Al-Salam–Carlitz Orthogonal Polynomials

• Al-Salam–Carlitz orthogonal polynomials are basic hypergeometric q-polynomials defined by explicit terminating series, three-term recurrences, and q-difference equations.
• They comprise two families, type I and type II, each with distinct representations, orthogonality relations, and spectral behaviors on real and complex lattices.
• Recent studies extend these polynomials to multivariable and operator-theoretic frameworks, enhancing applications in q-ensembles, generating functions, and zero monotonicity analyses.

Searching arXiv for relevant papers on Al-Salam–Carlitz orthogonal polynomials and related orthogonality results. Looking up additional arXiv records for Al-Salam–Carlitz type I/II orthogonality, zeros, and generalized $q$-difference equations. Al-Salam–Carlitz orthogonal polynomials are classical basic hypergeometric $q$-polynomials introduced by Al-Salam and Carlitz and usually organized into two families: Al-Salam–Carlitz I, commonly written $U_n^{(a)}(x;q)$, and Al-Salam–Carlitz II, commonly written $V_n(x;a\mid q)$. They are characterized by explicit terminating basic hypergeometric representations, three-term recurrences, second-order $q$-difference equations, and orthogonality relations on real $q$-lattices, discrete exponential lattices, and, in the nonclassical regime, contours in the complex plane. Later work has emphasized quasi-definite orthogonality for complex parameters, spectral realizations of type II polynomials, generalized generating functions, and operator-theoretic extensions to multivariable families (Cohl et al., 2016, Groenevelt, 2013, Cao et al., 2020).

1. Definitions and notation

For Al-Salam–Carlitz I, one basic hypergeometric form is

$$U_n^{(a)}(x;q) = (-a)^n q^{\binom n2}\, {}_2\phi_1\!\Bigl(q^{-n},x^{-1};0;q,\frac{q x}{a}\Bigr),$$

equivalently,

$$U_n(x;a;q) = (-a)^n q^{\tfrac{n(n-1)}2}\, {}_2\phi_1\!\Bigl(q^{-n},x^{-1};0;q,\frac{q x}{a}\Bigr) = \sum_{k=0}^n \begin{bmatrix} n\ k\end{bmatrix}_q (-a)^{\,n-k} q^{\,\tfrac{k(k-1)}2} x^k.$$

The sequence has full degree, $\deg_x U_n^{(a)}=n$, so it is normal and may be rescaled to monic form (Cohl et al., 2016, Forrester et al., 2022).

For Al-Salam–Carlitz II, one standard representation is

$$V_n(x;a\mid q) = (-a)^n q^{-\tfrac{n(n-1)}2}\, {}_2\phi_0\!\Bigl(q^{-n},x^{-1};-;q,a^{-1}q^n x\Bigr),$$

and an equivalent terminating form is

$q$0

A notational variant used in the study of zero monotonicity is $q$1, also represented by a terminating $q$2 series on the $q$3-linear lattice $q$4, $q$5, with parameter range $q$6 (Groenevelt, 2013, Castillo et al., 2020).

The literature also contains two-variable and higher-parameter extensions. The classical two-variable families are

$q$7

and

$q$8

A three-parameter generalization is

$q$9

and a five-parameter extension is introduced through homogeneous $U_n^{(a)}(x;q)$0-exponential operators (Cao et al., 2020).

2. Recurrence relations and $U_n^{(a)}(x;q)$1-difference equations

For $U_n^{(a)}(x;q)$2, the three-term recurrence recorded in the complex-parameter orthogonality study is

$U_n^{(a)}(x;q)$3

with

$U_n^{(a)}(x;q)$4

By Favard’s theorem, this recurrence together with the nonvanishing of the off-diagonal coefficients $U_n^{(a)}(x;q)$5 is equivalent to quasi-definiteness of the corresponding linear functional (Cohl et al., 2016).

A second-order $U_n^{(a)}(x;q)$6-difference equation for the monic Al-Salam–Carlitz I polynomials is

$U_n^{(a)}(x;q)$7

where

$U_n^{(a)}(x;q)$8

An equivalent form is

$U_n^{(a)}(x;q)$9

This places the family in the standard eigenfunction setting for second-order $V_n(x;a\mid q)$0-difference operators (Forrester et al., 2022).

