Higher Jacobi Polynomials and Extensions

• Higher Jacobi polynomials are generalized extensions of classical Jacobi polynomials characterized by higher-order differential equations, extended recurrence relations, and modified orthogonality conditions.
• They encompass distinct variants—Jacobi-type, exceptional, multi-indexed, generalized, and coding-theoretic—that preserve different components of the classical framework through methods like Darboux transformations and finite-term recurrences.
• Recent research highlights their applications in spectral theory, bispectrality, multivariate analysis, and coding theory, showcasing the structural richness and practical implications of these polynomial systems.

Searching arXiv for recent and foundational papers on higher Jacobi polynomials to ground the article. {"query":"higher Jacobi polynomials exceptional Jacobi Krall Jacobi Jacobi-type arXiv", "max_results": 10} The expression higher Jacobi polynomials is used for several extensions of the classical Jacobi system. At the classical level, the monic Jacobi polynomials $P_n^{\alpha,\beta}(x)$ are eigenfunctions of the differential operator

$$L_{\alpha,\beta}[y](x)=(1-x^2)\,y''(x)+\bigl[\beta-\alpha-(\alpha+\beta+2)x\bigr]\,y'(x),$$

with eigenvalue $n(n+\alpha+\beta+1)$, and are orthogonal on $(-1,1)$ with respect to

$$w_{\alpha,\beta}(x)=(1-x)^\alpha(1+x)^\beta,\qquad \alpha,\beta>-1$$

(Garcia-Ferrero et al., 2024). The higher theories surveyed in the literature extend this setting in distinct directions: higher-order differential operators and finite-term recurrences, rational Darboux deformations with missing degrees, endpoint-mass perturbations of the measure, multivariate and $q\to -1$ limits, and coding-theoretic Jacobi invariants (Durán et al., 2020, Markett, 2017, Ho et al., 2012, Chakraborty et al., 16 Aug 2025).

1. Classical baseline and principal higher variants

Several non-equivalent notions of “higher” occur in the Jacobi literature. One line studies Jacobi-type polynomials obtained as suitable linear combinations of a fixed number of consecutive classical Jacobi polynomials; these are eigenfunctions of a higher-order differential operator and satisfy a higher-order finite-term recurrence relation (Durán et al., 2020). A second line studies exceptional Jacobi polynomials, i.e. polynomial eigenfunctions of a second-order operator with a finite set of missing degrees, together with their classification by spectral diagrams and rational Darboux transformations (Garcia-Ferrero et al., 2024). A third line studies multi-indexed Jacobi polynomials, produced by multiple Darboux transformations of the Pöschl–Teller Hamiltonian and represented by Wronskians; these satisfy a deformed second-order equation and typically have holes in their degrees (Ho et al., 2012, Odake et al., 2016). A fourth line concerns generalized Jacobi or Jacobi-type polynomials orthogonal with respect to the classical Jacobi measure plus one or two endpoint masses, and satisfying differential equations of arbitrarily high even order or of order $2\alpha+2\beta+6$ in the two-mass case (Markett, 2017, Markett, 2017). A fifth use of the term appears in coding theory, where higher Jacobi polynomials are generating functions built from higher subcode-weight distributions (Chakraborty et al., 16 Aug 2025).

Usage in the literature	Defining mechanism	Characteristic feature
Exceptional Jacobi polynomials	Rational Darboux transformations	Missing degrees
Jacobi-type polynomials	Linear combinations of consecutive Jacobi polynomials	Higher-order differential operator and finite-term recurrence
Multi-indexed Jacobi polynomials	Multiple Darboux transformations, Wronskians	Deformed weight and holes in the degree sequence
Generalized Jacobi polynomials	Jacobi weight plus endpoint masses	Higher-order differential equation
Higher Jacobi polynomials for codes	Higher subcode-weight enumerators	MacWilliams-type transforms

This multiplicity of meanings is structurally important. It shows that “higher” does not single out one canonical family, but rather a cluster of extensions that preserve different parts of the classical Jacobi package: orthogonality, spectral equations, recurrence, or hypergeometric structure.

