---
title: Noumi–Shiraishi Function in Integrable Systems
url: https://www.emergentmind.com/topics/noumi-shiraishi-function
type: topic
---

# Noumi–Shiraishi Function in 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 $(a;q)_n := \prod_{s=0}^{n-1}(1-aq^s)$ [2411.16517].

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 [2411.16517, 2503.07592]:
- **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 [2503.07592, 2412.19588].

## 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 $q^{\partial_{z_i}} f(z_1,...,z_i,...) = f(z_1,...,z_i+1,...)$ [2411.16517, 2503.07592]. 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 [2504.02022]. 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 [2504.02022].

## 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 [2601.17453]. 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 [2601.17453].

## 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 $\xi(1)=0$, $\xi(z^{-1})\propto z^{-1}\xi(z)$ [2002.12746]. 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 [1301.4912], leading to elliptic Ruijsenaars–Schneider polynomials, duality phenomena, Pieri rules, and novel Cauchy identities [2103.02508].

## 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 [2412.19588].
- Explicit integral representations and orthogonality relations in the elliptic and bi-elliptic settings.
- Full characterization of functorial properties within the Shiraishi functor framework [2002.12746].

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

Source: https://www.emergentmind.com/topics/noumi-shiraishi-function