For $V_n(x;a\mid q)$1 on the lattice $V_n(x;a\mid q)$2, the Nikiforov–Uvarov framework yields a second-order difference equation

$V_n(x;a\mid q)$3

with

$V_n(x;a\mid q)$4

and $V_n(x;a\mid q)$5 determined by $V_n(x;a\mid q)$6 up to an inessential additive constant. The same family satisfies

$V_n(x;a\mid q)$7

where

$V_n(x;a\mid q)$8

These formulas are the basis for the discrete Stieltjes-type analysis of zero variation (Castillo et al., 2020).

3. Orthogonality of Al-Salam–Carlitz I

The modern orthogonality theory for $V_n(x;a\mid q)$9 extends beyond the positive real regime. Let $q$0 be the unique linear functional satisfying

$q$1

If $q$2 and $q$3, then $q$4 is positive definite and is represented by a discrete $q$5-Jackson integral on $q$6. In the general complex-parameter case, with $q$7, the functional is only quasi-definite, but orthogonality still holds on a simple closed contour $q$8 formed by two $q$9-spirals joining $q$0: $q$1 With weight

$q$2

one has, for $q$3 and $q$4,

$q$5

where

$q$6

An analogous statement holds for $q$7 after replacing $q$8 and using the corresponding Jackson integral. In this formulation, the orthogonality relations characterize $q$9 up to a degree-dependent multiplicative constant (Cohl et al., 2016).

Other orthogonality realizations coexist in the literature. For the classical two-variable form $$U_n^{(a)}(x;q) = (-a)^n q^{\binom n2}\, {}_2\phi_1\!\Bigl(q^{-n},x^{-1};0;q,\frac{q x}{a}\Bigr),$$0, continuous orthogonality is recorded on $$U_n^{(a)}(x;q) = (-a)^n q^{\binom n2}\, {}_2\phi_1\!\Bigl(q^{-n},x^{-1};0;q,\frac{q x}{a}\Bigr),$$1 with weight

$$U_n^{(a)}(x;q) = (-a)^n q^{\binom n2}\, {}_2\phi_1\!\Bigl(q^{-n},x^{-1};0;q,\frac{q x}{a}\Bigr),$$2

namely

$$U_n^{(a)}(x;q) = (-a)^n q^{\binom n2}\, {}_2\phi_1\!\Bigl(q^{-n},x^{-1};0;q,\frac{q x}{a}\Bigr),$$3

with

$$U_n^{(a)}(x;q) = (-a)^n q^{\binom n2}\, {}_2\phi_1\!\Bigl(q^{-n},x^{-1};0;q,\frac{q x}{a}\Bigr),$$4

The same source records a discrete orthogonality on the lattice $$U_n^{(a)}(x;q) = (-a)^n q^{\binom n2}\, {}_2\phi_1\!\Bigl(q^{-n},x^{-1};0;q,\frac{q x}{a}\Bigr),$$5, $$U_n^{(a)}(x;q) = (-a)^n q^{\binom n2}\, {}_2\phi_1\!\Bigl(q^{-n},x^{-1};0;q,\frac{q x}{a}\Bigr),$$6, with a corresponding discrete measure (Cao et al., 2020).

These results show that the usual restriction to positive measures on real intervals is not exhaustive. In particular, orthogonality survives in the quasi-definite setting, where the support moves to logarithmic spirals in the complex plane and positivity is replaced by nondegeneracy of the moment functional (Cohl et al., 2016).