2. Higher-order differential operators and bispectrality

For Jacobi-type polynomials in the sense of Durán and de la Iglesia, one starts with two finite sets of positive integers

$$G=\{g_1<\cdots<g_{m_1}\},\qquad H=\{h_1<\cdots<h_{m_2}\},$$

chooses monic polynomials $R_g$ and $S_h$, and defines $$L_{\alpha,\beta}[y](x)=(1-x^2)\,y''(x)+\bigl[\beta-\alpha-(\alpha+\beta+2)x\bigr]\,y'(x),$$0 by a quasi-Casoratian determinant. For $$L_{\alpha,\beta}[y](x)=(1-x^2)\,y''(x)+\bigl[\beta-\alpha-(\alpha+\beta+2)x\bigr]\,y'(x),$$1, with $$L_{\alpha,\beta}[y](x)=(1-x^2)\,y''(x)+\bigl[\beta-\alpha-(\alpha+\beta+2)x\bigr]\,y'(x),$$2, each $$L_{\alpha,\beta}[y](x)=(1-x^2)\,y''(x)+\bigl[\beta-\alpha-(\alpha+\beta+2)x\bigr]\,y'(x),$$3 is a linear combination of

$$L_{\alpha,\beta}[y](x)=(1-x^2)\,y''(x)+\bigl[\beta-\alpha-(\alpha+\beta+2)x\bigr]\,y'(x),$$4

There exists an even integer $$L_{\alpha,\beta}[y](x)=(1-x^2)\,y''(x)+\bigl[\beta-\alpha-(\alpha+\beta+2)x\bigr]\,y'(x),$$5 and a differential operator

$$L_{\alpha,\beta}[y](x)=(1-x^2)\,y''(x)+\bigl[\beta-\alpha-(\alpha+\beta+2)x\bigr]\,y'(x),$$6

such that

$$L_{\alpha,\beta}[y](x)=(1-x^2)\,y''(x)+\bigl[\beta-\alpha-(\alpha+\beta+2)x\bigr]\,y'(x),$$7

and $$L_{\alpha,\beta}[y](x)=(1-x^2)\,y''(x)+\bigl[\beta-\alpha-(\alpha+\beta+2)x\bigr]\,y'(x),$$8. The same families satisfy higher-order recurrence relations: $$L_{\alpha,\beta}[y](x)=(1-x^2)\,y''(x)+\bigl[\beta-\alpha-(\alpha+\beta+2)x\bigr]\,y'(x),$$9 for any polynomial $n(n+\alpha+\beta+1)$0 with $n(n+\alpha+\beta+1)$1; in particular, $n(n+\alpha+\beta+1)$2 yields the minimal $n(n+\alpha+\beta+1)$3-term recurrence (Durán et al., 2020).

This pair of properties is the bispectral core of the theory: the same family diagonalizes a differential operator in $n(n+\alpha+\beta+1)$4 and a finite-term difference operator in the degree index $n(n+\alpha+\beta+1)$5. The paper proves that these higher-order recurrences always exist for Jacobi-type polynomials, and also proves a sharp restriction on orthogonality: the Krall–Jacobi families are the only Jacobi type polynomials which are orthogonal with respect to a measure on the real line (Durán et al., 2020). That statement separates algebraic bispectrality from positivity of the underlying measure.

A different higher-order picture appears in Koornwinder’s generalized Jacobi polynomials $n(n+\alpha+\beta+1)$6, orthogonal with respect to

$n(n+\alpha+\beta+1)$7

For $n(n+\alpha+\beta+1)$8 and $n(n+\alpha+\beta+1)$9, the differential equation has order $(-1,1)$0, and Markett gives it as a linear combination of four elementary components, proving symmetry of the operator with respect to the scalar product induced by the measure (Markett, 2017). In the single-mass case

$(-1,1)$1

Markett derives a completely elementary representation of the even-order Jacobi-type differential operator, as well as a new factorization yielding a recurrence with respect to the order of the equation (Markett, 2017).

