q-Racah Orthogonal Polynomial Ensemble

• The q-Racah orthogonal polynomial ensemble is a finite discrete system defined on a q-grid using q-Racah polynomials to encode recurrence relations and spectral measures.
• It connects determinantal point processes with finite Jacobi matrices, serving as a foundation for discrete quantum mechanics and integrable random tiling models.
• The ensemble underpins diverse applications including weighted lozenge tilings and spin chains, and admits extensions like multi-indexed and nonsymmetric q-Racah systems.

A q-Racah orthogonal polynomial ensemble is a finite discrete orthogonal polynomial system built from q-Racah polynomials and their associated weight, together with the equivalent determinantal, Jacobi-matrix, and spectral-measure structures that those polynomials encode. In the probabilistic sense, it is a discrete point process whose correlation kernel is generated by q-Racah orthogonal polynomials on a finite q-grid; in the operator-theoretic sense, it is the finite Jacobi matrix of the q-Racah recurrence and its spectral decomposition. This dual viewpoint places q-Racah ensembles simultaneously in the Askey scheme, discrete quantum mechanics with real shifts, weighted lozenge tilings, and exactly solvable finite systems such as inhomogeneous XX chains (Duits et al., 2023, Crampe et al., 13 Dec 2025).

1. Defining data and finite orthogonality

A standard q-Racah realization uses the spectral variable

$$\nu_n(y)=\mathbf q^{-y}+\gamma_n\delta\,\mathbf q^{y+1},$$

with q-Racah polynomials

$$R_j(\nu_n(y))= {}_4\phi_3\!\left( \begin{matrix} \mathbf q^{-j},\ \alpha\beta \mathbf q^{j+1},\ \mathbf q^{-y},\ \delta\gamma_n \mathbf q^{y+1}\ \alpha\mathbf q,\ \delta\beta\mathbf q,\ \gamma_n\mathbf q \end{matrix} ;\mathbf q,\mathbf q \right),$$

orthogonal on the finite lattice $y=0,1,\dots,M$. The weight is

$$w(y)= \frac{ (\alpha\mathbf{q},\beta\delta\mathbf{q},\gamma_n\mathbf{q},\gamma_n\delta \mathbf{q};\mathbf{q})_y \bigl(1-\gamma_n\delta \mathbf{q}^{2y+1}\bigr) } { (\mathbf{q},\alpha^{-1}\gamma_n\delta\mathbf{q},\beta^{-1}\gamma_n\mathbf{q},\delta\mathbf{q};\mathbf{q})_y\, (\alpha\beta\mathbf{q})^y\,(1-\gamma_n\delta\mathbf{q}) },$$

and the finite support is enforced by $\gamma_n=\mathbf q^{-(M+1)}$ (Duits et al., 2023).

Passing to monic and then orthonormal polynomials produces the three-term recurrence

$$\nu_n(y)\, r_j(y)=a_{j,n}r_{j+1}(y)+b_{j,n}r_j(y)+a_{j-1,n}r_{j-1}(y),$$

with

$$a_{j,n}=\sqrt{A_jC_{j+1}},\qquad b_{j,n}=1+\delta\gamma_n\mathbf q-A_j-C_j,$$

where

$$A_j= \frac{ (1-\gamma_n\mathbf q^{j+1}) (1-\alpha\mathbf q^{j+1}) (1-\alpha\beta\mathbf q^{j+1}) (1-\delta\beta\mathbf q^{j+1}) } { (1-\alpha\beta\mathbf q^{2j+1}) (1-\alpha\beta\mathbf q^{2j+2}) },$$

$$C_j= \frac{ \mathbf q(1-\mathbf q^j)(1-\beta\mathbf q^j) (\delta-\alpha\mathbf q^j) (\gamma_n-\alpha\beta\mathbf q^j) } { (1-\alpha\beta\mathbf q^{2j}) (1-\alpha\beta\mathbf q^{2j+1}) }.$$

These coefficients are the canonical finite Jacobi data of the ensemble (Duits et al., 2023).

