides a universal, explicit series expansion for eigenfunctions of Macdonald–Ruijsenaars systems, unifies symmetric and nonsymmetric integrable bases, and underpins generalizations along twisted and elliptic rays in DIM algebra. Open directions include:

• Canonical elliptic generalization of the defining linear equations for BA functions in the elliptic triad (Mironov et al., 2024).
• Explicit integral representations and orthogonality relations in the elliptic and bi-elliptic settings.
• Full characterization of functorial properties within the Shiraishi functor framework (Awata et al., 2020).

The NS series and its extensions structure a hierarchy of bispectral objects central to current developments in symmetric function theory and quantum integrable systems.