3. Exceptional and multi-indexed Jacobi polynomials

Exceptional Jacobi operators retain a second-order principal part but admit polynomial eigenfunctions of every degree except for a finite exceptional set. In the rational gauge, such an operator may be written as

$(-1,1)$2

where $(-1,1)$3 is a polynomial that does not vanish at $(-1,1)$4. Their eigenfunctions are quasi-rational functions

$(-1,1)$5

and every exceptional Jacobi operator is obtained from a classical Jacobi operator by a finite chain of rational Darboux transformations (Garcia-Ferrero et al., 2024). The classification theorem organizes these operators into six degeneracy classes according to whether $(-1,1)$6 or $(-1,1)$7 assume integer values, and establishes a one-to-one correspondence between exceptional Jacobi operators and combinatorial spectral diagrams (Garcia-Ferrero et al., 2024).

The explicit construction uses Wronskians. If $(-1,1)$8 are distinct quasi-rational seed eigenfunctions of $(-1,1)$9, then the exceptional polynomials are

$$w_{\alpha,\beta}(x)=(1-x)^\alpha(1+x)^\beta,\qquad \alpha,\beta>-1$$0

Orthogonality survives in deformed form: $$w_{\alpha,\beta}(x)=(1-x)^\alpha(1+x)^\beta,\qquad \alpha,\beta>-1$$1 with generalized orthogonality on $$w_{\alpha,\beta}(x)=(1-x)^\alpha(1+x)^\beta,\qquad \alpha,\beta>-1$$2 (Garcia-Ferrero et al., 2024).

Multi-indexed Jacobi polynomials arise from an allied but not identical Darboux scheme. Starting from the Pöschl–Teller Hamiltonian

$$w_{\alpha,\beta}(x)=(1-x)^\alpha(1+x)^\beta,\qquad \alpha,\beta>-1$$3

one applies $$w_{\alpha,\beta}(x)=(1-x)^\alpha(1+x)^\beta,\qquad \alpha,\beta>-1$$4 Darboux transformations with virtual-state seed solutions. The transformed eigenfunctions have the Wronskian form

$$w_{\alpha,\beta}(x)=(1-x)^\alpha(1+x)^\beta,\qquad \alpha,\beta>-1$$5

and after removing universal factors one obtains the polynomial part $$w_{\alpha,\beta}(x)=(1-x)^\alpha(1+x)^\beta,\qquad \alpha,\beta>-1$$6 or $$w_{\alpha,\beta}(x)=(1-x)^\alpha(1+x)^\beta,\qquad \alpha,\beta>-1$$7, with $$w_{\alpha,\beta}(x)=(1-x)^\alpha(1+x)^\beta,\qquad \alpha,\beta>-1$$8 (Ho et al., 2012, Odake et al., 2016). The resulting equation is still second order, but with additional apparent singularities at the zeros of the denominator polynomial. In fine-tuned cases, the Fuchsian equation has an extra apparent singularity of exponents $$w_{\alpha,\beta}(x)=(1-x)^\alpha(1+x)^\beta,\qquad \alpha,\beta>-1$$9 and $q\to -1$0, and the orthogonality weight takes the form

$q\to -1$1

(Ho et al., 2012).

The multi-indexed theory also admits simplified determinant expressions. Odake and Sasaki derive polynomial determinant formulas for both the denominator polynomial $q\to -1$2 and the multi-indexed polynomial $q\to -1$3, and prove the parity relation

$q\to -1$4

extends to the multi-indexed setting in a reflected-index form (Odake et al., 2016).

4. Orthogonality, missing degrees, and recurrence structure

A central structural distinction among higher Jacobi families concerns orthogonality. Classical Jacobi polynomials are orthogonal on $q\to -1$5 with respect to $q\to -1$6 (Garcia-Ferrero et al., 2024). Exceptional Jacobi families remain orthogonal on $q\to -1$7, but the weight is rationally deformed by a squared denominator polynomial, and in degenerate cases the norms of a finite set of low-degree eigenfunctions vanish, reflecting the removal of degrees from the polynomial flag (Garcia-Ferrero et al., 2024). Multi-indexed Jacobi polynomials are orthogonal on $q\to -1$8 with

$q\to -1$9

provided $2\alpha+2\beta+6$0 has no zeros in $2\alpha+2\beta+6$1 (Odake et al., 2016). Generalized Jacobi and Jacobi-type systems with endpoint masses are orthogonal with respect to a bilinear form that adds one or two Dirac masses to the classical measure (Markett, 2017, Markett, 2017).