4. Al-Salam–Carlitz II and spectral completion

For Al-Salam–Carlitz II, Groenevelt’s construction fixes two real parameters $$U_n^{(a)}(x;q) = (-a)^n q^{\binom n2}\, {}_2\phi_1\!\Bigl(q^{-n},x^{-1};0;q,\frac{q x}{a}\Bigr),$$7 and two nonzero parameters $$U_n^{(a)}(x;q) = (-a)^n q^{\binom n2}\, {}_2\phi_1\!\Bigl(q^{-n},x^{-1};0;q,\frac{q x}{a}\Bigr),$$8, and considers the discrete support

$$U_n^{(a)}(x;q) = (-a)^n q^{\binom n2}\, {}_2\phi_1\!\Bigl(q^{-n},x^{-1};0;q,\frac{q x}{a}\Bigr),$$9

The associated Jackson-sum integral is

$$U_n(x;a;q) = (-a)^n q^{\tfrac{n(n-1)}2}\, {}_2\phi_1\!\Bigl(q^{-n},x^{-1};0;q,\frac{q x}{a}\Bigr) = \sum_{k=0}^n \begin{bmatrix} n\ k\end{bmatrix}_q (-a)^{\,n-k} q^{\,\tfrac{k(k-1)}2} x^k.$$0

with weight

$$U_n(x;a;q) = (-a)^n q^{\tfrac{n(n-1)}2}\, {}_2\phi_1\!\Bigl(q^{-n},x^{-1};0;q,\frac{q x}{a}\Bigr) = \sum_{k=0}^n \begin{bmatrix} n\ k\end{bmatrix}_q (-a)^{\,n-k} q^{\,\tfrac{k(k-1)}2} x^k.$$1

Under the stated nonvanishing and positivity conditions, including the sample sufficient conditions

$$U_n(x;a;q) = (-a)^n q^{\tfrac{n(n-1)}2}\, {}_2\phi_1\!\Bigl(q^{-n},x^{-1};0;q,\frac{q x}{a}\Bigr) = \sum_{k=0}^n \begin{bmatrix} n\ k\end{bmatrix}_q (-a)^{\,n-k} q^{\,\tfrac{k(k-1)}2} x^k.$$2

the inner product

$$U_n(x;a;q) = (-a)^n q^{\tfrac{n(n-1)}2}\, {}_2\phi_1\!\Bigl(q^{-n},x^{-1};0;q,\frac{q x}{a}\Bigr) = \sum_{k=0}^n \begin{bmatrix} n\ k\end{bmatrix}_q (-a)^{\,n-k} q^{\,\tfrac{k(k-1)}2} x^k.$$3

is positive definite. Writing

$$U_n(x;a;q) = (-a)^n q^{\tfrac{n(n-1)}2}\, {}_2\phi_1\!\Bigl(q^{-n},x^{-1};0;q,\frac{q x}{a}\Bigr) = \sum_{k=0}^n \begin{bmatrix} n\ k\end{bmatrix}_q (-a)^{\,n-k} q^{\,\tfrac{k(k-1)}2} x^k.$$4

one obtains

$$U_n(x;a;q) = (-a)^n q^{\tfrac{n(n-1)}2}\, {}_2\phi_1\!\Bigl(q^{-n},x^{-1};0;q,\frac{q x}{a}\Bigr) = \sum_{k=0}^n \begin{bmatrix} n\ k\end{bmatrix}_q (-a)^{\,n-k} q^{\,\tfrac{k(k-1)}2} x^k.$$5

with

$$U_n(x;a;q) = (-a)^n q^{\tfrac{n(n-1)}2}\, {}_2\phi_1\!\Bigl(q^{-n},x^{-1};0;q,\frac{q x}{a}\Bigr) = \sum_{k=0}^n \begin{bmatrix} n\ k\end{bmatrix}_q (-a)^{\,n-k} q^{\,\tfrac{k(k-1)}2} x^k.$$6

where

$$U_n(x;a;q) = (-a)^n q^{\tfrac{n(n-1)}2}\, {}_2\phi_1\!\Bigl(q^{-n},x^{-1};0;q,\frac{q x}{a}\Bigr) = \sum_{k=0}^n \begin{bmatrix} n\ k\end{bmatrix}_q (-a)^{\,n-k} q^{\,\tfrac{k(k-1)}2} x^k.$$7

