Noncommutative Bi-orthogonal Polynomial Systems

• Non-commutative multiple bi-orthogonal polynomial systems are algebraic frameworks defined over noncommutative algebras that impose several orthogonality constraints via quasideterminants.
• The formalism unifies bi-orthogonality with multiple orthogonality and links them to integrable dynamics, including noncommutative discrete Toda lattice equations.
• This framework bridges abstract polynomial constructions with practical applications in matrix models, ASEP, and noncommutative quantum mechanics.

Non-commutative multiple bi-orthogonal polynomial systems are polynomial systems over a non-commutative coefficient algebra, typically a division ring or a matrix algebra, subject simultaneously to multiple orthogonality constraints and left-right bi-orthogonality. In the formal framework introduced by Adam Doliwa, the basic data are $r$ families of formal bi-moments $(\nu_{ij}^{(k)})_{i,j\ge 0}$, $k=1,\dots,r$, in a free division ring $\mathcal{R}$, from which one defines $r$ bilinear forms and monic polynomials indexed by multi-indices $s=(s_1,\dots,s_r)$. These polynomials satisfy $(x^i,Q_s(x))_{(k)}=0$ for $i=0,\dots,s_k-1$, admit quasideterminantal expressions, and their normalization functions satisfy non-commutative Hirota equations; after a Hankel-type reduction and a standard discrete-time shift of moments, the same formalism yields a non-commutative multidimensional discrete-time Toda system (Doliwa, 2 Oct 2025). In this sense, the subject unifies three previously distinct lines of development: non-commutative orthogonality, bi-orthogonality, and multiple orthogonality.

1. Genealogy of the subject

The non-commutative multiple theory rests on an earlier algebraic theory of non-commutative bi-orthogonality. Emily Sergel formulated noncommutative biorthogonal polynomials over a division ring $R$ with center $C$, using a biadditive pairing

$(\nu_{ij}^{(k)})_{i,j\ge 0}$0

determined by bimoments $(\nu_{ij}^{(k)})_{i,j\ge 0}$1. In that setting, a biorthogonal system is a pair $(\nu_{ij}^{(k)})_{i,j\ge 0}$2 with $(\nu_{ij}^{(k)})_{i,j\ge 0}$3 and $(\nu_{ij}^{(k)})_{i,j\ge 0}$4 for $(\nu_{ij}^{(k)})_{i,j\ge 0}$5, with quasideterminants replacing determinants throughout. Sergel also proved a broad extension of Favard’s theorem: any two sequences of degree $(\nu_{ij}^{(k)})_{i,j\ge 0}$6 polynomials can be made biorthogonal for a uniquely constructed system of bimoments (Sergel, 2010).

That framework was not yet “multiple” in the classical sense. This distinction is explicit in the study of deformed complex Hermite polynomials in noncommutative quantum mechanics: the $(\nu_{ij}^{(k)})_{i,j\ge 0}$7-deformed families $(\nu_{ij}^{(k)})_{i,j\ge 0}$8 and $(\nu_{ij}^{(k)})_{i,j\ge 0}$9 are biorthogonal with respect to a single measure on $k=1,\dots,r$0, but the construction is stated not to be of the multiple orthogonal type, because it uses one measure and two dual families rather than several functionals or weights (Balogh et al., 2013).

The genuinely multiple non-commutative setting therefore emerges only when several bilinear forms or moment functionals are imposed simultaneously. Doliwa’s formal construction makes that unification explicit by combining the multi-indexed constraints of multiple orthogonality with the left-right asymmetry of bi-orthogonality and the quasideterminantal machinery required by non-commutativity (Doliwa, 2 Oct 2025).

2. Formal algebraic structure

In the formal theory, one starts from $k=1,\dots,r$1 matrices of formal bi-moments

$k=1,\dots,r$2

with entries in a free division ring $k=1,\dots,r$3. These define $k=1,\dots,r$4 bilinear forms on $k=1,\dots,r$5: $k=1,\dots,r$6 For a multi-index $k=1,\dots,r$7, with $k=1,\dots,r$8, one forms a block matrix $k=1,\dots,r$9 from the first $\mathcal{R}$0 rows of the $\mathcal{R}$1-th bi-moment array. The monic non-commutative multiple bi-orthogonal polynomial is then defined by a quasideterminant

$\mathcal{R}$2

and satisfies the orthogonality conditions

$\mathcal{R}$3

(Doliwa, 2 Oct 2025).

