Zeros of GKP sequences of polynomials

Technical Summary: Zeros of GKP Sequences of Polynomials

Problem Statement
This paper investigates the properties of the zeros of a specific class of polynomial sequences defined by a two-parameter recurrence relation. These sequences, termed "GKP sequences" in honor of Graham, Knuth, and Patashnik (who posed the general problem of solving such recurrences in Concrete Mathematics), are generated by the formula:
$$p_n(x) = (ax^2 + bx + c)p'_{n-1}(x) + (\phi_n + \psi_n x)p_{n-1}(x)$$
where $ax^2 + bx + c$ has two distinct real zeros, and $\phi = (\phi_i)$ and $\psi = (\psi_i)$ are sequences of real numbers. The authors aim to determine conditions under which the zeros of these polynomials are real, simple, and interlaced, and to derive asymptotic behaviors for these zeros. The study encompasses well-known polynomial families such as Tangent, Secant, Eulerian, and Jacobi polynomials as specific instances of GKP sequences.

Methodology
The authors employ a combination of recurrence analysis, differential operator theory, and asymptotic analysis of polynomial coefficients.

1. Normalization and Reduction: The paper first establishes that any GKP sequence with a general quadratic $ax^2 + bx + c$ can be transformed via a linear change of variables and normalization into a sequence $s_n(x)$ defined by the specific recurrence:
   $$s_{n+1}(x) = (1-x^2)s'_n(x) + (\phi_{n+1} + \psi_{n+1}x)s_n(x)$$
   This allows the authors to focus on the interval $(-1, 1)$ and the polynomial $1-x^2$.

2. Interlacing and Realness Proofs: The core of the zero-analysis relies on lemmas concerning the operator $q(x) = (1-x^2)p'(x) + (a+bx)p(x)$. By analyzing the sign changes of $q(x)$ relative to the zeros of $p(x)$, the authors prove that if $b + |a| < 0$, the operator preserves the reality of zeros and induces an interlacing property. This is applied inductively to the GKP sequence.

3. Symmetry and Parameter Dependence: The authors investigate the symmetry of the coefficients of $s_n$ with respect to the parameters $\phi$ and $\psi$. They prove that the coefficients are symmetric functions of $\phi_1, \dots, \phi_n$ if and only if the sequence $\psi$ is constant. This leads to the study of monotonicity of zeros with respect to the parameters $\phi$.

4. Asymptotic Analysis: For the case where $\phi$ and $\psi$ are constant sequences (denoted $s_{v,u}^n$), the authors analyze the coefficients of the transformed polynomials $s_{v,u}^n(2x+1)$. They demonstrate that these coefficients behave asymptotically as $F(j)G(j)^{n-K}\tau_j(n)$, where $F$ and $G$ satisfy specific log-concavity and log-convexity conditions. This allows the application of a general theorem (Theorem 5.2) regarding the asymptotic distribution of zeros for polynomials with such coefficients.

5. Linear Combinations: The study extends to finite linear combinations of these polynomials, $q_n(x) = \sum \gamma_j s_{v,u}^{n-j}(2x+1)$, which correspond to cases where $\phi$ is eventually constant. The authors relate the location of the zeros of $q_n$ to the zeros of an auxiliary polynomial $P(x) = \sum \gamma_j x^{K-j}$.

Key Contributions and Results

• General Zero Properties (Theorem 1.1): Under the mild assumption $|\phi_n| + \psi_n < 0$, the authors prove that all zeros of $s_n$ are real, simple, and lie within the interval $(-1, 1)$. Furthermore, the zeros of $s_{n+1}$ strictly interlace the zeros of $s_n$. This result unifies the zero properties of Tangent, Secant, and Jacobi polynomials under a single framework.

• Monotonicity and Symmetry (Section 4): When $\psi$ is constant, the coefficients of $s_n$ are symmetric in $\phi$. The authors prove that the zeros of $s_n$ are increasing functions of the parameters $\phi_j$. They also establish that if the condition $|\phi_i| < -u$ is violated for a finite number of parameters, the number of real zeros in $(-1, 1)$ is at least $n - N$, where $N$ is the count of violating parameters.

• Asymptotics for Constant Parameters (Theorem 1.2): For constant sequences $\phi_i = u$ and $\psi_i = v$ (with $|v| + u < 0$), the authors derive precise asymptotic formulas for the rightmost and leftmost zeros of $s_{v,u}^n(2x+1)$. Specifically, they show that the $j$-th rightmost zero $\zeta_{n,j}$ behaves as:
  $$\lim_{n \to \infty} \frac{\zeta_{n,j}}{j} \left( \frac{u+v-2j+2}{u+v-2j} \right)^n = 1$$
  (with a corresponding formula for the leftmost zeros).

• Linear Combinations (Theorem 1.3): For linear combinations $q_n$ defined by a polynomial $P$, the authors characterize the distribution of zeros based on the roots of $P$.

  • If the roots of $P$ lie within a specific interval determined by $u$ and $v$, all zeros of $q_n$ are real, simple, and interlaced for $n \ge K$.
  • If $P$ has roots outside this interval, the authors prove that for sufficiently large $n$, $q_n$ still possesses only real zeros. They provide exact counts for the number of zeros in the intervals $(-1, 0)$, $(0, +\infty)$, and $(-\infty, -1)$, denoted by $n - N^+ - N^-$, $N^+$, and $N^-$ respectively, where $N^+$ and $N^-$ are determined by sign changes in $P$ evaluated at specific arithmetic progressions.

• Asymptotics for Linear Combinations (Theorem 6.2): The paper provides asymptotic formulas for the extreme zeros of these linear combinations, showing they converge to values determined by the roots of $P$ and the parameters $u, v$.

Significance and Claims
The paper claims to provide a "general theory" for the zeros of a broad class of polynomial sequences that includes many classical orthogonal and combinatorial polynomials. By establishing the reality and interlacing of zeros under mild conditions, the authors unify the study of these diverse families.

The work is significant for:

1. Unification: It demonstrates that Tangent, Secant, Eulerian, and Jacobi polynomials are specific cases of a single GKP recurrence structure.
2. Rigorous Asymptotics: It offers new, precise asymptotic descriptions for the extreme zeros of these sequences, which were previously not fully characterized in this generality.
3. Stability of Real Zeros: The authors show that even when the parameters deviate from the "safe" region (where all zeros are guaranteed to be in $(-1, 1)$), the polynomials retain a high degree of real zeros, and for linear combinations, the zeros eventually become real for large $n$ provided the auxiliary polynomial $P$ does not vanish at specific critical points.

The authors conclude with a conjecture suggesting that the operator generating these sequences is a "real zero increasing operator" in a deeper sense: even if the initial polynomial has non-real zeros, repeated application of the operator (corresponding to increasing $n$) will eventually yield a polynomial with only real zeros, provided the auxiliary polynomial $P$ satisfies non-vanishing conditions. This remains a conjecture and is not proven in the paper.