---
title: Al-Salam–Carlitz Orthogonal Polynomials
url: https://www.emergentmind.com/topics/al-salam-carlitz-orthogonal-polynomials
type: topic
---

# Al-Salam–Carlitz Orthogonal Polynomials

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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 [1601.08178], [1309.7569], [2011.05848].

## 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 [1601.08178], [2206.08633].

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
\[
V_n(x;a\mid q)
=
\sum_{k=0}^n
\begin{bmatrix} n\\ k\end{bmatrix}_q
(-a)^k q^{\tfrac{k(k-1)}2} x^k.
\]
A notational variant used in the study of zero monotonicity is \(V_n^{(\alpha)}(x;q)\), also represented by a terminating \({}_2\phi_0\) series on the \(q\)-linear lattice \(x=q^{-s}\), \(s=0,1,2,\dots\), with parameter range \(0<\alpha<q^{-1}\) [1309.7569], [2006.07884].

The literature also contains two-variable and higher-parameter extensions. The classical two-variable families are
\[
\phi_n(x,y;a\mid q)
=
\sum_{k=0}^n
\binom{n}{k}_q (a;q)_k\,x^{n-k}y^k,
\]
and
\[
\psi_n(x,y;a\mid q)
=
\sum_{k=0}^n
\binom{n}{k}_q
q^{k(k-n)}(aq^{1-k};q)_k\,x^k y^{n-k}.
\]
A three-parameter generalization is
\[
\Phi_n^{(a,b;c)}(x,y\mid q)
=
\sum_{k=0}^n
\binom{n}{k}_q
\frac{(a,b;q)_k}{(c;q)_k}\,
x^{n-k}y^k,
\]
and a five-parameter extension is introduced through homogeneous \(q\)-exponential operators [2011.05848].

## 2. Recurrence relations and \(q\)-difference equations

For \(U_n^{(a)}(x;q)\), the three-term recurrence recorded in the complex-parameter orthogonality study is
\[
x\,U_n^{(a)}(x;q)
=
U_{n+1}^{(a)}(x;q)
+
(a+1)q^n U_n^{(a)}(x;q)
-
a q^{n-1}(1-q^n)\,U_{n-1}^{(a)}(x;q),
\qquad n\ge 1,
\]
with
\[
U_0^{(a)}(x;q)=1,
\qquad
U_1^{(a)}(x;q)=x-(a+1).
\]
By Favard’s theorem, this recurrence together with the nonvanishing of the off-diagonal coefficients \(a q^{n-1}(1-q^n)\) is equivalent to quasi-definiteness of the corresponding linear functional [1601.08178].

A second-order \(q\)-difference equation for the monic Al-Salam–Carlitz I polynomials is
\[
\bigl((1-x)(x-a)D_q-\bigl[(1+a)-q x\bigr]\bigr)\,U_n(x)
=
-(1-q^n)\,U_n(x),
\]
where
\[
D_q f(x)=\frac{f(x)-f(qx)}{(1-q)x}.
\]
An equivalent form is
\[
(1-x)(x-a)\,U_n(qx)
+
(qx-1-a)\,U_n(x)
+
a\,U_n(x/q)
=
-q^n(1-q^n)\,U_n(x).
\]
This places the family in the standard eigenfunction setting for second-order \(q\)-difference operators [2206.08633].

For \(V_n^{(\alpha)}(x;q)\) on the lattice \(x(s)=q^{-s}\), the Nikiforov–Uvarov framework yields a second-order difference equation
\[
A(s;\alpha)\,y(s-1)+B(s;\alpha)\,y(s+1)+C(s;\alpha)\,y(s)=0,
\]
with
\[
A(s;\alpha)=(1-q^{-s})(\alpha-q^{-s}),
\qquad
B(s;\alpha)=\alpha q,
\]
and \(C(s;\alpha)\) determined by \(C=-A-B\) up to an inessential additive constant. The same family satisfies
\[
x\,V_n^{(\alpha)}(x)
=
V_{n+1}^{(\alpha)}(x)
+\beta_n V_n^{(\alpha)}(x)
+\gamma_n V_{n-1}^{(\alpha)}(x),
\]
where
\[
\beta_n=\alpha q^{1-n}+(1-q^{-n}),
\qquad
\gamma_n=\alpha(1-q^{-n})q^{-n+1}.
\]
These formulas are the basis for the discrete Stieltjes-type analysis of zero variation [2006.07884].