The formalism also has a dual right-action version: one may define polynomials $\mathcal{R}$4 by requiring $\mathcal{R}$5 for the corresponding ranges of $\mathcal{R}$6, with dual normalization potentials $\mathcal{R}$7 (Doliwa, 2 Oct 2025). This left-right asymmetry is intrinsic. It is the direct analogue, in a multiple setting, of the separate polynomial families in the earlier noncommutative biorthogonal construction over $\mathcal{R}$8 and $\mathcal{R}$9 (Sergel, 2010).

Conceptually, the structure generalizes three distinct theories at once. It is “multiple” because the constraints are indexed by $r$0; it is “bi-orthogonal” because the bilinear forms are not assumed symmetric; and it is “non-commutative” because coefficients and moments take values in a non-commutative algebra, so order matters and quasideterminants replace determinants (Doliwa, 2 Oct 2025).

3. Quasideterminants and normalization data

The normalization functions are as central as the polynomials themselves. For each $r$1, Doliwa defines

$r$2

These functions play the role that squared norms, leading principal minors, or $r$3-functions play in commutative theories (Doliwa, 2 Oct 2025).

The use of quasideterminants is structurally unavoidable in non-commutative settings. Sergel’s original construction of noncommutative biorthogonal polynomials is already phrased entirely in quasideterminantal form, with the polynomials $r$4 and $r$5 extracted from moment matrices by noncommutative analogues of Cramer-type formulas (Sergel, 2010). Matrix-valued variants show the same pattern in a more concrete realization: matrix-valued Cauchy bi-orthogonal polynomials and matrix-valued $r$6-deformed bi-orthogonal polynomials are both represented by quasideterminants built from block moment matrices (Li et al., 2022, Gilson et al., 2023).

A distinguished reduction occurs when the bi-moments are of Hankel type,

$r$7

Then the formal multiple bi-orthogonal system reduces to a theory of non-commutative multiple orthogonal polynomials, with bilinear forms

$r$8

and the quasideterminants become Hankel-type expressions in the moments (Doliwa, 2 Oct 2025). This reduction is the precise non-commutative analogue of passing from general bi-moment arrays to moment sequences in commutative orthogonal polynomial theory.

4. Integrable-system content

The normalization functions $r$9 satisfy the non-commutative Hirota system

$s=(s_1,\dots,s_r)$0

together with the three-index relation

$s=(s_1,\dots,s_r)$1

for distinct $s=(s_1,\dots,s_r)$2. The associated linear system is

$s=(s_1,\dots,s_r)$3

The proof is based on quasideterminant versions of the Sylvester and homological identities, so integrability is not appended externally; it is encoded in the same algebraic identities that define the polynomials (Doliwa, 2 Oct 2025).

Under the Hankel reduction, one may introduce a discrete-time variable by shifting moments,

$s=(s_1,\dots,s_r)$4

Then the polynomials satisfy

$s=(s_1,\dots,s_r)$5

and the compatibility conditions become the non-commutative multidimensional discrete-time Toda lattice

$s=(s_1,\dots,s_r)$6

(Doliwa, 2 Oct 2025).

This integrable interpretation is consistent with several earlier non-commutative bi-orthogonal models. Matrix-valued Cauchy bi-orthogonal polynomials satisfy a four-term recurrence whose coefficients obey a noncommutative C-Toda lattice with a Lax pair built from fractional differential operators with non-abelian variables (Li et al., 2022). Matrix-valued $s=(s_1,\dots,s_r)$7-deformed bi-orthogonal polynomials generate non-commutative Toda-type hierarchies, and Wronski quasi-determinants are used there as non-commutative $s=(s_1,\dots,s_r)$8-functions; moment modification implements Bäcklund transformations (Gilson et al., 2023). Adjacent families of matrix-valued $s=(s_1,\dots,s_r)$9-deformed bi-orthogonal polynomials likewise produce a fully discrete non-commutative hungry Toda lattice, which is then used as a pre-precessing algorithm for block Hessenberg matrices (Wang et al., 2024).

