Ladder operators, bispectral algebra, and degenerations of the biorthogonal rational functions on elliptic grids: completing the rational-function analogue of the Askey-Wilson framework

The biorthogonal rational functions (BRF) expressible as very-well-poised ${}_{12}E_{11}$ elliptic hypergeometric series on elliptic grids form the top tier of the rational-function analogue of the Askey-Wilson polynomial framework. Spiridonov and Zhedanov (Ramanujan J. 13 (2007) 285-310) showed that a first-order divided-difference operator lowering the number of prescribed poles exists precisely on elliptic grids and outlined Padé interpolation schemes with prescribed poles and zeros there, deferring the derivation of the associated BRF to a companion work. That companion (Zhedanov, Rokko Lect. Math. 18 (2005) 323-363) realized the interpolants as ${}_{12}E_{11}$ series and established two biorthogonality relations: the Kronecker-string system, which reproduces the earlier finite discrete BRF with their generalized-eigenvalue spectral characterization (realized in Sklyanin-algebra representations by Rosengren, 2007), and a new vertical-string infinite system biorthogonal on a possibly dense set of grid points, for which no spectral theory exists. The literature status is open: the narrowly deferred constructions were published in 2005, but the operator-theoretic completion of the framework - the interpretation of the divided-difference operator $\mathcal{D}$ and companion multiplication operator $\mathcal{M}$ as ladder operators for the ${}_{12}E_{11}$ BRF, the quadratic (Askey-Wilson-type) algebra they generate, the spectral and completeness theory of the dense-grid system, and the classification of degeneration cases to lower levels - remains unbuilt. The corresponding algebraic program was completed only through the Hahn, $q$-Hahn, and Wilson levels (2020-2026), with the $q$-Racah level explicitly named as the next target, two levels below the elliptic system.

Background

The Askey-Wilson polynomials stand at the top of the $q$-Askey scheme of basic hypergeometric orthogonal polynomials. They are bispectral - satisfying both a three-term recurrence in the degree and a second-order $q$-difference equation in the variable - and the two operators generating these equations form the Askey-Wilson algebra, a quadratic algebra whose representation theory controls the polynomials' connection coefficients. A parallel hierarchy exists one step up in transcendentality for biorthogonal rational functions (BRF): pairs of rational functions $R_n$ and $R_n^*$ with prescribed poles satisfying a biorthogonality sum $\sum_s w_s R_n(x_s) R_m^*(x_s) = h_n \delta_{nm}$ over a discrete grid, or an analogous integral relation; the continuous hypergeometric case was introduced by Rahman (1986) through very-well-poised ${}_{10}\phi_9$ series (hypergeometric series whose parameters satisfy a maximally symmetric balancing pattern). The elliptic level of this hierarchy consists of BRF expressible through the very-well-poised elliptic hypergeometric series ${}_{12}E_{11}$ (the notation ${}_{12}V_{11}$ is also used), biorthogonal on elliptic grids - grids $x_s = z(s)$ parametrized by even order-two elliptic functions $z$, so that the grid points obey a symmetric biquadratic (Euler-Baxter) relation.

The source paper (Spiridonov-Zhedanov, Ramanujan J. 13 (2007) 285-310) approaches these objects from approximation theory. It characterizes the grids on which a first-order divided-difference operator $\mathcal{D}$, defined by $\mathcal{D}F(x) = (F(x(s+1))-F(x(s)))/(x(s+1)-x(s))$, maps rational functions with $n$ prescribed poles to rational functions with $n-1$ poles - the rational-function analogue of the lowering operators of the classical theory of orthogonal polynomials of a discrete variable. The answer is: precisely the elliptic grids, and the admissibility question proves equivalent to the Poncelet closure problem for two conics. On these grids the paper outlines explicit Padé-interpolation schemes with prescribed poles and zeros (the PPZ scheme, in which poles and zeros of the interpolants are fixed in advance and interpolation conditions are imposed at the grid points), constructs raising and lowering operators, and defers to a companion work the detailed derivation of the associated biorthogonal rational functions, stating that the correspondence between these operators and the discrete BRF of the authors' earlier papers - and with it the complete rational-function analogue of the Askey-Wilson framework, including spectral properties and the ladder-operator interpretation of the operators $\mathcal{D}$ and $\mathcal{M}$ - will be addressed elsewhere.

