Finite-term recurrences in a generalized Bochner--Krall family

Technical Summary: Finite-Term Recurrences in a Generalized Bochner–Krall Family

Problem Statement
This paper addresses the algebraic higher-order Bochner–Krall problem, which seeks to classify differential operators that possess polynomial eigenfunctions satisfying a recurrence relation with a fixed number of terms (finite lower bandwidth). Specifically, the authors investigate the three-parameter family of differential operators defined by:
$$T = T_{j,m,\ell} := z^j \partial_z^j + z^m \partial_z^\ell$$
where $0 \le m < \ell$ and $1 \le j < \ell$. The operator $T$ is exactly solvable, preserving the degree filtration of $\mathbb{C}[z]$ and acting triangularly on the monomial basis. The study focuses on the sequence of monic eigenpolynomials $\{P_n\}_{n \ge 0}$ associated with $T$. While such a sequence always exists, the central question is determining the specific conditions on the parameters $j, m, \ell$ under which $\{P_n\}$ satisfies a recurrence of the form:
$$z P_n = P_{n+1} + \sum_{s=0}^{N-1} \alpha_{n, n-s} P_{n-s}$$
for some fixed integer $N$ (the optimal bandwidth).

Methodology
The authors employ a combination of spectral analysis, combinatorial recurrence relations, and hypergeometric summation techniques. The methodology proceeds through the following steps:

1. Eigenpolynomial Construction: The authors establish the existence and uniqueness of monic eigenpolynomials $P_n$ for $n \ge j$. They derive explicit recurrence relations for the coefficients of $P_n$ and identify a crucial congruence constraint: the support of $P_n$ is restricted to powers $z^s$ where $s \equiv n \pmod k$, with $k = \ell - m$. This implies the sequence is $(k-1)$-symmetric.
2. Subspace Decomposition: The space of polynomials $\mathbb{C}[z]$ is decomposed into $k$ invariant subspaces $V_r$ spanned by monomials with exponents congruent to $r \pmod k$. The action of $T$ and multiplication by $z$ is analyzed within these subspaces.
3. Bidiagonal Model: On each invariant subspace $V_c$, the authors construct a bidiagonal model. By analyzing the coordinate functionals of the eigenbasis, they derive explicit formulas for the action of $T$ and the shift operator.
4. Asymptotic Analysis for Necessity: To prove that finite-term recurrences are impossible when $j \ge 2$, the authors assume a fixed bandwidth $N$ and derive a contradiction. They analyze the asymptotic behavior of the coefficients in the recurrence relation as $n \to \infty$. They show that for $j \ge 2$, the left-hand side of the derived identity grows unboundedly (as a polynomial of degree $\ell - j \ge 1$), while the right-hand side remains bounded, leading to a contradiction.
5. Explicit Evaluation for Sufficiency: In the case $j=1$, the authors explicitly evaluate the recurrence coefficients using hypergeometric series (specifically the Gauss hypergeometric series and the Chu–Vandermonde identity). This allows them to determine the exact closed form of the coefficients and the conditions under which the series terminates.

Key Contributions and Results

• Complete Classification (Theorem 1.1): The paper provides a necessary and sufficient condition for the existence of a finite-term recurrence for the family $T_{j,m,\ell}$. Such a recurrence exists if and only if:

  1. $j = 1$
  2. $k = \ell - m$ divides $\ell$ (i.e., $k \mid \ell$).

  If these conditions are met, the optimal lower bandwidth is exactly $N = \ell$, independent of $m$.

• Closed-Form Coefficients: For the admissible cases ($j=1, k \mid \ell$), the authors derive an explicit formula for the recurrence coefficients $\gamma_n^{(q)}$:
  $$z P_n = P_{n+1} + \sum_{q=1}^{\sigma} \gamma_n^{(q)} P_{n+1-qk}$$
  where $\sigma = \ell/k$. The coefficients are given by:
  $$\gamma_n^{(q)} = (-1)^q \frac{1}{q! k^q} \left( \prod_{a=0}^{q-1} (\ell - ak) \right) \left( \prod_{c=0}^{q-1} (n - ck)_{\ell-1} \right)$$
  where the falling factorial notation $(x)_r$ is defined as $(x)_r := x(x - 1) \cdots (x - r + 1)$.

• Clarification on Symmetry and Recurrence Structure: The authors clarify a potential misconception regarding symmetric sequences. While $(k-1)$-symmetry implies a specific congruence in the recurrence indices (specifically, $R \equiv n+1 \pmod k$), it does not guarantee a recurrence with a single lower term. The paper explicitly demonstrates that symmetric sequences can possess multiple lower diagonals while still maintaining a finite total number of terms. For instance, Example 2.3 exhibits a 1-symmetric sequence with two lower diagonals, proving that symmetry alone does not force the recurrence into the special shape $P_{n+k} = zP_{n+k-1} - \gamma_n P_n$ often associated with $(k-1)$-orthogonal sequences.

• Connection to Conjectures: The results are linked to the Horozov–Shapiro–Tater conjectures (specifically Conjectures 1.9 and 1.10 in [5]):

  • Conjecture 1.10: The authors prove that every admissible operator in this family belongs to "Type (2)" of the conjecture, providing an explicit factorization $T = z\partial_z + q'(G)G$.
  • Conjecture 1.9: The authors confirm that the associated difference operator has order $\ell$, which equals the order of the differential operator $T$. This validates the predicted equality of orders for this specific family without assuming irreducibility.

Significance
The paper's significance lies in its rigorous classification of a generalized Bochner–Krall family, resolving the existence of finite-term recurrences through a sharp dichotomy based on the parameter $j$. It moves beyond the classical case of orthogonal polynomials (which correspond to specific parameter choices) to a broader class of exactly solvable operators. By providing explicit formulas for the recurrence coefficients and proving the order equality predicted by recent conjectures, the work offers concrete evidence supporting the structural hypotheses in the theory of polynomial eigenfunctions. The authors note that the smallest "new" case not previously covered by extreme divisors is $(\ell, m, j) = (4, 2, 1)$, corresponding to $T = z\partial_z + z^2\partial_z^4$.

The paper concludes that the equality of differential and difference orders holds for this entire admissible family, reinforcing the structural connection between the differential operator and its associated recurrence relation.