Asymptotics for Some q-Hypergeometric Polynomials

We tackle the study of a type of local asymptotics, known as Mehler–Heine asymptotics, for some q–hypergeometric polynomials. Some consequences about the asymptotic behavior of the zeros of these polynomials are discussed. We illustrate the results with numerical examples.


Introduction
The basic q-hypergeometric function r φ s is defined by the series (see, for example, [2] or [15, where (a 1 , . . . , a r ; q) k = (a 1 ; q) k (a 2 ; q) k · · · (a r ; q) k . For our purposes we assume throughout the paper 0 < q < 1. The expressions (a j ; q) k and (b j ; q) k denote the q-analogues of the Pochhammer symbol, i.e., given a complex number a Obviously, the series (1) is well-defined when the quantities, known as q-shifted factorials or q-Pochhammer symbols, (b j ; q) k = 0, for j = 0, . . . , s. It is well known that the radius of convergence ρ of the q−hypergeometric functions (1) is given by (see for example [15, p. 15]) ρ = ⎧ ⎨ ⎩ ∞, if r < s + 1; 1, if r = s + 1; 0, if r > s + 1.
In particular, for our interest s φ s is always convergent. In this paper we consider r = s because it is the context where we can establish our main goal (see Theorem 1).
When one of the parameters a j in (1) is equal to q −n , where n is a nonnegative integer, the basic q-hypergeometric function is a polynomial of degree at most n in the variable z. Thus, our objective is to obtain a type of asymptotics for these q-polynomials. Concretely, by scaling adequately these polynomials we intend to get a limit relation between them and a q-analogue of the Bessel function of the first kind. In the framework of orthogonal polynomials, this type of asymptotics is known as Mehler-Heine asymptotics (also as Mehler-Heine formula). Originally, this type of local asymptotics was introduced for a special case of orthogonal polynomials (OP), Legendre polynomials, by the German mathematicians H. E. Heine and G. F. Mehler in the 19th century. Later, it was extended to the families of classical OP: Jacobi, Laguerre, Hermite (see, for example, [19]). More recently, these formulae were obtained for other families of polynomials such as OP in the Nevai's class [5], discrete OP [7], generalized Freud polynomials [1], multiple OP [20,21] or Sobolev OP (in this context there are many papers in this century, being [17] one of the first), among others.
In our case, we will show that the variable is scaled z → a n z by a sequence a n satisfying a n → 0 when n → +∞. So, we can establish asymptotic relations between these q-polynomials and a q-Bessel function. Thus, the novelty of our approach is to extend the classical Mehler-Heine formulae to these qhypergeometric polynomials. Now, we establish the notation that we will use in the following sections and we will show our main result. We denote by [z] q the well-known q-number given by for 0 < q < 1, it is easy to prove that lim q→1 [z] q = z.
In addition, we will use the q-Gamma function given by (see, for example, [3, This function is a q-analogue of the Gamma function. It is a meromorphic function, without zeros and with poles in z = −n ± 2πik/ log(q) where n and k are nonnegative integers. This function verifies the relation (see [3,Pag. 495 and satisfies An important role in this paper is played by the q-Bessel function J (2) α (z; q) given by (see, for example, [15, f. (1.14.8)]) which is an extension of the Bessel function of the first kind J α (z), i.e.
With this notation, we will prove in the Theorem 1 that, for s ≥ 2, lim n→+∞ s φ s q −n , q asn+bs q α , q csn+ds ; q, where q asn+bs = q a1n+b1 , q a2n+b2 , . . . , q as−1n+bs−1 , [n] q as = [n] q a 1 [n] q a 2 · · · [n] q a s−1 , The above result is also true when s = 1, getting in this case the following relation: We also discuss the case r − 1 ≤ s in Proposition 4. In addition, one of the referees asked us what result would be obtained when we take the scale z → q n z. These results are shown in Propositions 5 and 6. They are very nice but, as far as we know, they cannot be used to deduce the result obtained in [6], so we have included both scaling.
We structure the paper as follows. Section 2 is devoted to technical results which will be necessary in Sect. 3 to prove the main result, Theorem 1, as well as Propositions 4, 5 and 6. Finally, in Sect. 4 we discuss the consequences of the Mehler-Heine formula on the asymptotic behavior of the zeros of these q-hypergeometric polynomials. We illustrate this discussion with a variety of numerical examples and we leave some questions open, concretely one about the zeros of the q-function z 1−α J (2) α−1 (z(1 − q); q) .