5. Model realizations and representative families

One major source of non-commutative bi-orthogonal systems is the asymmetric simple exclusion process. In the tensor-algebraic formulation of ASEP, a linear map $(x^i,Q_s(x))_{(k)}=0$0 on a tensor algebra generated by non-commuting symbols $(x^i,Q_s(x))_{(k)}=0$1 gives a bi-moment matrix

$(x^i,Q_s(x))_{(k)}=0$2

From the determinant and $(x^i,Q_s(x))_{(k)}=0$3 decomposition of this matrix, one constructs monic polynomials $(x^i,Q_s(x))_{(k)}=0$4 and $(x^i,Q_s(x))_{(k)}=0$5 satisfying

$(x^i,Q_s(x))_{(k)}=0$6

In one version, the polynomials satisfy first-order recurrence relations and the second moment defines a tridiagonal matrix linked to Chebyshev-like orthogonal polynomials; in the five-parameter case, the boundary basis diagonalizes the bi-moment matrix, while $(x^i,Q_s(x))_{(k)}=0$7 becomes tridiagonal and yields a family having the same moments as the Askey–Wilson polynomials (Brak et al., 2014, Brak et al., 2019).

A very different realization comes from noncommutative quantum mechanics. The $(x^i,Q_s(x))_{(k)}=0$8-deformation of complex Hermite polynomials produces families $(x^i,Q_s(x))_{(k)}=0$9 and $i=0,\dots,s_k-1$0 with

$i=0,\dots,s_k-1$1

This is a biorthogonal family adapted to non-unitary mode mixing, but it is explicitly distinguished from multiple orthogonality because the construction still uses a single measure (Balogh et al., 2013).

Another direction is provided by non-commutative Laurent bi-orthogonal polynomials. Bao Wang and Shi-Hao Li define a moment pairing

$i=0,\dots,s_k-1$2

construct $i=0,\dots,s_k-1$3 and $i=0,\dots,s_k-1$4 by quasideterminants, and derive a non-commutative three-term recurrence. Their recurrence coefficients realize the non-commutative relativistic Toda dynamics underlying the non-commutative leapfrog map, whose integrability is expressed by a discrete zero-curvature equation and a compatible network Poisson structure (Wang et al., 2023).

6. Distinctions, reductions, and significance

A recurrent source of confusion is the relation among “matrix-valued,” “bi-orthogonal,” and “multiple.” These notions overlap but are not identical. Matrix-valued orthogonal or bi-orthogonal polynomials are non-commutative because coefficients and weights are matrices, yet they need not be multiple. Bi-orthogonal systems involve two dual families or asymmetric pairings, yet they need not involve several functionals. The deformed complex Hermite families provide a clear example of bi-orthogonality without multiple orthogonality (Balogh et al., 2013), while Sergel’s theory provides noncommutative biorthogonality without a multi-indexed family of constraints (Sergel, 2010).

The formal theory of non-commutative multiple bi-orthogonal polynomials is significant because it subsumes all three aspects in a single algebraic object. Its normalization functions solve non-commutative Hirota equations, its polynomials furnish the associated linear problem, and its Hankel reduction gives a standard route to a multidimensional non-commutative Toda system (Doliwa, 2 Oct 2025). This establishes a direct bridge between quasideterminantal polynomial algebra and discrete integrable hierarchies.

The subject also has concrete analytic and computational ramifications. In matrix-valued settings, non-commutative bi-orthogonal recurrences lead to block Hessenberg operators and to integrable numerical algorithms such as the generalized block qd-algorithm (Wang et al., 2024). In statistical mechanics, the same machinery explains matrix product representations and boundary bases in ASEP (Brak et al., 2014, Brak et al., 2019). In discrete geometry and integrable maps, Laurent bi-orthogonal polynomials encode the leapfrog dynamics and its zero-curvature structure (Wang et al., 2023).

The present theory remains explicitly formal in its most general multiple form, being built from formal bi-moments in a free division ring (Doliwa, 2 Oct 2025). This suggests that analytic realizations—measure-theoretic, spectral, or Riemann–Hilbert theoretic—are likely to remain model-dependent. A plausible implication is that future work will proceed by importing the formal multiple framework into concrete non-commutative models, much as matrix orthogonal polynomials have already been connected to IIKS kernels, Riemann–Hilbert problems, and non-commutative Painlevé equations in the orthogonal, rather than bi-orthogonal, setting (Cafasso et al., 2013).