Al-Salam–Carlitz Moment Problem

• The Al-Salam–Carlitz moment problem is a framework for classifying moment sequences and associated orthogonality structures arising from q-orthogonal polynomial families.
• It employs discrete q-Jackson integrals, q-hypergeometric identities, and recurrence relations to distinguish between determinate and indeterminate moment settings.
• The analysis extends to maximum-entropy solutions and q-deformed random matrix ensembles, revealing deep connections in spectral theory and modern mathematical physics.

The Al-Salam–Carlitz moment problem concerns the description and classification of moment sequences, corresponding measures, and associated orthogonality structures arising from the Al-Salam–Carlitz (ASC) polynomials—both type I and type II. The subject interfaces $q$-orthogonal polynomials, discrete and complex analysis, indeterminate moment problems, and probabilistic models including $q$-deformed random matrix ensembles. Treatments span the study of positive and complex-valued functionals (quasi-definite but not positive definite), spectral completeness in discrete and continuous settings, entropy-maximization among admissible measures, and limiting spectral distributions. The ASC moment problem illustrates both classical determinacy phenomena and the multitude of solutions in the indeterminate regime.

1. Formulation and Functional Setting

The ASC polynomials $U_n^{(a)}(x;q)$ (type I) are determined by the parameters $a, q\in \mathbb{C}\setminus\{0,1\}$ and are orthogonal with respect to a linear functional $u$ defined via a discrete $q$–Jackson integral. For $0<|q|<1$, the support consists of two geometric progressions: $\{q^k:k=0,1,2,\dots\}\cup\{a q^k:k=0,1,2,\dots\}$, viewed as interlaced geometric spirals in $\mathbb{C}$ joining $1\to0$ and $0\to a$. The functional is explicitly

$$\langle u, p\rangle = \int_a^1 p(x) w(x;a,q) d_qx = \int_0^1 p(x) w(x;a,q) d_qx - \int_0^a p(x) w(x;a,q) d_qx,$$

where $w(x;a,q) = (q x;q)_\infty (q x/a;q)_\infty$ and

$$\int_0^b f(x) d_qx = b(1-q)\sum_{k=0}^{\infty} f(b q^k) q^k.$$

For $|q|>1$, the case is analogous with $q$ replaced by $p=q^{-1}$. The polynomials and weight then exhibit orthogonality on a single contour $C$ in $\mathbb{C}$ formed by the union of the two spirals, with the exact measure possibly complex-valued except in classical real cases (Cohl et al., 2016).

For the type II ASC polynomials $P_n(x;c,d;q)$, the orthogonality is defined on a support $$R_q = \{z_+ q^k : k\in\mathbb{Z}\}\cup\{z_- q^k : k\in\mathbb{Z}\}$$ (for real $z_-<0<z_+$) with Jackson measure (Groenevelt, 2013).

2. Moment Sequences and Explicit Formulae

The moments associated to the ASC-I polynomials are given by

$$\mu_k = \langle u, x^k\rangle = (1-q)\sum_{j=0}^\infty \left[q^{jk+j} w(q^j;a,q) - a^{k+1} q^{jk+j} w(a q^j;a,q)\right],$$

with $$w(q^j;a,q) = (q^{j+1};q)_\infty (q^{j+1}/a;q)_\infty$$. For $k=0$, normalization yields $\mu_0 = (a;q)_\infty (q/a;q)_\infty$ (Cohl et al., 2016).

For ASC-II, moments take the form

$$m_k = \int_{R_q} x^k w(x) d_qx = B(cq^k, dq^k; z_-, z_+),$$

where $B$ is an explicit function of its arguments representing normalization and the weight function (Groenevelt, 2013).

The moment sequences for both types can be generated by $q$-hypergeometric function identities, and in the ASC-II case, by Heine’s summation (Groenevelt, 2013).

3. Orthogonality, Quasi-Definiteness, and Indeterminacy

ASC-I polynomials satisfy a three-term recurrence

$$x U_n^{(a)}(x;q) = U_{n+1}^{(a)}(x;q) + (a+1) q^n U_n^{(a)}(x;q) - a q^{n-1}(1-q^n) U_{n-1}^{(a)}(x;q),$$

with non-degenerate coefficients for $a\neq 0,1$, so by Favard’s theorem, the functional $u$ is quasi-definite. In the classical regime ($a < 0$, $0 < q < 1$), $w(x;a,q)$ is nonnegative, rendering $u$ positive definite and the associated moment problem determinate. For complex $a$ or $q$, $u$ is typically only quasi-definite, yielding a unique orthogonality structure for the polynomials up to scaling, but with complex-valued measure and zeros distributed along the spiral support $C$ (Cohl et al., 2016).