In the discrete orthogonal polynomial ensemble formulation, the joint law is

$$\mathbb P(y_1,\dots,y_n)\propto \prod_{1\le i<j\le n}\bigl(\nu_n(y_i)-\nu_n(y_j)\bigr)^2 \prod_{j=1}^n w_n(y_j),$$

with $$R_j(\nu_n(y))= {}_4\phi_3\!\left( \begin{matrix} \mathbf q^{-j},\ \alpha\beta \mathbf q^{j+1},\ \mathbf q^{-y},\ \delta\gamma_n \mathbf q^{y+1}\ \alpha\mathbf q,\ \delta\beta\mathbf q,\ \gamma_n\mathbf q \end{matrix} ;\mathbf q,\mathbf q \right),$$0. This is the q-Racah OPE in the strict probabilistic sense used in random tilings and discrete integrable probability (Duits et al., 2023).

2. Jacobi matrices, spectral measures, and discrete quantum mechanics

The same structure admits an operator realization as a finite Jacobi matrix. In discrete quantum mechanics with real shifts, one works with

$$R_j(\nu_n(y))= {}_4\phi_3\!\left( \begin{matrix} \mathbf q^{-j},\ \alpha\beta \mathbf q^{j+1},\ \mathbf q^{-y},\ \delta\gamma_n \mathbf q^{y+1}\ \alpha\mathbf q,\ \delta\beta\mathbf q,\ \gamma_n\mathbf q \end{matrix} ;\mathbf q,\mathbf q \right),$$1

which factorizes as $$R_j(\nu_n(y))= {}_4\phi_3\!\left( \begin{matrix} \mathbf q^{-j},\ \alpha\beta \mathbf q^{j+1},\ \mathbf q^{-y},\ \delta\gamma_n \mathbf q^{y+1}\ \alpha\mathbf q,\ \delta\beta\mathbf q,\ \gamma_n\mathbf q \end{matrix} ;\mathbf q,\mathbf q \right),$$2. The eigenfunctions take the form

$$R_j(\nu_n(y))= {}_4\phi_3\!\left( \begin{matrix} \mathbf q^{-j},\ \alpha\beta \mathbf q^{j+1},\ \mathbf q^{-y},\ \delta\gamma_n \mathbf q^{y+1}\ \alpha\mathbf q,\ \delta\beta\mathbf q,\ \gamma_n\mathbf q \end{matrix} ;\mathbf q,\mathbf q \right),$$3

with $$R_j(\nu_n(y))= {}_4\phi_3\!\left( \begin{matrix} \mathbf q^{-j},\ \alpha\beta \mathbf q^{j+1},\ \mathbf q^{-y},\ \delta\gamma_n \mathbf q^{y+1}\ \alpha\mathbf q,\ \delta\beta\mathbf q,\ \gamma_n\mathbf q \end{matrix} ;\mathbf q,\mathbf q \right),$$4 determined by the ratio $$R_j(\nu_n(y))= {}_4\phi_3\!\left( \begin{matrix} \mathbf q^{-j},\ \alpha\beta \mathbf q^{j+1},\ \mathbf q^{-y},\ \delta\gamma_n \mathbf q^{y+1}\ \alpha\mathbf q,\ \delta\beta\mathbf q,\ \gamma_n\mathbf q \end{matrix} ;\mathbf q,\mathbf q \right),$$5. For q-Racah, one choice of sinusoidal coordinate is

$$R_j(\nu_n(y))= {}_4\phi_3\!\left( \begin{matrix} \mathbf q^{-j},\ \alpha\beta \mathbf q^{j+1},\ \mathbf q^{-y},\ \delta\gamma_n \mathbf q^{y+1}\ \alpha\mathbf q,\ \delta\beta\mathbf q,\ \gamma_n\mathbf q \end{matrix} ;\mathbf q,\mathbf q \right),$$6

while related papers use equivalent normalizations such as $$R_j(\nu_n(y))= {}_4\phi_3\!\left( \begin{matrix} \mathbf q^{-j},\ \alpha\beta \mathbf q^{j+1},\ \mathbf q^{-y},\ \delta\gamma_n \mathbf q^{y+1}\ \alpha\mathbf q,\ \delta\beta\mathbf q,\ \gamma_n\mathbf q \end{matrix} ;\mathbf q,\mathbf q \right),$$7. This parameter dependence of $$R_j(\nu_n(y))= {}_4\phi_3\!\left( \begin{matrix} \mathbf q^{-j},\ \alpha\beta \mathbf q^{j+1},\ \mathbf q^{-y},\ \delta\gamma_n \mathbf q^{y+1}\ \alpha\mathbf q,\ \delta\beta\mathbf q,\ \gamma_n\mathbf q \end{matrix} ;\mathbf q,\mathbf q \right),$$8 is a distinctive feature of the q-Racah case (Odake, 2017, Odake, 2017).

