Mixed-type twisted Pieri formulas for Jack polynomials

Shibukawa's 2020 work established raising-type twisted Pieri formulas — explicit identities for the action of iterated commutators of the multiplication operator $|\mathbf{z}|=z_1+\cdots+z_r$ through the Sekiguchi operator $S_r^{(d)}(u;\mathbf{z})$ on two normalized families $\Phi_{\mathbf{k}}^{(d)}$ and $\Psi_{\mathbf{k}}^{(d)}$ of Jack polynomials — complementing his earlier falling-type formulas built from $|\partial_{\mathbf{z}}|=\sum_j\partial/\partial z_j$. The paper's concluding remarks explicitly display, with undetermined right-hand sides, the mixed-type identities combining both commutators, and express the expectation that they would contribute to formulas for root systems of type $BC$. This problem asks for closed-form expressions for $\big[\tfrac{(\operatorname{ad}|\mathbf{z}|)^m}{m!}\tfrac{(\operatorname{ad}|\partial_{\mathbf{z}}|)^n}{n!}S_r^{(d)}(u;\mathbf{z})\big]\Phi_{\mathbf{k}}^{(d)}(\mathbf{z})$ and the analogous expression on $\Psi_{\mathbf{k}}^{(d)}(\mathbf{z})$ for nonnegative integers $m,n$. The question remains open.

Background

Jack polynomials $P_{\mathbf{m}}(\mathbf{z};d/2)$ form a one-parameter family of symmetric polynomials in $r$ variables $\mathbf{z}=(z_1,\ldots,z_r)$, indexed by partitions $\mathbf{m}$ with at most $r$ parts; they specialize to Schur functions at a limiting value of the parameter $d$ and are eigenfunctions of a commuting family of differential operators. Sekiguchi (1977) and Debiard constructed such a commuting family $H_{r,p}^{(d)}(\mathbf{z})$, $p=0,1,\ldots,r$, of second-order differential operators conjugated by the Vandermonde-type factor $\Delta(\mathbf{z})=\prod_{i<j}(z_i-z_j)$, and Shibukawa packages them into the generating Sekiguchi operator $S_r^{(d)}(u;\mathbf{z})=\sum_{p=0}^{r}H_{r,p}^{(d)}(\mathbf{z})\,u^{r-p}$, where $u$ is an auxiliary indeterminate. The cited papers use two explicit normalizations of the Jack polynomial called kernel functions, $\Phi_{\mathbf{k}}^{(d)}(\mathbf{z})$ and $\Psi_{\mathbf{k}}^{(d)}(\mathbf{z})$, which are adapted to the binomial formula for Jack polynomials: the exponential generating operator $e^{\mathbf{z}}$ acts on them with coefficients given by evaluations of Jack polynomials at shifted points, in analogy with the classical binomial theorem. Writing $\operatorname{ad}(A)(B)=AB-BA$ for the commutator, $|\mathbf{z}|=z_1+\cdots+z_r$ for multiplication by the sum of the variables, and $|\partial_{\mathbf{z}}|=\sum_{j=1}^{r}\partial/\partial z_j$ for the total differentiation operator, a twisted Pieri formula is an explicit closed-form expression for the action of the 'twisted' operator $\big[\tfrac{(\operatorname{ad} X)^l}{l!}S_r^{(d)}(u;\mathbf{z})\big]$, with $X$ equal to $|\partial_{\mathbf{z}}|$ (falling type, lowering the degree) or $-|\mathbf{z}|$ (raising type), on $\Phi_{\mathbf{k}}^{(d)}(\mathbf{z})$ or $\Psi_{\mathbf{k}}^{(d)}(\mathbf{z})$. Shibukawa derived the falling-type formulas for $l=0,1,\ldots,r$ in a companion paper (SIGMA 16 (2020) 118) and the raising-type formulas in arXiv:2010.04241 (Theorem 1.3): both take the shape of finite sums over subsets $J\subseteq[r]=\{1,\ldots,r\}$ of size $l$, with explicit coefficients built from deformed elementary symmetric functions $I_J^{(d)}(u;\mathbf{k})$ and from the Pieri coefficients $A_{\pm,J}^{(d)}(\mathbf{k})$ that govern multiplication by $|\mathbf{z}|$. As applications, both papers recover, by a unified commutator method, the Knop–Sahi difference equations and Pieri formulas for interpolation Jack polynomials — the inhomogeneous deformations of Jack polynomials characterized, for the symmetric cone setting, by their zeros and binomial-type behavior (Knop and Sahi, Int. Math. Res. Not. 1996, no. 10, 473–486). The concluding section of arXiv:2010.04241 leaves the explicit evaluation of the mixed iterated commutator as the stated next stage, expected to also contribute to formulas for root systems of type $BC$ (the non-reduced crystallographic root systems whose associated $BC$-symmetric polynomial families, such as Koornwinder's multivariate orthogonal polynomials, generalize the Jack family).