The companion work appeared: Zhedanov (Rokko Lect. Math. 18 (2005) 323-363) realizes the interpolants of the PPZ Padé table on the elliptic grid as ${}_{12}E_{11}$ series and derives two biorthogonality relations. The Kronecker (anti-diagonal) string of the Padé table reproduces the known finite discrete BRF of the earlier papers, for which a spectral characterization exists: the functions solve a generalized eigenvalue problem $Y_1 v = \lambda Y_2 v$ for a pair of tridiagonal (Jacobi-type) matrices, a structure Rosengren (Ramanujan J. 13 (2007) 133-168) realized inside representations of the Sklyanin algebra, the elliptic deformation of $sl(2)$ used in integrable lattice models; in his Introduction Rosengren remarks that classifying all limit cases of the family, along with their continuous relatives, is a desirable open task. The vertical string instead yields a new infinite BRF system biorthogonal on a set of grid points that may be dense in a real interval; for this system no spectral characterization, completeness theory, or asymptotic analysis exists. What has never been constructed, for either string, is the operator-theoretic core of the Askey-Wilson analogue: the exhibition of $\mathcal{D}$ and the multiplication operator $\mathcal{M}$ as ladder operators mapping $T_n$ to $T_{n\pm1}$, and the quadratic (Askey-Wilson-type) algebra they generate. At the levels below, this program is complete for Hahn-type and $q$-Hahn-type BRF - Bussiere-Gaboriaud-Vinet-Zhedanov (J. Math. Anal. Appl. 516 (2022) 126443) built the corresponding Askey-Wilson-type algebra from a bispectral triplet of difference operators - and the 2026 Leonard-trio machinery of Crampe et al. (Leonard pairs are the dual tridiagonal operator pairs underlying the bispectral classification; a Leonard trio is their three-operator extension) covers the Wilson rational functions and a dual-$q$-Hahn family; both works explicitly name the $q$-Racah level as the next target, two levels below the elliptic ${}_{12}E_{11}$ system, which remains the unclaimed top tier of the rational-function analogue of the Askey-Wilson framework.

Problem Statement

Complete the operator-theoretic (bispectral) theory of the biorthogonal rational functions on elliptic grids - the rational-function analogue of the Askey-Wilson framework at its elliptic top tier. The objects are those constructed in Zhedanov (Rokko Lect. Math. 18 (2005) 323-363, §8.3) from the prescribed-poles-and-zeros Padé interpolation scheme: the Kronecker-string pair $T_n(x;N)$, $T_n^*(x;N)$ of very-well-poised ${}_{12}E_{11}$ elliptic hypergeometric BRF, which coincide with the discrete elliptic BRF of Spiridonov-Zhedanov and admit the GEVP characterization, and the vertical-string infinite system $U_n(x;m)$, $U_n^*(x;m)$ biorthogonal on the grid $x_s = \mathrm{sn}(\omega(s-s_0);k)$ (with $\omega$ the real step of the grid, distinct from the deformation base $q$), which may be dense in $[-1,1]$; the operator $\mathcal{D}$ is the first-order pole-lowering divided-difference operator shown in Spiridonov-Zhedanov (Ramanujan J. 13 (2007) 285-310) to exist precisely on elliptic grids. The corresponding bispectral program is complete only through the Hahn, $q$-Hahn, and Wilson levels (Bussière et al. 2022; Crampé et al. 2026), with $q$-Racah named as the next target, and the limit-case classification of the elliptic family is the task Rosengren (Ramanujan J. 13 (2007), Introduction) remarked as desirable. The problem has three jointly defining deliverables:

1. Ladder structure. For the ${}_{12}E_{11}$ biorthogonal rational functions on the elliptic grid arising from the prescribed-poles-and-zeros Padé interpolation schemes (the Kronecker-string system $T_n$, $T_n^*$), exhibit the divided-difference operator $\mathcal{D}$ and a precisely defined companion multiplication-type operator $\mathcal{M}$ as raising and lowering operators: derive the explicit ladder relations mapping $T_n$ to $T_{n\pm1}$ (with the corresponding relations for the dual functions $T_n^*$), and determine the algebra generated by $\mathcal{D}$ and $\mathcal{M}$ - in particular, whether it is a finitely presented quadratic algebra, the elliptic analogue of the Askey-Wilson algebra - together with the bispectral equations (difference equations in the spectral variable and recurrence relations in the degree) that characterize the functions.

2. Spectral theory of the dense-grid system. For the vertical-string infinite BRF system $U_n$, $U_n^*$, biorthogonal on a possibly dense set of elliptic-grid points of an interval, establish a spectral characterization (a generalized-eigenvalue form or an explicit three-term recurrence in the degree) and the accompanying analytic theory: completeness in an explicitly identified function space appropriate to the dense discrete biorthogonality measure, and the asymptotic behavior of the functions and of the biorthogonality weights.

3. Degeneration scheme. Classify the limit cases of these elliptic-grid BRF under degeneration of the elliptic base ($q$ and $p$ denoting the base and the elliptic nome of the very-well-poised ${}_{12}E_{11}$ series; the limits are $p \to 0$ and $q \to 1$ with their admissible parameter scalings): determine the complete set of lower-level systems reached in each limit, and the fate of the ladder algebra in each limit.

The unifying objective is a verified elliptic top tier of the rational-function analogue of the Askey-Wilson framework: the ladder algebra (deliverable 1) is its core, the dense-grid spectral theory (deliverable 2) extends it to the genuinely new infinite system that the source program produced, and the degeneration scheme (deliverable 3) embeds it in the known hierarchy.

References

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