In one standard q-Racah parametrization,

$$R_j(\nu_n(y))= {}_4\phi_3\!\left( \begin{matrix} \mathbf q^{-j},\ \alpha\beta \mathbf q^{j+1},\ \mathbf q^{-y},\ \delta\gamma_n \mathbf q^{y+1}\ \alpha\mathbf q,\ \delta\beta\mathbf q,\ \gamma_n\mathbf q \end{matrix} ;\mathbf q,\mathbf q \right),$$9

$y=0,1,\dots,M$0

and the eigenvalues are

$y=0,1,\dots,M$1

The corresponding orthogonality measure is $y=0,1,\dots,M$2 on $y=0,1,\dots,M$3 (Odake, 2017).

This Jacobi-matrix picture is also the basis of deterministic q-Racah “ensembles.” In the one-excitation sector of an open XX spin chain, the Hamiltonian reduces to a tridiagonal matrix whose recurrence coefficients are identified with those of q-Racah polynomials. In that setting, the chain realizes a finite q-Racah orthogonal polynomial Jacobi ensemble: the eigenvalues lie on the q-Racah grid, and the eigenvector components are normalized q-Racah polynomials evaluated on that grid (Crampe et al., 13 Dec 2025).

3. Probabilistic realizations in lozenge tilings and dynamic ensembles

A principal probabilistic realization occurs in weighted lozenge tilings of a hexagon. For each vertical slice $y=0,1,\dots,M$4, the white-dot positions $y=0,1,\dots,M$5 have joint distribution

$y=0,1,\dots,M$6

with

$y=0,1,\dots,M$7

and $y=0,1,\dots,M$8 equal to the q-Racah weight above. Thus each fixed-time slice of the tiling model is exactly a q-Racah orthogonal polynomial ensemble (Duits et al., 2023).

The same paper studies a double-scaling regime in which $y=0,1,\dots,M$9, $$w(y)= \frac{ (\alpha\mathbf{q},\beta\delta\mathbf{q},\gamma_n\mathbf{q},\gamma_n\delta \mathbf{q};\mathbf{q})_y \bigl(1-\gamma_n\delta \mathbf{q}^{2y+1}\bigr) } { (\mathbf{q},\alpha^{-1}\gamma_n\delta\mathbf{q},\beta^{-1}\gamma_n\mathbf{q},\delta\mathbf{q};\mathbf{q})_y\, (\alpha\beta\mathbf{q})^y\,(1-\gamma_n\delta\mathbf{q}) },$$0, and the recurrence coefficients converge to slowly varying limits $$w(y)= \frac{ (\alpha\mathbf{q},\beta\delta\mathbf{q},\gamma_n\mathbf{q},\gamma_n\delta \mathbf{q};\mathbf{q})_y \bigl(1-\gamma_n\delta \mathbf{q}^{2y+1}\bigr) } { (\mathbf{q},\alpha^{-1}\gamma_n\delta\mathbf{q},\beta^{-1}\gamma_n\mathbf{q},\delta\mathbf{q};\mathbf{q})_y\, (\alpha\beta\mathbf{q})^y\,(1-\gamma_n\delta\mathbf{q}) },$$1. This yields a law of large numbers for linear statistics, an explicit density formula derived from the limiting recurrence coefficients, and a central limit theorem in which global fluctuations are governed by the Gaussian Free Field. In the hexagon model, the liquid-region coordinates are encoded by

$$w(y)= \frac{ (\alpha\mathbf{q},\beta\delta\mathbf{q},\gamma_n\mathbf{q},\gamma_n\delta \mathbf{q};\mathbf{q})_y \bigl(1-\gamma_n\delta \mathbf{q}^{2y+1}\bigr) } { (\mathbf{q},\alpha^{-1}\gamma_n\delta\mathbf{q},\beta^{-1}\gamma_n\mathbf{q},\delta\mathbf{q};\mathbf{q})_y\, (\alpha\beta\mathbf{q})^y\,(1-\gamma_n\delta\mathbf{q}) },$$2

and the fluctuation field in these coordinates converges to the Dirichlet GFF on a strip (Duits et al., 2023).