Technical Results
This section is devoted to obtaining some properties about the q-Pochhammer symbol defined in (2) and the q-Gamma function given by (7). Actually, we are going to establish six technical statements which are indispensable to prove our main result in the following section. We do not claim that these results are new, but we have not found anything like them in our search in the literature.

Lemma 1. Let k be a nonnegative integer and z a complex number, such that
Proof. Using (2) and (9) it follows that

Lemma 2.
Let a be a positive real number and b a complex number. Then, we have, for any k ∈ Z fixed, Proof. First, we prove the result for a nonnegative integer k. Then, using (9) in a recursive way, we get When k is a negative integer, we can adapt the above proof easily applying (9) to the denominator instead of the numerator.

Lemma 3. Let k be a nonnegative integer. Then,
Now, using (15) and (16), the limit on the right side of the above expression can be computed as from where the result arises.
For the next results, we assume i j = 0 when i < j.

Lemma 5.
We have for n ≥ 1, Proof. For k = 0 the proof is trivial and the equality is reached.
which proves the result. In the next two statements we provide useful bounds for where b is a complex number with some restrictions (see Proposition 2) and a is a positive real number. These bounds allow us to prove the main result of this paper.

Proposition 1.
Let a be a positive real number and b = γ + iβ a complex number. Then, we have for n ≥ 1, Proof. We notice that 1 > q a ≥ q an > 0 for n ≥ 1, then We are going to use (5) and this inequality to prove the result, so Next, we use the notation Z − for the set formed by the number 0 and the negative integers, i.e. Z − = {0, −1, −2, . . .}.

Proposition 2.
Let a be a positive real number and b = γ + iβ a complex number. We assume that an + γ / ∈ Z − for all n positive integer. Then, it exists ε > 0 such that for n ≥ 1 and k = 0, 1, . . . , n, we have Proof. To prove this lower bound, we will use the well-known inequality |z − w| ≥ |z| − |w| where z and w are complex numbers, and the equality q an+γ+j+iβ = q an+γ+j .
Then, we have Now, we distinguish two cases: • When γ ≥ 0, taking into account q an ≥ q an+γ+j , we obtain that and the result follows. • When γ < 0, we have Now, on the one hand, assuming a and b fixed, it exits a positive integer n 0 that depends on γ such that for all n > n 0 , we have Thus, we can affirm that for n > n 0 , On the other hand, for n ∈ {1, 2, . . . , n 0 }, k = 0, . . . , n, and assuming an + γ / ∈ Z − with n any positive integer, we have that −γ = an+j for any j nonnegative integer. Thus, we define Taking δ := min Δ > 0, we obtain Finally, if we define ε := min{ q −γ 2 , δ} > 0 the result holds for n ≥ 1. (1−q) k [n] k q a . For our purposes, and without loss of generality, we can affirm that there are two constants, C a and D a , independent of n, so that

Main Result
In this section we obtain the main goal of this paper: the Mehler-Heine asymptotics for some q-hypergeometric polynomials. Before stating this result, we still have to take a further step on this issue giving a relation for the q-Bessel function (10).