The derivation uses a specialization of Askey’s $$U_n(x;a;q) = (-a)^n q^{\tfrac{n(n-1)}2}\, {}_2\phi_1\!\Bigl(q^{-n},x^{-1};0;q,\frac{q x}{a}\Bigr) = \sum_{k=0}^n \begin{bmatrix} n\ k\end{bmatrix}_q (-a)^{\,n-k} q^{\,\tfrac{k(k-1)}2} x^k.$$8-beta integral together with the $$U_n(x;a;q) = (-a)^n q^{\tfrac{n(n-1)}2}\, {}_2\phi_1\!\Bigl(q^{-n},x^{-1};0;q,\frac{q x}{a}\Bigr) = \sum_{k=0}^n \begin{bmatrix} n\ k\end{bmatrix}_q (-a)^{\,n-k} q^{\,\tfrac{k(k-1)}2} x^k.$$9-binomial theorem (Groenevelt, 2013).

A corresponding second-order $\deg_x U_n^{(a)}=n$0-difference operator is

$\deg_x U_n^{(a)}=n$1

with

$\deg_x U_n^{(a)}=n$2

This operator is symmetric and, with a suitable dense domain, self-adjoint on $\deg_x U_n^{(a)}=n$3. The Al-Salam–Carlitz II polynomials are eigenfunctions: $\deg_x U_n^{(a)}=n$4 The spectral analysis shows that the polynomial sector is not complete by itself: besides the discrete eigenvalues $\deg_x U_n^{(a)}=n$5, there is a continuous spectrum accumulating at $\deg_x U_n^{(a)}=n$6. To obtain a complete orthogonal basis, one introduces supplementary nonpolynomial functions

$\deg_x U_n^{(a)}=n$7

where

$\deg_x U_n^{(a)}=n$8

Then

$\deg_x U_n^{(a)}=n$9

is an orthogonal basis of $$V_n(x;a\mid q) = (-a)^n q^{-\tfrac{n(n-1)}2}\, {}_2\phi_0\!\Bigl(q^{-n},x^{-1};-;q,a^{-1}q^n x\Bigr),$$0 (Groenevelt, 2013).

A separate orthogonality model for the same “second” family uses the $$V_n(x;a\mid q) = (-a)^n q^{-\tfrac{n(n-1)}2}\, {}_2\phi_0\!\Bigl(q^{-n},x^{-1};-;q,a^{-1}q^n x\Bigr),$$1-linear grid $$V_n(x;a\mid q) = (-a)^n q^{-\tfrac{n(n-1)}2}\, {}_2\phi_0\!\Bigl(q^{-n},x^{-1};-;q,a^{-1}q^n x\Bigr),$$2, $$V_n(x;a\mid q) = (-a)^n q^{-\tfrac{n(n-1)}2}\, {}_2\phi_0\!\Bigl(q^{-n},x^{-1};-;q,a^{-1}q^n x\Bigr),$$3, and the positive weight

$$V_n(x;a\mid q) = (-a)^n q^{-\tfrac{n(n-1)}2}\, {}_2\phi_0\!\Bigl(q^{-n},x^{-1};-;q,a^{-1}q^n x\Bigr),$$4

giving

$$V_n(x;a\mid q) = (-a)^n q^{-\tfrac{n(n-1)}2}\, {}_2\phi_0\!\Bigl(q^{-n},x^{-1};-;q,a^{-1}q^n x\Bigr),$$5

This formulation is the one used in the monotonicity analysis of zeros (Castillo et al., 2020).

5. Generating functions and operator generalizations

The classical generating function for $$V_n(x;a\mid q) = (-a)^n q^{-\tfrac{n(n-1)}2}\, {}_2\phi_0\!\Bigl(q^{-n},x^{-1};-;q,a^{-1}q^n x\Bigr),$$6 is

$$V_n(x;a\mid q) = (-a)^n q^{-\tfrac{n(n-1)}2}\, {}_2\phi_0\!\Bigl(q^{-n},x^{-1};-;q,a^{-1}q^n x\Bigr),$$7