The q-Racah tiling model is also dynamic. The full space-time process is an extended OPE with transition kernels built from the orthogonal basis at successive times, and the asymptotic ratio of time-dependent coefficients is exponential in $$w(y)= \frac{ (\alpha\mathbf{q},\beta\delta\mathbf{q},\gamma_n\mathbf{q},\gamma_n\delta \mathbf{q};\mathbf{q})_y \bigl(1-\gamma_n\delta \mathbf{q}^{2y+1}\bigr) } { (\mathbf{q},\alpha^{-1}\gamma_n\delta\mathbf{q},\beta^{-1}\gamma_n\mathbf{q},\delta\mathbf{q};\mathbf{q})_y\, (\alpha\beta\mathbf{q})^y\,(1-\gamma_n\delta\mathbf{q}) },$$3. This is the mechanism by which the single-time q-Racah ensemble extends to a determinantal dynamic q-Racah process, with the same recurrence-coefficient asymptotics controlling both the limit shape and the fluctuation theory (Duits et al., 2023).

4. Deformations: multi-indexed, dual, nonsymmetric, and rational q-Racah-type systems

Several nonclassical deformations enlarge the notion of q-Racah ensemble beyond the standard finite OPE. Multi-indexed q-Racah polynomials are produced by multi-step Darboux transformations in discrete quantum mechanics. They are given by Casoratian determinant formulas built from virtual-state polynomials, remain orthogonal on the same finite lattice, and have missing low degrees. For the q-Racah case, the determinant entries retain explicit $$w(y)= \frac{ (\alpha\mathbf{q},\beta\delta\mathbf{q},\gamma_n\mathbf{q},\gamma_n\delta \mathbf{q};\mathbf{q})_y \bigl(1-\gamma_n\delta \mathbf{q}^{2y+1}\bigr) } { (\mathbf{q},\alpha^{-1}\gamma_n\delta\mathbf{q},\beta^{-1}\gamma_n\mathbf{q},\delta\mathbf{q};\mathbf{q})_y\, (\alpha\beta\mathbf{q})^y\,(1-\gamma_n\delta\mathbf{q}) },$$4-dependence because the sinusoidal coordinate $$w(y)= \frac{ (\alpha\mathbf{q},\beta\delta\mathbf{q},\gamma_n\mathbf{q},\gamma_n\delta \mathbf{q};\mathbf{q})_y \bigl(1-\gamma_n\delta \mathbf{q}^{2y+1}\bigr) } { (\mathbf{q},\alpha^{-1}\gamma_n\delta\mathbf{q},\beta^{-1}\gamma_n\mathbf{q},\delta\mathbf{q};\mathbf{q})_y\, (\alpha\beta\mathbf{q})^y\,(1-\gamma_n\delta\mathbf{q}) },$$5 depends on parameters; this is the exceptional feature singled out in the determinant formulas (Odake, 2017).

The dual multi-indexed q-Racah polynomials reverse the roles of degree and position. They satisfy an ordinary three-term recurrence and various higher-order difference equations; the paper characterizes them as ordinary orthogonal polynomials of Krall type. This yields new exactly solvable discrete quantum-mechanical systems with real shifts whose eigenvectors are described by dual multi-indexed q-Racah polynomials, and whose Hamiltonians are band matrices rather than tridiagonal Jacobi matrices (Odake, 2018).

