Twisted Pieri formulas beyond type $A$: extensions to $BC$-type root systems, $(q,t)$-Macdonald polynomials, and elliptic analogues

Shibukawa (SIGMA 2020) proved twisted Pieri formulas for Jack polynomials of type $A$ — explicit closed forms for the action of the iterated commutators of the total differentiation operator with the generating series of the Sekiguchi differential operators on suitably normalized Jack polynomials — and used them, via a finite combinatorial identity (a 'mysterious summation'), to rederive the Knop–Sahi higher-order difference equations for interpolation Jack polynomials, to prove Pieri formulas for interpolation Jack polynomials, and to obtain intertwining relations for the Jack kernel function ${}_0F_0{}^{(d)}(z,w)$. The paper explicitly poses the extension of this commutator-based machinery to three further settings, none of which has been carried out in the literature as of August 2026: (i) the $BC$-type root system (Koornwinder and $BC$-type interpolation polynomials), (ii) the $(q,t)$-deformed world of Macdonald and interpolation Macdonald polynomials, where a twisted-Pieri derivation of the difference equations originally due to Okounkov is asked for, and (iii) elliptic analogues built on Rains' elliptic interpolation theory.

Background

Jack polynomials $P_\lambda(z;d/2)$ are homogeneous symmetric polynomials in $r$ variables, indexed by partitions $\lambda$, characterized as eigenfunctions of the Calogero–Sutherland operator $D(z)=\sum_j z_j^2\partial_{z_j}^2+d\sum_{j\neq l}\frac{z_j^2}{z_j-z_l}\partial_{z_j}$, the second-order differential operator associated with the root system $A_{r-1}$. They are joint eigenfunctions of a commuting family of differential operators $H^{(d)}_{r,p}(z)$ (Sekiguchi operators, $p=0,\dots,r$), collected in the generating series $S^{(d)}_r(u;z)=\sum_{p=0}^r H^{(d)}_{r,p}(z)u^{r-p}$. Classical Pieri type formulas, due to Stanley and Lassalle, describe the action of $|\partial_z|:=\sum_j\partial_{z_j}$ and of multiplication by $|z|:=\sum_j z_j$ on two normalizations of the Jack polynomials, $\Phi^{(d)}_\lambda(z):=P_\lambda(z;d/2)/P_\lambda(\mathbf{1};d/2)$ and $\Psi^{(d)}_\lambda(z):=P_\lambda(z;d/2)/P^{ip}_\lambda(\lambda+\tfrac{d}{2}\delta;d/2)$, expressing them as sums over partitions obtained by lowering or raising one part, with explicit product coefficients $A^{(d)}_{\pm,i}(\lambda)$; here $\delta=(r-1,r-2,\dots,1,0)$ and $P^{ip}_\lambda$ denotes the interpolation (or shifted) Jack polynomial.

Interpolation Jack polynomials $P^{ip}_\lambda(z;d/2)$, introduced by Sahi, by Knop–Sahi, and by Okounkov–Olshanski, are the inhomogeneous symmetric polynomials defined by the vanishing conditions $P^{ip}_k(\lambda+\tfrac{d}{2}\delta;d/2)=0$ for $k\not\subseteq\lambda$ together with the condition that the top homogeneous part equals the Jack polynomial $P_\lambda$; they are a multivariate analogue of falling factorials and appear in binomial-type formulas for multivariate hypergeometric functions. Knop and Sahi proved that these polynomials are eigenfunctions of an explicit higher-order difference operator $D^{(d)ip}_r(u;x)=\sum_{J\subseteq[r]}(-1)^{|J|}I^{(d)}_{J^c}(u;x)A^{(d)}_{-,J}(x)\prod_{j\in J}(x_j+\tfrac{d}{2}(r-j))\,T^J_x$ (where $T^j_x$ shifts $x_j\mapsto x_j-1$ and $I^{(d)}_{J^c}(u;x)$ is an explicit product), with eigenvalues $I^{(d)}_r(u;\lambda)$ — the higher-order difference equations for interpolation Jack polynomials.