Proposition 3.
Let α ∈ R\Z − be. Then, Proof. From (10), we have Then, making the change x 2 = 4z and introducing the factor (1 − q) 1−α , we have the following identity, Now, using (1) and (7), we get Finally, it is enough to make the change of variable √ z → √ z(1 − q) to end the proof.
We have all the ingredients to establish our main result.
Theorem 1 (Mehler-Heine asymptotics). We use the notation from (12)- (14), assuming that α ∈ R\Z − , a j > 0, c j > 0 and that b j and d j are complex numbers satisfying a j n + Re(b j ) / ∈ Z − and c j n + Re(d j ) / ∈ Z − with j ∈ {1, 2, . . . , s − 1} and s ≥ 2. Then, lim n→+∞ s φ s q −n , q asn+bs q α , q csn+ds ; q, uniformly on compact subsets of the complex plane. For s = 1, it holds Proof. First, we observe that the quantities a j n + b j and c j n + d j , for j = 1, . . . , s − 1 and s ≥ 2, satisfy the hypothesis posed in Proposition 2.
Now, scaling the variable z in (1) in the following way z → Using (17) and (18), we have for k fixed uniformly on compact subsets of the complex plane. Furthermore, we take z on a compact subset Ω of the complex plane, so |z| ≤ C Ω . Then, for n ≥ 0 and 0 ≤ k ≤ n, we get where we have used (19) and (20). Thus, we have found a dominant for n k=0 g n,k (z). This dominant is convergent, i.e. the series +∞ k=0 g k (z) converges. We can see this by applying the D'Alembert (or quotient) criterion for series of nonnegative terms. Now, to apply the Lebesgue's dominated convergence theorem, for n, k ≥ 0 we define F n,k (z) as Then, using (23) and (24) and taking z on a compact subset Ω of the complex plane, we have for each k Thus, we can write where dμ(k) is the discrete measure with support on the nonnegative integers (k = 0, 1, . . . ) and with a mass equal to one in each point of the support. Then, we apply the Lebesgue's dominated convergence theorem and (23), obtaining Finally, using Proposition 3 we get (21). The proof of (22) is similar, but now it is not necessary to use either the limit (4) or the bounds (20). Now, we can tackle the case r − 1 ≤ s. As we have commented previously, in this case, as far as we know, the limit function cannot be expressed as a known q-hypergeometric function except when r = s, then we get the same result as in Theorem 1.
In addition, Proof. Observe that when r ≥ 2 and s ≥ 2, we can write n,k (z).
Moreover, we can prove for k fixed that uniformly on compact subsets of the complex plane. Then, acting like in the proof of Theorem 1 we can apply the Lebesgue's dominated convergence theorem, getting the result. Finally, the asymptotic relations (25) and (26) can be obtained in the same way by handling the notation adequately.

Remark 2.
It is worth noting that the condition r −1 ≤ s is necessary to apply the Lebesgue's dominated convergence theorem in the two previous proofs. Then, using (4), (8), (10) and (11) in Theorem 1, we deduce Theorem 1 in [6] when r = s.
As we have mentioned previously in the introduction, one of the referees proposed us to use the scaling z → q n z. We have obtained the following statements.
Proposition 5. We use the notation from (12)- (14), assuming that α ∈ R\Z − , a j > 0, c j > 0 and that b j and d j are complex numbers satisfying a j n+Re(b j ) / ∈ Z − and c j n + Re(d j ) / ∈ Z − with j ∈ {1, 2, . . . , s − 1} and s ≥ 2. Then, For s = 1, we have uniformly on compact subsets of the complex plane.
In addition, under the assumptions posed in Propositions 1 and 2, we use the same technique to establish that there are two constants, C a and D a , independent of n, satisfying From (28-31) we can prove the result in the same way as in Theorem 1. The proof of (27) is similar, but now it is not necessary to use either the limit (29) or the bounds (31).

Remark 4.
Notice that the result in Proposition 5 does not depend on s. This is due to the type of scaling and to the fact that all q-numbers disappear in the limits (28)-(29). Now, we discuss the case r − 1 ≤ s making the scaling z → q n z. Proposition 6. We take r ≥ 1, s ≥ 1, r − 1 ≤ s, and α ∈ R\Z − . We consider b j and d complex numbers satisfying a j n + Re(b j ) / ∈ Z − , c n + Re(d ) / ∈ Z − where a j > 0, c > 0 with j ∈ {1, 2, . . . , r − 1} and ∈ {1, 2, . . . , s − 1}. Then, for r ≥ 2 and s ≥ 2, In addition, Proof. The proof is totally similar to the one of Proposition 4 but in this case we use (28)-(31).

Two Classical Examples
Now, we use Theorem 1 and Proposition 4 to obtain the Mehler-Heine formula for two important families of basic hypergeometric polynomials.
After some simple algebraic computations we obtain L (α) n (z; q) = q α+1 ; q n (q; q) n 1 φ 1 q −n q α+1 ; q, −q n+α+1 z . and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons. org/licenses/by/4.0/.