The Hahn-Exton $q$-Bessel function as the characteristic function of a Jacobi matrix

A family $\mathcal{T}^{(\nu)}$, $\nu\in\mathbb{R}$, of semiinfinite positive Jacobi matrices is introduced with matrix entries taken from the Hahn-Exton $q$-difference equation. The corresponding matrix operators defined on the linear hull of the canonical basis in $\ell^{2}(\mathbb{Z}_{+})$ are essentially self-adjoint for $|\nu|\geq1$ and have deficiency indices $(1,1)$ for $|\nu|<1$. A convenient description of all self-adjoint extensions is obtained and the spectral problem is analyzed in detail. The spectrum is discrete and the characteristic equation on eigenvalues is derived explicitly in all cases. Particularly, the Hahn-Exton $q$-Bessel function $J_{\nu}(z;q)$ serves as the characteristic function of the Friedrichs extension. As a direct application one can reproduce, in an alternative way, some basic results about the $q$-Bessel function due to Koelink and Swarttouw.


Introduction
There exist three commonly used q-analogues of the Bessel function J ν (z). Two of them were introduced by Jackson in the beginning of the 20th century and are mutually closely related, see [6] for a basic overview and original references. Here we shall be concerned with the third analogue usually named after Hahn and Exton. Its most important features like properties of the zeros and the associated Lommel polynomials including orthogonality relations were studied not so long ago [11,10,9]. The Hahn-Exton q-Bessel function is defined as follows J ν (z; q) ≡ J (3) ν (z; q) = (q ν+1 ; q) ∞ (q; q) ∞ z ν 1 φ 1 (0; q ν+1 ; q, qz 2 ).
In order to keep notations simple we will also suppress the superscript (ν) provided this cannot lead to misunderstanding. Our main goal in this paper is to provide a detailed analysis of those operators T in ℓ 2 ≡ ℓ 2 (Z + ) (with Z + standing for nonnegative integers) whose matrix in the canonical basis equals T . This example has that interesting feature that it exhibits a transition between the indeterminate and determinate cases depending on ν. In more detail, denote by C ∞ the linear space of all complex sequences indexed by Z + and by D the subspace of those sequences having at most finitely many nonvanishing entries. One may also say that D is the linear hull of the canonical basis in ℓ 2 . It turns out that the matrix operator induced by T on the domain D is essentially self-adjoint in ℓ 2 if and only if |ν| ≥ 1. For |ν| < 1 there exists a one-parameter family of self-adjoint extensions.
Another interesting point is a close relationship between the spectral data for these operators T and the Hahn-Exton q-Bessel function. It turns out that, for an appropriate (Friedrichs) self-adjoint extension, J ν (q −1/2 √ x; q) serves as the characteristic function of T in the sense that its zero set on R + exactly coincides with the spectrum of T . There also exists an explicit formula for corresponding eigenvectors. Moreover, T −1 can be shown to be compact. This makes it possible to reproduce, in a quite straightforward but alternative way, some results originally derived in [11,9].
Finally we remark that recently we have constructed, in [14,15], a number of examples of Jacobi operators with discrete spectra and characteristic functions explicitly expressed in terms of special functions, a good deal of them comprising various combinations of q-Bessel functions. That construction confines, however, only to a class of Jacobi matrices characterized by a convergence condition imposed on the matrix entries. For this condition is readily seen to be violated in the case of T , as defined in (3) and (4), in the present paper we have to undertake another approach whose essential part is a careful asymptotic analysis of formal eigenvectors of T .
2 Self-adjoint operators induced by T

A * -algebra of semiinfinite matrices
Denote by M fin the set of all semiinfinite matrices indexed by Z + × Z + such that each row and column of a matrix has only finitely many nonzero entries. For instance, M fin comprises all band matrices and so all finite-order difference operators. Notice that M fin is naturally endowed with the structure of a * -algebra, matrices from M fin act linearly on C ∞ and D is M fin -invariant.
Choose A ∈ M fin and let A H stand for its Hermitian adjoint. Let us introduce, in a fully standard manner, operatorsȦ, A min and A max on ℓ 2 , all of them being restrictions of A to appropriate domains. Namely,Ȧ is the restriction A D , A min is the closure ofȦ and Dom A max = {f ∈ ℓ 2 ; Af ∈ ℓ 2 }.