For ASC-II, the moment problem is classically indeterminate: a one-parameter family of discrete orthogonality measures, parametrized by endpoints $z_-,z_+$, exhausts all $N$-extremal solutions, with no single determinacy unless endpoints degenerate (Groenevelt, 2013). Spectral completion is realized by supplementing the polynomial eigenfunctions with a complementary family of (generally non-polynomial) eigenfunctions so that the $L^2$ space is fully spanned.

4. Extremal Measures and the Nevanlinna Family

In indeterminate cases (notably for Stieltjes moment problems with $0$q$-geometric lattices. For general solutions, the Nevanlinna parameterization produces a continuum of analytic densities: $$f_{\beta+i\gamma}(x) = \frac{\gamma}{\pi}\left[(\beta B(x) - D(x))^2 + (\gamma B(x))^2\right]^{-1}, \quad x\in\mathbb{R},$$ where $B(x), D(x)$ are entire functions associated to ASC orthonormal and second-kind polynomials. The specific member for $(\beta,\gamma)=(0,1)$ yields a canonical density

$$\nu(q,a)(x) = \frac{(a-1)}{\pi a} [q]_\infty [a q]_\infty [q/a]_\infty \left([ (1+x)/a ]_\infty^2 + [1+x]_\infty^2 \right)^{-1}.$$

This density and its parameterized relatives form the full analytic solution set to the indeterminate moment sequences (Berg, 16 Nov 2025).

For ASC-II, the family of discrete measures parametrized by $(z_-, z_+)$ is exhaustive, with no single probability measure associated to the moments (excluding boundary degeneration) (Groenevelt, 2013).

5. Entropy and Maximum-Entropy Solutions

Within the analytic Nevanlinna family, each density $f_{\beta+i\gamma}$ possesses finite Shannon entropy

$H[f] = -\int_{\mathbb{R}} f(x) \log f(x) dx,$

due to exponential-type bounds on $B(x), D(x)$. While the existence of a maximal entropy density $g_{h_{\max}}$ is established by convexity arguments, identification of the maximizing Nevanlinna parameter remains open. The explicit canonical density $\nu(q,a)$ provides concrete instances of finite-entropy solutions, but the entropy maximizer for the ASC moment problem has no closed-form at present (Berg, 16 Nov 2025).

6. Random Matrix Ensembles and Limiting Behavior

The ASC-I moment problem connects with $q$-deformed unitary random matrix ensembles. For $a<0$, the ensemble measures points $x_j$ on the geometric lattice $(a,1)$ with weight $\omega_U^{(a)}(x;q)$. By combinatorial enumeration via Flajolet–Viennot theory and matchings, moments admit manifestly positive hypergeometric sums over Motzkin paths and generalized matchings.

In the double-scaling limit $q = e^{-\lambda/N}, N \to \infty$, explicit closed-form formulas for spectral moments and limiting density are derived. The limiting spectral density $\rho^{(a)}(x;\lambda)$ exhibits successive phase transitions: as the parameter $\lambda$ increases, the number of soft edges in the support reduces from two, to one, to none. The density is given, for example, in the two-soft-edge regime $0<\lambda<\log(1-a)$, as

$$\rho^{(a)}(x) = \frac{2}{\pi\,\lambda\,x} \arctan\!\left[ \sqrt{\frac{1-x_0-x_1}{1-x_0+x_1} \frac{1-e^{-\lambda}-x_0+x_1}{x_0+x_1-1+e^{-\lambda}} }\right],$$

with $x_0$ and $x_1$ explicit algebraic functions of $x,a$, and soft-edge boundaries determined by $a, \lambda$. This limiting density coincides with the limiting zero distribution of the appropriately rescaled orthogonal polynomials, confirming a classical result (Kuijlaars–Van Assche) linking moment convergence and zero distributions (Byun et al., 24 Jul 2025).

7. Generating Functions and Spectral Analysis

The generating function for ASC-I polynomials is: $$(x t;q)_\infty (a t;q)_\infty = \sum_{n=0}^\infty (-1)^n q^{n(n-1)/2} (q;q)_n U_n^{(a)}(x;q) t^n,$$ with complex-parameter generalizations available through connection relations and basic hypergeometric transformations. Such relations extend the moment problem analysis to the setting of generalized analytic functionals, and, together with recurrence properties and explicit moments, supply all necessary constructive ingredients for both classical and non-classical settings (Cohl et al., 2016).

For ASC-II, the three-term recurrence and the explicit moment and generating function formulas arise from $q$-beta integral evaluations. Full spectral analysis reveals a complementary set of eigenfunctions outside the polynomial sector, which together with the polynomials span the corresponding $L^2$-space, with the non-polynomial sector explaining the spectral origin of indeterminacy (Groenevelt, 2013).