The recurrence behavior is equally diagnostic. Classical Jacobi polynomials satisfy a three-term recurrence, but exceptional and multi-indexed families generally do not. In the exceptional theory, the “missing” degrees are encoded by spectral diagrams (Garcia-Ferrero et al., 2024). In the multi-indexed theory, the lowest degree is shifted, there are “holes” below that threshold, and one cannot write a standard three-term recurrence with coefficients independent of $2\alpha+2\beta+6$2; instead the recurrence involves more than three terms, in fact $2\alpha+2\beta+6$3 terms (Odake et al., 2016). Jacobi-type polynomials of Durán and de la Iglesia satisfy finite-term recurrences of systematically controlled length, while orthogonality on the real line forces the family into the Krall–Jacobi subclass (Durán et al., 2020).

A common misconception is that every higher Jacobi family is still governed by a second-order Sturm–Liouville operator together with a three-term recurrence. The literature does not support that statement. Exceptional and multi-indexed families preserve second-order differential equations but lose the classical degree sequence or the classical three-term recurrence; Jacobi-type and generalized Jacobi families recover bispectrality by moving to higher-order differential operators and longer recurrences (Garcia-Ferrero et al., 2024, Durán et al., 2020, Odake et al., 2016).

5. Classification results and hypergeometric organization

One classification program concerns exceptional Jacobi operators. The 2024 classification gives six mutually exclusive degeneracy classes: $2\alpha+2\beta+6$4 determined by whether $2\alpha+2\beta+6$5 or $2\alpha+2\beta+6$6 are integers. Spectral diagrams encode the quasi-rational spectrum, including endpoint asymptotics and missing degrees, and rational Darboux transformations act by flipping exactly one asymptotic label while shifting $2\alpha+2\beta+6$7 by $2\alpha+2\beta+6$8 (Garcia-Ferrero et al., 2024). In the fully degenerate class $2\alpha+2\beta+6$9, exceptional Jacobi operators may carry an arbitrary number of continuous parameters through confluent Darboux transforms (Garcia-Ferrero et al., 2024).

A second classification program concerns Jacobi-type families in a coefficient-ratio sense. For a monic polynomial family $$G=\{g_1<\cdots<g_{m_1}\},\qquad H=\{h_1<\cdots<h_{m_2}\},$$0, one defines Jacobi type by requiring

$$G=\{g_1<\cdots<g_{m_1}\},\qquad H=\{h_1<\cdots<h_{m_2}\},$$1

with $$G=\{g_1<\cdots<g_{m_1}\},\qquad H=\{h_1<\cdots<h_{m_2}\},$$2, $$G=\{g_1<\cdots<g_{m_1}\},\qquad H=\{h_1<\cdots<h_{m_2}\},$$3, so that the step-ratio

$$G=\{g_1<\cdots<g_{m_1}\},\qquad H=\{h_1<\cdots<h_{m_2}\},$$4

is polynomial in $$G=\{g_1<\cdots<g_{m_1}\},\qquad H=\{h_1<\cdots<h_{m_2}\},$$5 with rational-in-$$G=\{g_1<\cdots<g_{m_1}\},\qquad H=\{h_1<\cdots<h_{m_2}\},$$6 coefficients (Bernstein et al., 2024). Up to affine change of variable and normalization, there are exactly five hypergeometric families of Jacobi type: the classical Jacobi, Laguerre, and Bessel polynomials, together with two one-parameter families $$G=\{g_1<\cdots<g_{m_1}\},\qquad H=\{h_1<\cdots<h_{m_2}\},$$7 and $$G=\{g_1<\cdots<g_{m_1}\},\qquad H=\{h_1<\cdots<h_{m_2}\},$$8 (Bernstein et al., 2024). The last two can be written through Lommel polynomials, are orthogonal with respect to a positive discrete measure for $$G=\{g_1<\cdots<g_{m_1}\},\qquad H=\{h_1<\cdots<h_{m_2}\},$$9 and $R_g$0 respectively, and satisfy fourth-order differential equations rather than second-order ones (Bernstein et al., 2024).

These classification results show that the higher Jacobi landscape is not merely an unstructured collection of deformations. In the exceptional setting, spectral diagrams give a combinatorial parameterization of all admissible rational Darboux deformations. In the coefficient-ratio setting, the classification theorem isolates a rigid hypergeometric core consisting of exactly five families (Garcia-Ferrero et al., 2024, Bernstein et al., 2024).