Clearly,Ȧ ⊂ A max . Straightforward arguments based just on systematic application of definitions show that Hence A max is closed and A min ⊂ A max . Lemma 1. Suppose p, w ∈ C and let A ∈ M fin be defined by A n,n = p n , A n+1,n = −wp n+1 for all n ∈ Z + , A m,n = 0 otherwise.
Then A min = A max if and only if 1/|p| < |w| < 1, and in that case Since both f and g in (6) are supposed to belong to ℓ 2 this condition is obviously fulfilled if |p| ≤ 1. Furthermore, the situation becomes fully transparent for w = 0.
In that case the sequences {p n g n } and {p n f n } are square summable and (6) is always fulfilled. In the remainder of the proof we assume that |p| > 1 and w = 0. Consider first the case when |w| ≥ 1. Relation Af = h can readily be inverted even in C ∞ and one finds that Denote temporarily byh the sequence withh n = (pw) −n . It is square summable since, by our assumptions, |pw| > 1. For f ∈ Dom A max one has h ∈ ℓ 2 and Assumption f ∈ ℓ 2 clearly implies h , h = 0 and then, by the Schwarz inequality, Whence A n,n g n f n → 0 as n → ∞ for all g ∈ Dom A H max and so f ∈ Dom A min . Suppose now that |w| < 1. If A H g = h in C ∞ and h is bounded then, as an easy computation shows, for all n and some constant γ. Observe that, by the Schwarz inequality, and this expression tends to zero as n tends to infinity provided h ∈ ℓ 2 . In the case when |pw| ≤ 1 the property g ∈ ℓ 2 and A H g = h ∈ ℓ 2 implies that the constant γ in (7) is zero, and from (8) one infers that A n,n g n → 0 as n → ∞. Thus one finds condition (6) to be always fulfilled meaning that f ∈ Dom A min .
If |pw| > 1 then the sequence g defined in (7) is square summable whatever γ ∈ C and h ∈ ℓ 2 are. Condition (6) is automatically fulfilled, however, for γ = 0. Hence (6) can be reduced to the single nontrivial case when we chooseg ∈ Dom A H max with g n = (pw) −n . Then A Hg = 0 and condition g, Af = 0 means that w −n f n → 0 as n → ∞. It remains to show that there exists f ∈ Dom A max not having this property. However the sequencef , withf n = w n , does the job since Af = (1, 0, 0, . . .) ∈ ℓ 2 .
Since we are primarily interested in spectral properties of T (ν) in the Hilbert space ℓ 2 we may restrict ourselves, without loss of generality, to nonnegative values of the parameter ν. The value ν = 0 turns out to be somewhat special and will be discussed separately later on, in Subsection 3.2. Thus, if not stated otherwise, we assume from now on that ν > 0.
Given T ∈ M fin we again introduce the operatorsṪ , T min , T max as explained in Subsection 2.1. Notice that It follows at once that the operatorsṪ and consequently T min are positive. In fact, for any real sequence {f n } ∈ D one has This is equivalent to the factorization T =A H A where the matrix A ≡ A (ν) ∈ M fin is defined by the prescription: ∀f ∈ C ∞ , That is, ∀n ≥ 0, and A m,n = 0 otherwise. Thus T induces a positive form on the domain D with values f, T f = Af 2 , ∀f ∈ D. Let us call t its closure. Then Dom t = Dom A min and t(x) = A min x 2 , ∀x ∈ Dom t. The positive operator T F associated with t according to the representation theorem is the Friedrichs extension of the closed positive operator T min . One has It is known that T F has the smallest form-domain among all self-adjoint extensions of T min and also that this is the only self-adjoint extension of T min with its domain contained in Dom t, see [8, Chapter VI].