Shibukawa's twisted Pieri formulas give, for every $l=0,1,\dots,r$, the explicit action of the iterated commutator $\frac{(\operatorname{ad}|\partial_z|)^l}{l!}S^{(d)}_r(u;z)$ (with $(\operatorname{ad}A)(B):=AB-BA$) on $\Phi^{(d)}_\lambda$ and $\Psi^{(d)}_\lambda$: the result is a sum over partitions $\lambda-\epsilon_J$ obtained by lowering a subset $J$ of parts, with factorized coefficients. The proof is an induction on $l$ that closes precisely because of a finite combinatorial identity (Lemma 1 of the paper, the 'mysterious summation'). From the twisted Pieri formulas, Shibukawa rederived the Knop–Sahi difference equations, proved Pieri formulas for the interpolation Jack polynomials, and derived intertwining relations for the kernel function ${}_0F_0{}^{(d)}(z,w):=\sum_{\lambda\in\mathcal{P}}\Psi^{(d)}_\lambda(z)\Phi^{(d)}_\lambda(w)$, a multivariate analogue of $e^{zw}$.

All of this takes place in type $A$ at the Jack (differential) level. The neighboring settings are well developed but by different methods. Macdonald polynomials are the $(q,t)$-deformation of Jack polynomials, eigenfunctions of commuting $q$-difference operators; Koornwinder polynomials are their $BC_n$-type analogue. Sahi introduced the interpolation Macdonald polynomials, and Okounkov established the binomial formula for Macdonald polynomials together with the corresponding difference equations for the interpolation Macdonald polynomials (in Shibukawa's reading, with an 'Idea of Proof'); Okounkov also constructed $BC$-type interpolation Macdonald polynomials, proved Pieri-type rules and a binomial formula for Koornwinder polynomials, and showed that these $BC$ interpolation polynomials satisfy no rational $q$-difference equation — a structural constraint on any $BC$ extension. Rains then produced a family of difference equations satisfied by the $BC$ interpolation polynomials together with branching and Pieri generalizations, and, in the elliptic direction, constructed $BC_n$-symmetric abelian functions including elliptic interpolation functions with associated difference equations. van Diejen and Emsiz proved Pieri formulas for Macdonald's spherical functions and polynomials in the $BC$ setting, and Noumi, Shiraishi, and Komori developed a bispectral direct approach and kernel functions for Ruijsenaars-type difference operators of types $A$ and $BC$. Koornwinder revisited the $BC$-type interpolation Macdonald polynomials and their $q=1$ limit, which is the natural $BC$-type interpolation Jack family. Thus ordinary Pieri formulas and several difference equations already exist in the $BC$ and elliptic worlds; what does not exist anywhere is the twisted, commutator-based version of these identities and the derivation of the interpolation-polynomial difference equations and kernel intertwining relations from it.

Problem Statement

Extend Shibukawa's twisted Pieri formulas and their applications from type $A$ Jack polynomials in each of the following three settings, all three of which are requested; the three are independent extensions, so each may be pursued and resolved on its own:

1. BC-type. Generalize the twisted Pieri formulas from type $A$ to type $BC$: explicit closed-form expressions for the action of the iterated commutators of the total shift (or, at the differential level, total derivative) operator with the generating series of the commuting family of difference (van Diejen/Sekiguchi-type) operators on appropriate normalizations of the polynomials of type $BC$. The formulas are required for each of the following families: the Koornwinder (BC-type Macdonald) polynomials, the related but distinct inhomogeneous family of $BC$-type interpolation polynomials of Okounkov and Koornwinder, and the $q=1$ limits of these two families. Two known structural facts bear on any such generalization: Okounkov proved that the $BC$ interpolation Macdonald polynomials satisfy no rational $q$-difference equation, and Rains produced a family of difference equations that these polynomials do satisfy. In type $A$ the twisted Pieri formulas lead to Pieri rules and difference equations for interpolation polynomials and to kernel intertwining relations; $BC$ analogues of these consequences, where they exist, would be natural targets of the same generalization.

2. $(q,t)$-Macdonald case. Establish the $q,t$-analogues of the twisted Pieri formulas for Macdonald polynomials, and use them to (i) give a complete derivation (alternative to Okounkov's 'Idea of Proof') of the higher-order difference equations for the interpolation Macdonald polynomials, (ii) prove the Pieri formulas for these interpolation polynomials, and (iii) derive the intertwining relations for the Macdonald kernel.

3. Elliptic case. Explore a new method extending the twisted Pieri technique to the elliptic setting, i.e., to the elliptic interpolation functions of Rains' $BC_n$-symmetric abelian function theory and the Coskun–Gustafson well-poised Macdonald functions, with the concrete target of establishing or refuting the elliptic analogue of the twisted Pieri formulas.

References

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