A related basic-$$V_n(x;a\mid q) = (-a)^n q^{-\tfrac{n(n-1)}2}\, {}_2\phi_0\!\Bigl(q^{-n},x^{-1};-;q,a^{-1}q^n x\Bigr),$$8 generating function is

$$V_n(x;a\mid q) = (-a)^n q^{-\tfrac{n(n-1)}2}\, {}_2\phi_0\!\Bigl(q^{-n},x^{-1};-;q,a^{-1}q^n x\Bigr),$$9

where $q$00 is a slight variant of $q$01. The complex-parameter orthogonality paper also derives a connection relation between $q$02 and $q$03, with $q$04, and from it obtains the generalized generating function

$q$05

These identities show that parameter changes can be encoded by explicit connection coefficients rather than by ad hoc renormalization (Cohl et al., 2016).

The operator-theoretic extension developed in the generalized $q$06-difference paper begins with the $q$07-derivatives

$q$08

and introduces the five-parameter homogeneous $q$09-exponential operators

$q$10

$q$11

A seven-variable analytic function $q$12 expands in the family $q$13 if and only if it satisfies the stated functional equation, and then

$q$14

An analogous characterization holds for the $q$15-family $q$16, with

$q$17

Within the same framework, the paper derives $q$18-type generating functions, Ramanujan-type integrals involving generalized Al-Salam–Carlitz polynomials, and transformation formulas that generalize classical Heine–Euler–Cauchy identities by inserting Al-Salam–Carlitz polynomials in place of monomials (Cao et al., 2020).

6. Zeros, special parameter regimes, and limiting cases

For the type II family $q$19, the zero-monotonicity analysis is based on the function

$q$20

On the support of the zeros, $q$21, one has $q$22, $q$23, and $q$24. The discrete Stieltjes theorem then implies that every zero

$q$25

is a strictly increasing function of $q$26. In the terminology of the paper, this is Proposition 3.3: the zeros of $q$27 are strictly increasing in $q$28 (Castillo et al., 2020).

For $q$29, the real positive-definite regimes include $q$30 and $q$31, and also $q$32 with $q$33. In these cases the polynomials are orthogonal on the real $q$34-lattice in $q$35; all zeros lie in $q$36 and interlace in the usual way. When $q$37 and $q$38, the support of the discrete measure is no longer a subset of the real line but lies on two intertwined logarithmic spirals, while the functional remains quasi-definite. In that regime, the zeros spread through the complex plane. Thus the loss of positivity does not imply the loss of orthogonality; what changes is the geometry of the support and the status of the moment functional (Cohl et al., 2016).

The root-of-unity case $q$39, $q$40, requires a different mechanism. The usual three-term recurrence persists, but one replaces a single linear functional by a finite Gaussian quadrature and then restores nondegeneracy beyond degree $q$41 by adding higher-order $q$42-difference terms to the bilinear form. This produces quasi-definite bilinear forms for all degrees. At the degenerate parameter value $q$43, one recovers a simpler one-parameter family satisfying

$q$44

but this family no longer has a nontrivial three-term recurrence and is therefore not covered by Favard’s theorem. Under suitable scaling, the limit $q$45 yields Hermite-, Laguerre-, or Charlier-type limits of the Carlitz–Al-Salam family (Cohl et al., 2016).

7. Appearance in discrete $q$46-ensembles

In the construction of discrete orthogonal ensembles on exponential lattices, the monic Al-Salam–Carlitz I polynomials are taken on

$q$47

with Jackson integral

$q$48

and weight

$q$49

The orthogonality relation is

$q$50

with

$q$51

In the same setting, one builds a Pfaffian ensemble by introducing a skew-symmetric kernel $q$52 and the skew inner product

$q$53

The skew-orthogonal polynomials can be chosen as

$q$54

and the $q$55-point correlation functions take the Pfaffian form

$q$56

where the $q$57 matrix kernel is built from skew Christoffel–Darboux sums of $q$58. In this probabilistic realization, the limit $q$59 gives $q$60 and degenerates to the unitary-symmetry $q$61-ensemble, while $q$62, after suitable rescaling, recovers continuous orthogonal and skew-orthogonal ensembles of Hermite- or Charlier-type (Forrester et al., 2022).