In [5] one finds a clear explicit description of the domain of the Friedrichs extension of a positive Jacobi matrix which can be applied to our case. To this end, consider the homogeneous three-term recurrence equation on Z. It simplifies to a recurrence equation with constant coefficients, One can distinguish two independent solutions, {Q n } and {Q (2) n }, where Notice that {Q (1) n } satisfies the initial conditions Q  where W n (f, g) := α n (f n g n+1 − g n f n+1 ). Theorem 4 in [5] tells us that where we put for f, g ∈ C ∞ provided the limit exists. It is useful to note, however, that discrete Green's formula implies existence of the limit whenever f, g ∈ Dom T max , and then We wish to determine all self-adjoint extensions of the closed positive operator T min . This is a standard general fact that the deficiency indices of T min for any real symmetric Jacobi matrix T of the form (3), with all α n 's nonzero, are either (0, 0) or (1, 1). The latter case happens if and only if for some x ∈ C all solutions of the second-order difference equation are square summable on Z + , and in that case this is true for any value of the spectral parameter x (see, for instance, a detailed discussion in Section 2.6 of [16]). Let us remark that a convenient description of the one-parameter family of all selfadjoint extensions is also available if the deficiency indices are (1,1). Fix x ∈ R and any couple Q (1) , Q (2) of independent solutions of (18). Then all self-adjoint extensions of T min are operatorsT (κ) defined on the domains In our case we know, for x = 0, a couple of solutions of (18) explicitly, cf. (16). From their form it becomes obvious that T min = T max is self-adjoint if and only if ν ≥ 1. With this choice of Q (1) , Q (2) and sticking to notation (19), it is seen from (17) that the Friedrichs extension T F coincides withT (∞).
Then every sequence f ∈ Dom T max has the asymptotic expansion where C 1 , C 2 ∈ C are some constants. Proof Hence A H g = h and, as already observed in the course of the proof of Lemma 1, there exists a constant γ such that Furthermore, the relation Af = g can be inverted, Bearing in mind that h ∈ ℓ 2 one concludes, with the aid of the Schwarz inequality, that ζ n → 0 as n → ∞.
With the knowledge of the asymptotic expansion established in Lemma 2 one can formulate a somewhat simpler and more explicit description of self-adjoint extensions of T min .
In particular, T (∞) equals the Friedrichs extension T F .
One finds at once that W ∞ (g (1) , h) = W ∞ (g (2) , h) = 0 and After a simple computation one deduces from (19) that f ∈ Dom T max belongs to is one-to-one, P 1 (R) ∋ κ → T (κ) is another parametrization of self-adjoint extensions of T min . Particularly,κ = ∞ maps to κ = ∞ and so T (∞) = T F . Remark 4. One can also describe Dom T min . For ν ≥ 1 we simply have T min = T max = T F . In the case when 0 < ν < 1 it has been observed in [5] that a sequence f ∈ Dom T max belongs to Dom T min if and only if W ∞ (f, g) = 0 for all g ∈ Dom T max . But this is equivalent to the requirement C 1 (f ) = C 2 (f ) = 0. Thus one has

The Green function and spectral properties
For Proposition 5. The matrix (G j,k ) defined in (23) represents a Hilbert-Schmidt operator G ≡ G (ν) on ℓ 2 with the Hilbert-Schmidt norm The operator G is positive and one has, ∀f ∈ ℓ 2 , Moreover, the inverse G −1 exists and equals T , if ν ≥ 1, or T F , if 0 < ν < 1.
Proof. As is well known, if T min is not self-adjoint then the resolvent of any of its self-adjoint extensions is a Hilbert-Schmidt operator [16,Lemma 2.19]. But in our case the resolvent is claimed to be Hilbert-Schmidt for ν ≥ 1 as well. One can directly compute the Hilbert-Schmidt norm of G for any ν > 0, Thus one obtains (24). Hence the Green matrix unambiguously defines a self-adjoint compact operator G on ℓ 2 . Concerning the formula for the quadratic form one has to verify that, for all m, n ∈ Z + , m ≤ n, But this can be carried out in a straightforward manner. A simple computation shows that for any f ∈ C ∞ and n, N ∈ Z + , n < N, Considering the limit N → ∞ one finds that, for a given f ∈ Dom T max , the equality GT f = f holds iff W ∞ (f, Q (2) ) = 0. According to (17), this condition determines the domain of the Friedrichs extension T F . Hence GT F ⊂ I (the identity operator). Furthermore, one readily verifies that, for all f ∈ ℓ 2 , T Gf = f . We still have to check that Ran G ⊂ Dom T F . But using the equality W ∞ (Q (1) , Q (2) ) = 1 one computes, for f ∈ ℓ 2 and n ∈ Z + , Considering the case ν ≥ 1, the fact that the Jacobi