Problem Statement

Derive explicit closed-form expressions for the mixed-type twisted Pieri operator identities displayed as open in Section 5 of Shibukawa (arXiv:2010.04241): for all nonnegative integers $m$ and $n$, find explicit formulas for

$$\left[\frac{(\operatorname{ad}|\mathbf{z}|)^{m}}{m!}\frac{(\operatorname{ad}|\partial_{\mathbf{z}}|)^{n}}{n!}S_r^{(d)}(u;\mathbf{z})\right]\Phi_{\mathbf{k}}^{(d)}(\mathbf{z}) \quad\text{and}\quad \left[\frac{(\operatorname{ad}|\mathbf{z}|)^{m}}{m!}\frac{(\operatorname{ad}|\partial_{\mathbf{z}}|)^{n}}{n!}S_r^{(d)}(u;\mathbf{z})\right]\Psi_{\mathbf{k}}^{(d)}(\mathbf{z}),$$

where $r\geq 1$ is the number of variables, $\mathbf{k}$ ranges over partitions with at most $r$ parts, $d\neq 0$ is the complex Jack parameter, $u$ is an indeterminate, $\operatorname{ad}(A)(B)=AB-BA$, $|\mathbf{z}|=z_1+\cdots+z_r$, $|\partial_{\mathbf{z}}|=\sum_{j=1}^{r}\partial/\partial z_j$, $S_r^{(d)}(u;\mathbf{z})=\sum_{p=0}^{r}H_{r,p}^{(d)}(\mathbf{z})\,u^{r-p}$ is the generating Sekiguchi operator, and $\Phi_{\mathbf{k}}^{(d)}$, $\Psi_{\mathbf{k}}^{(d)}$ are the two kernel-function normalizations of Jack polynomials defined in that paper (the sign convention $(\operatorname{ad}|\mathbf{z}|)^m$ versus $(-\operatorname{ad}|\mathbf{z}|)^m$ is immaterial). At $m=0$ with $0\leq n\leq r$ and at $n=0$ with $0\leq m\leq r$ these formulas coincide with the published falling-type (SIGMA 16 (2020) 118) and raising-type (arXiv:2010.04241, Theorem 1.3) twisted Pieri formulas, respectively. The source additionally expresses the expectation that such formulas contribute to descriptions of formulas for root systems of type $BC$ (the non-reduced crystallographic root systems whose associated $BC$-symmetric polynomial families, such as Koornwinder's multivariate orthogonal polynomials, generalize the Jack family) and to higher-order difference equations for multivariate Meixner polynomials (the multivariate discrete-orthogonality counterparts of the Jack-type families, associated with the negative multinomial distribution).

References

1. Genki Shibukawa, "Raising type twisted Pieri formulas for Jack polynomials and their applications to interpolation Jack polynomials", arXiv:2010.04241 (2020), https://arxiv.org/abs/2010.04241 (preprint)
2. Genki Shibukawa, "New Pieri type formulas for Jack polynomials and their applications to interpolation Jack polynomials", SIGMA, Symmetry Integrability Geom. Methods Appl. 16 (2020), Paper 118, 11 pp., DOI 10.3842/sigma.2020.118, https://arxiv.org/abs/2004.12875
3. Genki Shibukawa, "Higher order difference equations for interpolation Jack polynomials" (Japanese with English summary), RIMS Kôkyûroku Bessatsu B87 (2021), 7–26, https://repository.kulib.kyoto-u.ac.jp/dspace/handle/2433/265825
4. Friedrich Knop and Siddhartha Sahi, "Difference equations and symmetric polynomials defined by their zeros", International Mathematics Research Notices 1996, No. 10, 473–486, DOI 10.1155/S1073792896000311
5. Yosuke Kawamoto and Genki Shibukawa, "The intertwining property for β-Laguerre processes and integral operators for Jack polynomials", Latin American Journal of Probability and Mathematical Statistics (ALEA) 23 (2026), Paper 10, DOI 10.30757/alea.v23-10, https://arxiv.org/abs/2505.23139
6. Wolter Groenevelt and Joop Vermeulen, "Multivariate Meixner polynomials related to holomorphic discrete series representations of SU(1,d)", Indagationes Mathematicae, New Series 36 (2025), No. 1, 171–187, DOI 10.1016/j.indag.2024.04.010