A different extension is the nonsymmetric finite q-Racah system attached to a $$w(y)= \frac{ (\alpha\mathbf{q},\beta\delta\mathbf{q},\gamma_n\mathbf{q},\gamma_n\delta \mathbf{q};\mathbf{q})_y \bigl(1-\gamma_n\delta \mathbf{q}^{2y+1}\bigr) } { (\mathbf{q},\alpha^{-1}\gamma_n\delta\mathbf{q},\beta^{-1}\gamma_n\mathbf{q},\delta\mathbf{q};\mathbf{q})_y\, (\alpha\beta\mathbf{q})^y\,(1-\gamma_n\delta\mathbf{q}) },$$6-polynomial distance-regular graph of q-Racah type. There, an irreducible $$w(y)= \frac{ (\alpha\mathbf{q},\beta\delta\mathbf{q},\gamma_n\mathbf{q},\gamma_n\delta \mathbf{q};\mathbf{q})_y \bigl(1-\gamma_n\delta \mathbf{q}^{2y+1}\bigr) } { (\mathbf{q},\alpha^{-1}\gamma_n\delta\mathbf{q},\beta^{-1}\gamma_n\mathbf{q},\delta\mathbf{q};\mathbf{q})_y\, (\alpha\beta\mathbf{q})^y\,(1-\gamma_n\delta\mathbf{q}) },$$7-module of type $$w(y)= \frac{ (\alpha\mathbf{q},\beta\delta\mathbf{q},\gamma_n\mathbf{q},\gamma_n\delta \mathbf{q};\mathbf{q})_y \bigl(1-\gamma_n\delta \mathbf{q}^{2y+1}\bigr) } { (\mathbf{q},\alpha^{-1}\gamma_n\delta\mathbf{q},\beta^{-1}\gamma_n\mathbf{q},\delta\mathbf{q};\mathbf{q})_y\, (\alpha\beta\mathbf{q})^y\,(1-\gamma_n\delta\mathbf{q}) },$$8 produces a finite sequence of orthogonal Laurent polynomials. These are expressed through symmetric q-Racah data and identified with a finite nonsymmetric Askey–Wilson, hence nonsymmetric q-Racah, family. Orthogonality is discrete and supported on the finite spectrum of the Cherednik–Dunkl operator $$w(y)= \frac{ (\alpha\mathbf{q},\beta\delta\mathbf{q},\gamma_n\mathbf{q},\gamma_n\delta \mathbf{q};\mathbf{q})_y \bigl(1-\gamma_n\delta \mathbf{q}^{2y+1}\bigr) } { (\mathbf{q},\alpha^{-1}\gamma_n\delta\mathbf{q},\beta^{-1}\gamma_n\mathbf{q},\delta\mathbf{q};\mathbf{q})_y\, (\alpha\beta\mathbf{q})^y\,(1-\gamma_n\delta\mathbf{q}) },$$9 in the module (Lee, 2015).

The rational functions of q-Racah type introduced from $\gamma_n=\mathbf q^{-(M+1)}$0 and $\gamma_n=\mathbf q^{-(M+1)}$1 are not orthogonal polynomials in the usual Favard sense. They are overlap coefficients between solutions of a generalized eigenvalue problem and an eigenvalue problem, and they are expressed by terminating $\gamma_n=\mathbf q^{-(M+1)}$2 series of q-Racah type. Their natural structure is biorthogonal rather than orthogonal, and multivariate versions arise from coproduct constructions. This suggests a biorthogonal analogue of a q-Racah ensemble rather than a standard OPE (Groenevelt et al., 17 Jul 2025).

5. Algebraic and higher-rank frameworks

The q-Racah ensemble sits inside a broad algebraic landscape. In the higher-rank q-Bannai–Ito algebra $\gamma_n=\mathbf q^{-(M+1)}$3, multivariate $\gamma_n=\mathbf q^{-(M+1)}$4-Racah polynomials appear as connection coefficients between eigenbases of different maximal abelian subalgebras. They are orthogonal on the finite simplex

$\gamma_n=\mathbf q^{-(M+1)}$5

and satisfy commuting families of difference operators in both the position and degree variables. The paper also notes that, via the isomorphism with a higher-rank Askey–Wilson algebra under $\gamma_n=\mathbf q^{-(M+1)}$6, these results transfer directly to multivariate q-Racah-type structures (Bie et al., 2019).

A rank-one quantum-superalgebraic realization comes from the Racah problem for $\gamma_n=\mathbf q^{-(M+1)}$7. There, the intermediate Casimir operators generate a q-deformation of the Bannai–Ito algebra, and the Racah coefficients are expressed in terms of p-Racah polynomials with $\gamma_n=\mathbf q^{-(M+1)}$8. The resulting q-deformed Bannai–Ito polynomials are obtained from Askey–Wilson polynomials by a formal substitution $\gamma_n=\mathbf q^{-(M+1)}$9, and on finite grids they reduce to p-Racah, hence q-Racah-type, orthogonal systems (Genest et al., 2015).