operator T is positive and T −1 is compact has some well known consequences for its spectral properties. The same conclusions can be made for 0 < ν < 1 provided we replace T by T F . And from the general theory of self-adjoint extensions one learns that T (κ), for κ ∈ R, has similar properties as T F [18,Theorem 8.18]. Proposition 6. The spectrum of any of the operators T , if ν ≥ 1, or T (κ), with arbitrary κ ∈ P 1 (R), if 0 < ν < 1, is pure point and bounded from below, with all eigenvalues being simple and without finite accumulation points. Moreover, the operator T , for ν ≥ 1, or T F , for 0 < ν < 1, is positive definite and one has the following lower bound on the spectrum, i.e. on the smallest eigenvalue ξ 1 ≡ ξ (ν) 1 , Proof. This is a simple general fact that all formal eigenvectors of the Jacobi matrix T are unique up to a multiplier [3]. By Proposition 5, (T F ) −1 is compact and therefore the spectrum of T F is pure point and with eigenvalues accumulating only at infinity. For 0 < ν < 1, the deficiency indices of T min are (1, 1). Whence, by the general spectral theory, if T F has an empty essential spectrum then the same is true for all other self-adjoint extensions T (κ), κ ∈ R. Moreover, there is at most one eigenvalue of T (κ) below ξ 1 := min spec(T F ), see [18, § 8.3]. Referring once more to Proposition 5 one has min spec(T F ) = (max spec(G)) −1 ≥ G −1 HS . In view of (24), one obtains the desired estimate on ξ 1 .

More details on the indeterminate case
In this subsection we confine ourselves to the case 0 < ν < 1 and focus on some general spectral properties of the self-adjoint extensions T (κ), κ ∈ P 1 (R), in addition to those already mentioned in Proposition 6. The spectra of any two different self-adjoint extensions of T min are known to be disjoint (see, for instance, proof of Theorem 4.2.4 in [3]). Moreover, the eigenvalues of such a couple of self-adjoint extensions interlace (see [12] and references therein, or this can also be deduced from general properties of self-adjoint extensions with deficiency indices (1, 1) [18, § 8.3]). It is useful to note, too, that every x ∈ R is an eigenvalue of a unique self-adjoint extension T (κ), κ ∈ P 1 (R) [13,Theorem 4.11].
For positive symmetric operators there exists another powerful theory of selfadjoint extensions due to Birman, Krein and Vishik based on the analysis of associated quadratic forms. A clear exposition of the theory can be found in [2]. Its application to our case, with deficiency indices (1, 1), is as follows. A crucial role is played by the null space of T max = T * min which we denote by N := Ker T max = CQ (1) (recall that Q n =P n (0), ∀n ∈ Z + ). Let t ∞ = t be the quadratic form associated with the Friedrichs extension T F . Remember that the domain of t has been specified in (13). All other self-adjoint extensions of T min , except of T F , are in one-to-one correspondence with real numbers τ . The corresponding associated quadratic forms t τ , τ ∈ R, have all the same domain, (a direct sum), and for f ∈ Dom t ∞ , λ ∈ C, one has Our next task is to relate the self-adjoint extensions T (κ) described in Proposition 3 to the quadratic forms t τ .
Lemma 8. Let S and B be linear subspaces in a Hilbert space H such that S ∩B = {0}, and let s and b be positive quadratic forms on S and B, respectively. Denote bỹ s andb the extensions of these forms to S + B defined by ∀ϕ ∈ S , ∀η ∈ B,s(ϕ + η) = s(ϕ) andb(ϕ + η) = b(η), and assume that, for every ρ ∈ R, the forms + ρb is semibounded and closed. Then, for any τ ∈ C, the forms + τb is sectorial and closed. In particular, if S + B is dense in H thens + τb, τ ∈ C, is a holomorphic family of forms of type (a) in the sense of Kato.
But as already remarked above, the eigenvalues of T (κ) and T (∞) interlace and so we have (27).
One can admit complex values for the parameter τ in (25), (26). Then, according to Lemma 8, the family of forms t τ , τ ∈ C, is of type (a) in the sense of Kato. Referring once more to Proposition 7 one infers from [8, Theorem VII-4.2] that the family of self-adjoint operators T (κ), κ ∈ R, extends to a holomorphic family of operators on C. This implies that for any bounded interval K ⊂ R there exists an open neighborhood D of K in C and ρ ∈ R sufficiently large so that the resolvents (T (κ) + ρ) −1 , κ ∈ K, extend to a holomorphic family of bounded operators on D. In addition we know that, for every fixed n ∈ N and κ ∈ R, the nth eigenvalue of T (κ) is simple and isolated. By the analytic perturbation theory [8, § VII.3], ξ n (κ) is an analytic function on R.
