Dual Polynomials of the Multi-Indexed ($q$-)Racah Orthogonal Polynomials

We consider dual polynomials of the multi-indexed ($q$-)Racah orthogonal polynomials. The $M$-indexed ($q$-)Racah polynomials satisfy the second order difference equations and various $1+2L$ ($L\geq M+1$) term recurrence relations with constant coefficients. Therefore their dual polynomials satisfy the three term recurrence relations and various $2L$-th order difference equations. This means that the dual multi-indexed ($q$-)Racah polynomials are ordinary orthogonal polynomials and the Krall-type. We obtain new exactly solvable discrete quantum mechanics with real shifts, whose eigenvectors are described by the dual multi-indexed ($q$-)Racah polynomials. These quantum systems satisfy the closure relations, from which the creation/annihilation operators are obtained, but they are not shape invariant.

💡 Research Summary

The paper investigates the dual polynomials associated with the multi‑indexed (q‑)Racah orthogonal polynomials and uses them to construct new exactly solvable discrete quantum mechanical (rdQM) systems. Ordinary orthogonal polynomials satisfy a second‑order differential (or difference) equation and a three‑term recurrence relation. By contrast, the multi‑indexed (q‑)Racah polynomials, which arise from Darboux transformations with virtual‑state seed solutions, obey a second‑order difference equation but fail to satisfy a three‑term recurrence; instead they satisfy a family of 1 + 2L term recurrence relations with constant coefficients, where L ≥ M + 1 and M is the number of indices removed.

The authors first recapitulate the construction of the M‑indexed (q‑)Racah polynomials P_D,n(x;λ). The index set D={d₁,…,d_M} determines a denominator polynomial Ξ_D(η) and a degree shift ℓ_D. The polynomials P_D,n(η) have degree ℓ_D + n and are eigenfunctions of a tridiagonal Hamiltonian H_D acting on a finite lattice 0≤x≤N. Their orthogonality weight involves the ground‑state wavefunction ψ_D(x) and the factor Ξ_D(1). A key result (Theorem 1) shows that for any auxiliary polynomial Y(η) one can build a polynomial X(x) of degree L=ℓ_D+deg Y+1 such that X(x) P_D,n(x) expands into a linear combination of P_D,n+k(x) with k ranging from –L to +L. The coefficients r_{X,D}^{n,k} obey simple symmetry relations but have no closed‑form expression for general parameters.

The dual polynomials Q_D,x(n) are defined by normalising P_D,n(x) with P_D,0(x): Q_D,x(n)=P_D,n(x)/P_D,0(x). By interchanging the roles of the lattice variable x and the degree n, Q_D,x(n) becomes a polynomial in the spectral variable E_n, denoted Q_D,x(E). The three‑term recurrence relation for Q_D,x(E) follows directly from transposing the second‑order difference equation for P_D,n(x). Explicitly, \