Another related framework is matrix-valued harmonic analysis for the quantum analogue of $$\nu_n(y)\, r_j(y)=a_{j,n}r_{j+1}(y)+b_{j,n}r_j(y)+a_{j-1,n}r_{j-1}(y),$$0. The resulting matrix-valued orthogonal polynomials are orthogonal on $$\nu_n(y)\, r_j(y)=a_{j,n}r_{j+1}(y)+b_{j,n}r_j(y)+a_{j-1,n}r_{j-1}(y),$$1 with a positive matrix weight, and each scalar matrix entry is a finite sum involving continuous q-ultraspherical polynomials in the continuous variable and q-Racah polynomials in a discrete matrix index. In that setting, q-Racah polynomials govern internal discrete representation labels rather than the primary spectral variable (Aldenhoven et al., 2015).

6. Limits, finite-support features, and terminological scope

Within the Askey scheme, q-Racah is a top finite family, and many lower ensembles are obtained by degeneration. A limit formula that preserves orthogonal systems is

$$\nu_n(y)\, r_j(y)=a_{j,n}r_{j+1}(y)+b_{j,n}r_j(y)+a_{j-1,n}r_{j-1}(y),$$2

which realizes q-Racah $$\nu_n(y)\, r_j(y)=a_{j,n}r_{j+1}(y)+b_{j,n}r_j(y)+a_{j-1,n}r_{j-1}(y),$$3 big q-Jacobi as a genuine limit of orthogonal polynomial systems, not merely of individual polynomials. The same framework connects q-Racah to q-Hahn, little q-Jacobi, Hahn, Jacobi, and, under $$\nu_n(y)\, r_j(y)=a_{j,n}r_{j+1}(y)+b_{j,n}r_j(y)+a_{j-1,n}r_{j-1}(y),$$4, to classical Racah (Koornwinder, 2010).

Finite support also creates a distinctive factorization phenomenon. For the monic q-Racah polynomials of degrees $$\nu_n(y)\, r_j(y)=a_{j,n}r_{j+1}(y)+b_{j,n}r_j(y)+a_{j-1,n}r_{j-1}(y),$$5, the paper proves

$$\nu_n(y)\, r_j(y)=a_{j,n}r_{j+1}(y)+b_{j,n}r_j(y)+a_{j-1,n}r_{j-1}(y),$$6

where

$$\nu_n(y)\, r_j(y)=a_{j,n}r_{j+1}(y)+b_{j,n}r_j(y)+a_{j-1,n}r_{j-1}(y),$$7

Hence all higher-degree monic polynomials vanish on the finite lattice $$\nu_n(y)\, r_j(y)=a_{j,n}r_{j+1}(y)+b_{j,n}r_j(y)+a_{j-1,n}r_{j-1}(y),$$8; they are zero-norm eigenvectors of the associated $$\nu_n(y)\, r_j(y)=a_{j,n}r_{j+1}(y)+b_{j,n}r_j(y)+a_{j-1,n}r_{j-1}(y),$$9-dimensional Jacobi matrix. This factorization is the seed for the multi-indexed Darboux deformations described above (2207.14479).

A recurrent ambiguity concerns the word “ensemble.” In random-matrix and tiling theory, a q-Racah ensemble means a determinantal point process with joint density built from a Vandermonde factor and the q-Racah weight. In discrete quantum mechanics and spin-chain theory, the same phrase is often used more broadly for the finite Jacobi matrix, spectral measure, and orthogonal basis associated with q-Racah recurrence data. Both usages are present in the literature, and they are compatible because the Jacobi matrix and its orthogonality measure are precisely what generate the probabilistic OPE (Odake, 2017, Crampe et al., 13 Dec 2025).

A second ambiguity concerns related families. Para q-Racah systems, dual multi-indexed q-Racah systems, nonsymmetric q-Racah Laurent systems, and rational q-Racah-type functions are not identical objects. They share the same finite-grid or $$a_{j,n}=\sqrt{A_jC_{j+1}},\qquad b_{j,n}=1+\delta\gamma_n\mathbf q-A_j-C_j,$$0-type architecture, but they differ in whether the primary structure is orthogonal, biorthogonal, symmetric, nonsymmetric, or Krall-type. This suggests that “q-Racah orthogonal polynomial ensemble” is best treated as a core finite orthogonal framework with several adjacent, but not interchangeable, extensions (Crampe et al., 13 Dec 2025, Odake, 2018).