This relation already implies thatP −1 (x) = 0,P 0 (x) = 1. Notice also that the last term in the sum, with k = n, is zero and so (29) is in fact a recurrence for {P n (x)}. Equation (29) is pretty standard. Nevertheless, one may readily verify it by checking that this recurrence implies the original defining recurrence, i.e. the formal eigenvalue equation (11) which can be rewritten as follows Actually, from (29) one derives that and so Proposition 11. The sequence of polynomials {q (ν−1)n/2P n (x); n ∈ Z + } converges locally uniformly on C to an entire function and one has, ∀n ∈ Z, n ≥ −1, Proof. Denote (temporarily) H n (x) = q (ν−1)n/2P n (x), n ∈ Z + . Then (29) means that Proceeding by mathematical induction in n one can show that, ∀n ∈ Z + , is the q-Pochhammer symbol. This is obvious for n = 0. For the the induction step it suffices to notice that (33) implies Moreover, 1 + a n−1 k=0 q k (−a; q) k = (−a; q) n .
From the estimate (34) one infers that {H n (x)} is locally uniformly bounded on C. Consequently, from (33) it is seen that the RHS converges as n → ∞ and so H n (x) → Φ(x) pointwise. This leads to identity (31). Furthermore, one can rewrite (29) as followŝ Taking into account (31) one arrives at (32). Finally, from the locally uniform boundedness and Montel's theorem it follows that the convergence of {H n (x)} is even locally uniform and so Φ(x) is an entire function.
It turns out that Φ(x) may be called the characteristic function of the Jacobi operator T , if ν ≥ 1, or the Friedrichs extension T F , if 0 < ν < 1.
Notice that, by the assumptions, S = ∅ and the definition ofGf makes good sense.
Proposition 13. If ν ≥ 1, the spectrum of T coincides with the zero set of Φ(x). If 0 < ν < 1 then spec T (κ), κ ∈ P 1 (R), consists of the roots of the characteristic equation In particular, the spectrum of T F = T (∞) equals the zero set of Φ(x).
Assume now that 0 < ν < 1. This the indeterminate case meaning thatP (x) is square summable for all x ∈ C. Hence x is an eigenvalue of T (κ) iffP (x) ∈ Dom T (κ). Recall that T (κ) is defined in Proposition 3. From (32) one derives the asymptotic expansion From here it is seen thatP (x) fulfills the boundary condition in (21) if and only if x solves the equation Referring to (16)

Lemma 15.
For every m ∈ Z + and σ > 0, Proof. For a given m ∈ N, one derives from (30) the three-term inhomogeneous recurrence relation with the initial conditions Recall that, by Proposition 11, the sequence {q (ν−1)n/2P n (x)} converges on C locally uniformly and hence it is locally uniformly bounded. Combining this observation with Cauchy's integral formula one justifies that, for any m ∈ Z + fixed, the sequence {q (ν−1)n/2 d mP n (0)/dx m } is bounded as well. Therefore the LHS of (39) is well defined. Let us call it S m,σ . Applying summation in n to (40) and bearing in mind (41) one derives the recurrence Particularly, for m = 0 we know thatP n (0) = Q (1) n , n ∈ Z + . Whence (16)). A routine application of mathematical induction in m proves (39).
From (2) it is seen that the sequences {u k (x); k ∈ Z} and {v k (x); k ∈ Z}, where u k (x) = q k/2 J ν (q k/2 √ x; q) and v k (x) = q k/2 J −ν (q (k−ν)/2 √ x; q), obey both the difference equation (with α k , β k being defined in (4)). In the case of the former sequence, ν can be arbitrary positive, and in the case of the latter one we assume that 0 < ν < 1. Hence the sequence (u 0 (x), u 1 (x), u 2 (x), . . .) is a formal eigenvector of the Jacobi matrix T if and only if u −1 (x) = 0. A similar observation holds true if we replace u k (x) by u k (κ, x). In view of Proposition 14, it suffices to notice that u −1 (x) is proportional to Φ(x) and u −1 (κ, x) to κΦ(x) + Ψ(x).