## 3. Orthogonality of Al-Salam–Carlitz I

The modern orthogonality theory for \(U_n^{(a)}(x;q)\) extends beyond the positive real regime. Let \(\mathbf u\) be the unique linear functional satisfying
\[
\langle \mathbf u,\,U_m^{(a)}U_n^{(a)}\rangle
=
k_n\,\delta_{m,n},
\qquad
k_n\neq 0.
\]
If \(a<0\) and \(0<q<1\), then \(\mathbf u\) is positive definite and is represented by a discrete \(q\)-Jackson integral on \([a,1]\). In the general complex-parameter case, with \(a,q\in\mathbb C\setminus\{0,1\}\), the functional is only quasi-definite, but orthogonality still holds on a simple closed contour \(\Gamma\) formed by two \(q\)-spirals joining \(1\to 0\to a\):
\[
S_1:\ z(t)=|q|^t e^{i t\arg(q)},
\qquad
S_2:\ z(t)=|a|\,|q|^t e^{i(t\arg(q)+\arg(a))},
\qquad t\in[0,\infty).
\]
With weight
\[
w(x;a;q)=(q x;q)_\infty (q x/a;q)_\infty,
\]
one has, for \(a\neq 0,1\) and \(0<|q|<1\),
\[
\int_\Gamma
U_m^{(a)}(x;q)\,U_n^{(a)}(x;q)\,w(x;a;q)\,d_qx
=
d_n^2\,\delta_{m,n},
\]
where
\[
d_n^2
=
(-a)^n(1-q)(q;q)_n (q;q)_\infty (a;q)_\infty (q/a;q)_\infty q^{\binom n2}.
\]
An analogous statement holds for \(|q|>1\) after replacing \(q\mapsto q^{-1}\) and using the corresponding Jackson integral. In this formulation, the orthogonality relations characterize \(U_n^{(a)}(x;q)\) up to a degree-dependent multiplicative constant [1601.08178].

Other orthogonality realizations coexist in the literature. For the classical two-variable form \(\phi_n(x,1;a\mid q)\), continuous orthogonality is recorded on \([0,1]\) with weight
\[
w(x)=\frac{(q x;q)_\infty}{(a x;q)_\infty},
\]
namely
\[
\int_0^1
\phi_m(x,1;a\mid q)\,\phi_n(x,1;a\mid q)\,w(x)\,dx
=
h_n\,\delta_{m,n},
\]
with
\[
h_n
=
\frac{(1-a)(q;q)_n(aq;q)_n a^n}{(q;q)_\infty},
\qquad
0<|a|<1.
\]
The same source records a discrete orthogonality on the lattice \(x=q^{-k}\), \(k=0,1,2,\dots\), with a corresponding discrete measure [2011.05848].

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 [1601.08178].

## 4. Al-Salam–Carlitz II and spectral completion

For Al-Salam–Carlitz II, Groenevelt’s construction fixes two real parameters \(z_-<0<z_+\) and two nonzero parameters \(c,d\), and considers the discrete support
\[
R_q=\{z_-q^k,\ z_+q^k:\ k\in\mathbb Z\},
\qquad 0<q<1.
\]
The associated Jackson-sum integral is
\[
\int_{R_q} f(x)\,d_qx
=
(1-q)\sum_{k=-\infty}^{\infty}
\Bigl[f(z_+q^k)z_+q^k-f(z_-q^k)z_-q^k\Bigr],
\]
with weight
\[
w(x)=\frac{1}{(c x;q)_\infty(d x;q)_\infty}.
\]
Under the stated nonvanishing and positivity conditions, including the sample sufficient conditions
\[
q<1,\qquad c,d>0,\qquad q^k<z_\pm<q^{k-1}\ \text{for some integer }k,
\]
the inner product
\[
\langle f,g\rangle=\int_{R_q} f(x)\,g(x)\,w(x)\,d_qx
\]
is positive definite. Writing
\[
P_n(x)=V_n\!\Bigl(x;\frac cd\Bigm|q\Bigr),
\]
one obtains
\[
\sum_{x\in R_q} P_m(x)P_n(x)\,w(x)\,d_qx
=
\delta_{m,n}\,h_n,
\]
with
\[
h_n
=
B(c,d;z_-,z_+)\,
(q;q)_n (cd)^{-n} q^{-\tfrac{n(n-1)}2},
\]
where
\[
B(c,d;z_-,z_+)
=
(1-q)z_+
\frac{(q;q)_\infty}
{(z_-/z_+,\,cd\,z_-z_+;q)_\infty\,
(cz_-,dz_-,cz_+,dz_+;q)_\infty}.
\]
The derivation uses a specialization of Askey’s \(q\)-beta integral together with the \(q\)-binomial theorem [1309.7569].