6. Multivariate, analytic, and coding-theoretic extensions

The Jacobi framework also extends beyond the one-variable polynomial-eigenfunction setting. The paper “Two-variable $R_g$1 Jacobi polynomials” introduces bivariate polynomials

$R_g$2

depending on $R_g$3 and $R_g$4, defined as a coupled product of two univariate Big $R_g$5 Jacobi polynomials. For $R_g$6, they are orthogonal on the union of four triangles in the $R_g$7-plane, are simultaneous eigenfunctions of two commuting first-order differential/difference operators with reflections, satisfy a three-term relation in the $R_g$8-chain and a nine-term relation in the $R_g$9-chain, and arise as the $S_h$0 limit of Lewanowicz–Woźny’s two-variable Big $S_h$1-Jacobi polynomials (Genest et al., 2014). This provides a genuinely two-dimensional bispectral problem in the Bannai–Ito $S_h$2 hierarchy.

Another extension replaces the discrete polynomial degree $S_h$3 by a complex parameter $S_h$4. The Jacobi functions $S_h$5 and $S_h$6 are defined by Gauss $S_h$7 representations and admit multi-derivative and multi-integral formulas. When $S_h$8, the series terminate and one recovers the ordinary Jacobi polynomial (Cohl et al., 2023). This is not the exceptional or Krall notion of “higher,” but it extends the Jacobi family analytically in the degree parameter.

A different meaning of higher Jacobi polynomials appears in coding theory. For a linear code $S_h$9, a subset $$L_{\alpha,\beta}[y](x)=(1-x^2)\,y''(x)+\bigl[\beta-\alpha-(\alpha+\beta+2)x\bigr]\,y'(x),$$00, and an integer $$L_{\alpha,\beta}[y](x)=(1-x^2)\,y''(x)+\bigl[\beta-\alpha-(\alpha+\beta+2)x\bigr]\,y'(x),$$01, Chakraborty and Miezaki define

$$L_{\alpha,\beta}[y](x)=(1-x^2)\,y''(x)+\bigl[\beta-\alpha-(\alpha+\beta+2)x\bigr]\,y'(x),$$02

where $$L_{\alpha,\beta}[y](x)=(1-x^2)\,y''(x)+\bigl[\beta-\alpha-(\alpha+\beta+2)x\bigr]\,y'(x),$$03 counts $$L_{\alpha,\beta}[y](x)=(1-x^2)\,y''(x)+\bigl[\beta-\alpha-(\alpha+\beta+2)x\bigr]\,y'(x),$$04-dimensional subcodes with prescribed support sizes on $$L_{\alpha,\beta}[y](x)=(1-x^2)\,y''(x)+\bigl[\beta-\alpha-(\alpha+\beta+2)x\bigr]\,y'(x),$$05 and its complement (Chakraborty et al., 16 Aug 2025). They establish Jacobi analogues of the MacWilliams identity for both higher and extended weight enumerators, show that when the supports of $$L_{\alpha,\beta}[y](x)=(1-x^2)\,y''(x)+\bigl[\beta-\alpha-(\alpha+\beta+2)x\bigr]\,y'(x),$$06-subcodes of fixed weight form a $$L_{\alpha,\beta}[y](x)=(1-x^2)\,y''(x)+\bigl[\beta-\alpha-(\alpha+\beta+2)x\bigr]\,y'(x),$$07-design one has

$$L_{\alpha,\beta}[y](x)=(1-x^2)\,y''(x)+\bigl[\beta-\alpha-(\alpha+\beta+2)x\bigr]\,y'(x),$$08

and derive an alternative reconstruction from harmonic higher weight enumerators using Hahn polynomials (Chakraborty et al., 16 Aug 2025).

Taken together, these extensions indicate that the Jacobi paradigm supports several distinct generalization mechanisms: higher differential order, rational Darboux deformation, multivariate reflection operators, complex degree, and combinatorial generating functions. A plausible implication is that “higher Jacobi polynomials” is best read as a family resemblance term rather than a single definition, with each branch preserving a different subset of the classical Jacobi structure.