The case ν = 0
The case ν = 0 is very much the same thing as the case when 0 < ν < 1. First of all, this is again an indeterminate case, i.e. T min is not self-adjoint. On the other hand, there are some differences causing the necessity to modify several formulas, some of them rather substantially. Perhaps the main reason for this is the fact that the characteristic polynomial of the difference equation with constant coefficients, (15), has one double root if ν = 0 while it has two different roots if 0 < ν. Here we summarize the basic modifications but without going into details since the arguing remains quite analogous. For ν = 0 one has Dom t = {f ∈ ℓ 2 ; Af ∈ ℓ 2 }, and two distinguished solutions of (14) are Q (1) n = (n + 1)q n/2 , Q (2) n = q n/2 , n ∈ Z, where again Q (1) n =P n (0) for n ≥ 0 and {Q (2) n } is a minimal solution, W n (Q (1) , Q (2) ) = 1. The asymptotic expansion of a sequence f ∈ Dom T max reads f n = C 1 (n + 1) + C 2 q n/2 + o(q n ) as n → ∞, with C 1 , C 2 ∈ C. The one-parameter family of self-adjoint extensions of T min is again denoted T (κ), κ ∈ P 1 (R). Definition (21) of Dom T (κ) formally remains the same but the constants C 1 (f ), C 2 (f ) in the definition are now determined by the limits C 1 (f ) = lim n→∞ f n (n + 1) −1 q −n/2 , C 2 (f ) = lim n→∞ f n − C 1 (f ) (n + 1)q n/2 q −n/2 .
One still has T (∞) = T F . Similarly, f ∈ Dom T max belongs to Dom T min if and only if C 1 (f ) = C 2 (f ) = 0 meaning that (22) is true for ν = 0, too. Furthermore, everything what is claimed in Propositions 5 and 6 about the values 0 < ν < 1 is true for ν = 0 as well.
Relation (29) is valid for ν = 0 as well but more substantial modifications are needed in Proposition 11. One has and the convergence is locally uniform on C for one can estimate q −n/2 n + 1P n (x) ≤ n k=0 1 + (k + 1)q k |x| , n ∈ Z + .

Some applications to the q-Bessel functions
In this section we are going to only consider the Friedrichs extension if 0 < ν < 1. To simplify the formulations below we will unify the notation and use the same symbol T F for the corresponding self-adjoint Jacobi operator for all values of ν > 0, this is to say even in the case when ν ≥ 1. Making use of the close relationship between the spectral data for T F and the q-Bessel functions, as asserted in Propositions 13 and 14, we are able to reproduce in an alternative way some results from [11,9].
Remark. It is not difficult to show that the proposition remains valid also for −1 < ν ≤ 0. To this end, one can extend the values ν > 0 to ν = 0 following the lines sketched in Subsection 3.2 and employ Propositions 13 and 14 while letting κ = 0 in order to treat the values −1 < ν < 0. But we omit the details. An original proof of this proposition can be found in [11,Section 3].
Proof. All claims, except the simplicity of zeros and the normalization of eigenvectors, follow from the known spectral properties of T F . Namely, T F is positive definite, (T F ) −1 is compact, spec T F = {qw 2 n ; n ∈ N} and corresponding eigenvectors are given by formula (38); cf. Propositions 5, 6, 13 and 14.
In addition, one obtains at once an orthogonality relation for the sequence of orthogonal polynomials {P n (x)}. As is well known from the general theory [3] and Proposition 3, the orthogonality relation is unique if ν ≥ 1 and indeterminate if 0 < ν < 1. It was originally derived in [9,Theorem 3.6].
Remark 18. To complete the picture let us mention two more results which are known about the Hahn-Exton q-Bessel functions and the associated polynomials. First, denote again by w (ν) n ≡ w n , n ∈ N, the increasingly ordered positive zeros of J ν (z; q). In [1] it is proved that if q is sufficiently small, more precisely, if q ν+1 < (1 − q) 2 then q −m/2 > w m > q −m/2 1 − q m+ν 1 − q m , ∀m ∈ N.
More generally, in Theorem 2.2 and Remark 2.3 in [4] it is shown that for any q, 0 < q < 1, one has w m = q −m/2 (1 + O(q m )) as m → ∞.
Let us remark that a relative formula in terms of the Al-Salam-Chihara polynomials has been derived in [17,Theorem 2].