A corresponding second-order \(q\)-difference operator is
\[
(Lf)(x)
=
A(x)\,[f(qx)-f(x)]
+
B(x)\,[f(x/q)-f(x)],
\]
with
\[
A(x)=(1-cx)(1-dx),
\qquad
B(x)=-cd\,q\,x^2.
\]
This operator is symmetric and, with a suitable dense domain, self-adjoint on \(L^2(R_q,w\,d_qx)\). The Al-Salam–Carlitz II polynomials are eigenfunctions:
\[
L\,P_n(x)=(q^{-n}-1)\,P_n(x),
\qquad n=0,1,2,\dots.
\]
The spectral analysis shows that the polynomial sector is not complete by itself: besides the discrete eigenvalues \(\{q^{-n}-1:n\ge 0\}\), there is a continuous spectrum accumulating at \(-1\). To obtain a complete orthogonal basis, one introduces supplementary nonpolynomial functions
\[
Q_n(x)=\phi_{\,q^{\,n+1}/(cdz_-z_+)}(x),
\qquad n\in\mathbb Z,
\]
where
\[
\phi_y(x)
=
\frac{(d q y x;q)_\infty}{(cx;q)_\infty}
\,
{}_1\phi_1\!\Bigl(qy;\,d q y x;\,q,\frac{c q}{y x}\Bigr).
\]
Then
\[
\{P_n:n\ge 0\}\cup\{Q_n:n\in\mathbb Z\}
\]
is an orthogonal basis of \(L^2(R_q,w\,d_qx)\) [1309.7569].

A separate orthogonality model for the same “second” family uses the \(q\)-linear grid \(x(s)=q^{-s}\), \(s=0,1,2,\dots\), and the positive weight
\[
\omega(s)=\frac{\alpha^s}{(q;q)_s},
\]
giving
\[
\sum_{s=0}^{\infty}
V_m^{(\alpha)}(q^{-s};q)\,
V_n^{(\alpha)}(q^{-s};q)\,
\frac{\alpha^s}{(q;q)_s}\,
q^{-s}(1-q)
=0,
\qquad m\neq n.
\]
This formulation is the one used in the monotonicity analysis of zeros [2006.07884].

## 5. Generating functions and operator generalizations

The classical generating function for \(U_n^{(a)}(x;q)\) is
\[
\frac{(x t;q)_\infty}{(a t;q)_\infty}
=
\sum_{n=0}^\infty
(-1)^n q^{\binom n2}(q;q)_n\,U_n^{(a)}(x;q)\,t^n,
\qquad |q|<1.
\]
A related basic-\({}_1\phi_1\) generating function is
\[
(a t;q)_\infty\,
{}_1\phi_1(x;\,a t;\,q,t)
=
\sum_{n=0}^\infty
q^{n(n-1)}(q;q)_n\,V_n^{(a)}(x;q)\,t^n,
\]
where \(V_n^{(a)}\) is a slight variant of \(U_n^{(a)}\). The complex-parameter orthogonality paper also derives a connection relation between \(U_n^{(a)}(x;p)\) and \(U_k^{(b)}(x;p)\), with \(p=q^{-1}\), and from it obtains the generalized generating function
\[
(a t;p)_\infty\,{}_1\phi_1(x;\,a t;\,p,t)
=
\sum_{k=0}^\infty
p^{k(k-1)}(p;p)_k\,
{}_1\phi_1(b/a;\,0;\,p,\;a t p^k)\,
U_k^{(b)}(x;p)\,t^k,
\qquad |a t|<1.
\]
These identities show that parameter changes can be encoded by explicit connection coefficients rather than by ad hoc renormalization [1601.08178].

The operator-theoretic extension developed in the generalized \(q\)-difference paper begins with the \(q\)-derivatives
\[
D_x\{f(x)\}
=
\frac{f(x)-f(qx)}{(1-q)x},
\qquad
\theta_x\{f(x)\}
=
\frac{f(q^{-1}x)-f(x)}{(q^{-1}-1)x},
\]
and introduces the five-parameter homogeneous \(q\)-exponential operators
\[
T(a,b,c,d,e,\;yD_x)
=
\sum_{n=0}^\infty
\frac{(a,b,c;q)_n}{(q,d,e;q)_n}\,(yD_x)^n,
\]
\[
E(a,b,c,d,e,\;y\theta_x)
=
\sum_{n=0}^\infty
(-1)^n q^{\tfrac{n(n-1)}2}
\frac{(a,b,c;q)_n}{(q,d,e;q)_n}\,(y\theta_x)^n.
\]
A seven-variable analytic function \(F(a,b,c,d,e,x,y)\) expands in the family \(\{\Phi_n^{(a,b,c)}_{d,e}(x,y\mid q)\}\) if and only if it satisfies the stated functional equation, and then
\[
F(a,b,c,d,e,x,y)
=
T(a,b,c,d,e,\;yD_x)\{F(a,b,c,d,e,x,0)\}.
\]
An analogous characterization holds for the \(E\)-family \(\{U_n^{(a,b,c)}_{d,e}(x,y\mid q)\}\), with
\[
F(a,b,c,d,e,x,y)
=
E(a,b,c,d,e,\;y\theta_x)\{F(a,b,c,d,e,x,0)\}.
\]
Within the same framework, the paper derives \(U(n+1)\)-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 [2011.05848].

## 6. Zeros, special parameter regimes, and limiting cases

For the type II family \(V_n^{(\alpha)}(x;q)\), the zero-monotonicity analysis is based on the function
\[
f(s;\alpha)
=
\frac{B(s;\alpha)}{A(s;\alpha)}
=
\frac{\alpha q}{(1-q^{-s})(\alpha-q^{-s})}.
\]
On the support of the zeros, \(s>0\), one has \(f(s;\alpha)>0\), \(\partial f/\partial s<0\), and \(\partial f/\partial \alpha>0\). The discrete Stieltjes theorem then implies that every zero
\[
x_{n,k}(\alpha)=q^{-s_{n,k}(\alpha)}
\]
is a strictly increasing function of \(\alpha\in(0,q^{-1})\). In the terminology of the paper, this is Proposition 3.3: the zeros of \(V_n^{(\alpha)}(\cdot;q)\) are strictly increasing in \(\alpha\in(0,q^{-1})\) [2006.07884].

For \(U_n^{(a)}(x;q)\), the real positive-definite regimes include \(a<0\) and \(0<q<1\), and also \(0<aq<1\) with \(q>1\). In these cases the polynomials are orthogonal on the real \(q\)-lattice in \([a,1]\); all zeros lie in \([a,1]\) and interlace in the usual way. When \(a,q\in\mathbb C\setminus\{0,1\}\) and \(\arg(q)\neq 0\), 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 [1601.08178].

The root-of-unity case \(|q|=1\), \(q^N=1\), 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 \(N-1\) by adding higher-order \(q\)-difference terms to the bilinear form. This produces quasi-definite bilinear forms for all degrees. At the degenerate parameter value \(a\to 0\), one recovers a simpler one-parameter family satisfying
\[
U_{n+1}^{(0)}(x;q)=(x-q^n)\,U_n^{(0)}(x;q),
\]
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 \(q\to 1^-\) yields Hermite-, Laguerre-, or Charlier-type limits of the Carlitz–Al-Salam family [1601.08178].

## 7. Appearance in discrete \(q\)-ensembles

In the construction of discrete orthogonal ensembles on exponential lattices, the monic Al-Salam–Carlitz I polynomials are taken on
\[
\mathcal L_q=\{q^m\mid m=0,1,2,\dots\},
\qquad 0<q<1,
\]
with Jackson integral
\[
\int_0^\infty f(x)\,d_qx
=
(1-q)\sum_{m=0}^\infty f(q^m)\,q^m
\]
and weight
\[
w(q^m)=\frac{a^m q^{\,\tfrac{m(m-1)}2}}{(q;q)_m}.
\]
The orthogonality relation is
\[
\int_0^\infty
U_n(x;a;q)\,U_m(x;a;q)\,w(x)\,d_qx
=
h_n(a,q)\,\delta_{nm},
\]
with
\[
h_n(a,q)=\frac{(-a)^n q^{\,\tfrac{n(n-1)}2}}{(q;q)_n}.
\]
In the same setting, one builds a Pfaffian ensemble by introducing a skew-symmetric kernel \(s(x,y)\) and the skew inner product
\[
\langle f,g\rangle_s
=
\iint
\bigl(f(x)g(y)-f(y)g(x)\bigr)\,
s(x,y)\,w(x)\,w(y)\,d_qx\,d_qy.
\]
The skew-orthogonal polynomials can be chosen as
\[
Q_{2n}(x)=U_{2n}(x;a;q),
\qquad
Q_{2n+1}(x)=U_{2n+1}(x;a;q)+a q^{2n}(1-q)\,U_{2n-1}(x;a;q),
\]
and the \(k\)-point correlation functions take the Pfaffian form
\[
\rho_{N,k}(x_1,\dots,x_k)
=
\mathrm{Pf}\bigl[K_N(x_i,x_j)\bigr]_{1\le i,j\le k}
\prod_{i=1}^k w(x_i),
\]
where the \(2\times 2\) matrix kernel is built from skew Christoffel–Darboux sums of \(\{Q_n\}\). In this probabilistic realization, the limit \(a\to 0\) gives \(U_n(x;0;q)=x^n\) and degenerates to the unitary-symmetry \(q\)-ensemble, while \(q\to 1^-\), after suitable rescaling, recovers continuous orthogonal and skew-orthogonal ensembles of Hermite- or Charlier-type [2206.08633].

Source: https://www.emergentmind.com/topics/al-salam-carlitz-orthogonal-polynomials