The orthogonality of q-classical polynomials of the Hahn class: A geometrical approach

The idea of this review article is to discuss in a unified way the orthogonality of all positive definite polynomial solutions of the $q$-hypergeometric difference equation on the $q$-linear lattice by means of a qualitative analysis of the $q$-Pearson equation. Therefore, our method differs from the standard ones which are based on the Favard theorem, the three-term recurrence relation and the difference equation of hypergeometric type. Our approach enables us to extend the orthogonality relations for some well-known $q$-polynomials of the Hahn class to a larger set of their parameters. A short version of this paper appeared in SIGMA 8 (2012), 042, 30 pages http://dx.doi.org/10.3842/SIGMA.2012.042.


Introduction
The so-called q-polynomials are of great interest inside the class of special functions since they play an important role in the treatment of several problems such as Eulerian series and continued fractions [7,14], q-algebras and quantum groups [22,23,33] and q-oscillators [9,4,17], and references therein, among others.
A q-analog of the Chebychev's discrete orthogonal polynomials is due to Markov in 1884 [6, page 43], which can be regarded as the first example of a q-polynomial family. In 1949, Hahn introduced the q-Hahn class [18] including the big q-Jacobi polynomials, on the exponential lattice although he did not use this terminology. In fact, he did not give the orthogonality relations of the big q-Jacobi polynomials in [18] which was done by Andrew and Askey [6]. During the last decades the q-polynomials have been studied by many authors from different points of view. There are two most recognized approaches. The first approach, initiated by the work of Askey and Wilson [8] (see also Andrews and Askey [6]) is based on the basic hypergeometric series [7,16]. The second approach is due to Nikiforov and Uvarov [28,29] and uses the analysis of difference equations on non-uniform lattices. The readers are also referred to the surveys [10,27,30,32]. These approaches are associated with the so-called q-Askey scheme [20] and the Nikiforov-Uvarov scheme [30], respectively. Another approach was published in [26] where the authors proved several characterizations of the q-polynomials starting from the so-called distributional q-Pearson equation (for the non q-case see e.g. [15,25] and references therein).
In this paper, we deal with the orthogonality properties of the q-polynomials of the q-Hahn tableau from a different viewpoint than the one used in [20]. In [20], the authors presented a unified study of the orthogonality of q-polynomials based on the Favard Theorem. Here, the main idea is to provide a relatively simple geometrical analysis of the q-Pearson equation by taking into account every possible rational form of the polynomial coefficients of the q-difference equation. Roughly, our qualitative analysis is concerned with the examination of the behavior of the graphs of the ratio ρ(qx, q)/ρ(x, q) by means of the definite right hand side (r.h.s.) of (1.5) in order to find out a suitable q-weight function. Such a qualitative analysis implies all possible orthogonality relations among the polynomial solutions of the q-difference equation in question. Moreover, it allows us to extend the orthogonality relations for some well-known q-polynomials of the Hahn class to a larger set of their parameters (see sections 4.1 and 5.1). A first attempt of using a geometrical approach for studying the orthogonality of qpolynomials of the q-Hahn class was presented in [12]. However, the study is far from being complete and only some partial results were obtained. We will fill this gap in this review paper.
Our main goal is to study each orthogonal polynomial system or sequence (OPS), which is orthogonal with respect to (w.r.t.) a q-weight function ρ(x, q) > 0 satisfying the q-Pearson equation as well as certain boundary conditions (BCs) to be introduced in Section 2. For each family of polynomial solutions of (1.1) we search for the ones that are orthogonal in a suitable intervals depending on the range of the parameters coming from the coefficients of (1.1) and the corresponding q-Pearson equation. Hence, in Section 2, we present the candidate intervals by inspecting the BCs as well as some preliminary results. Theorems which help to calculate q-weight functions are given in Section 3. Section 4 deals with the qualitative analysis including the theorems stating the main results of our article. The last Section concludes the paper with some final remarks.

The orthogonality and preliminary results
We first introduce the so-called q-Jackson integrals and afterward a well known theorem for the orthogonality of polynomial solutions of (1.1) in order to make the article self-contained [1,11,27].
The q-Jackson integrals for q ∈ (0, 1) [16,19] are defined by f (x)d q x (2.2) when 0 < a < b and a < 0 < b, respectively. Furthermore, we make use of the improper q-Jackson integrals where the second one is sometimes called the bilateral q-integral. The q −1 -Jackson integrals are defined similarly. For instance, the improper q −1 -Jackson integral on (a, ∞) is given by q −j f (q −j a), a > 0 (2.4) provided that lim j→∞ q −j f (q −j a) = 0 and the series is convergent.
Theorem 2.1 Let ρ be a function satisfying the q-Pearson equation (1.5) in such a way that the BCs where d n (q) and δ mn denote the norm of the polynomials P n and the Kronecker delta, respectively. Analogously, if the conditions are fulfilled, the q-polynomials then satisfy the relation The relation (2.6) means that the polynomials P n (x, q) are orthogonal with respect to a measure supported on the set of points {q k a} k∈N 0 and {q k b} k∈N 0 . Since we are interested in the positive definite cases, i.e., when ρ(x, q) > 0, then, • when a = 0, the measure is supported on the set of points {q k b} k∈N 0 in (0, b].
• when a > 0, the measure should be supported on the finite set of points {q k b} N k=0 being a = q N +1 b.
• when a < 0, the measure is supported on the set A similar analysis can be done for the relation (2.8).
According to Theorem 2.1, we have to determine an interval (a, b) in which ρ is q-integrable and ρ > 0 on the lattice points of the types αq ±k and βq ±k for k ∈ N 0 and α, β ∈ R. Such a weight function will be a solution of the q-Pearson equation (1.5). To this end, a qualitative analysis of the q-Pearson equation is presented by a detailed inspection of the r.h.s. of (1.5). Note that the r.h.s. of (1.5) consists of the polynomial coefficients σ 1 and σ 2 of the q-EHT which can be made definite for possible forms of the coefficients. As a result, the possible behavior of ρ on the left hand side (l.h.s.) of (1.5) and the candidate intervals can be obtained accordingly.

OPSs on finite (a, b) intervals
First assume that (a, b) denotes a finite interval. We list the following possibilities for finding ρ which obeys the BCs in (2.5) or in (2.7).
PI. This is the simplest case where σ 1 vanishes at both x = a and b, i.e., σ 1 (a, q) = σ 1 (b, q) = 0. Using (1.5) rewritten of the form we see that the function ρ(x, q) becomes zero at the points q −k a and q −k b for k ∈ N. However, we have to take into consideration three different situations.
(i) Let a < 0 < b. Since the points q −k a and q −k b lie outside the interval (a, b) and BCs are fulfilled at x = a and b, there could be an OPS w.r.t. a measure supported on the union of the set of points (ii) Let 0 < a < b. In this case ρ(x, q) vanishes at the points q −k a in (a, b) and q −k b out of (a, b). Then, the only possibility to have an OPS on (a, b] depends on the existence of N such that q N +1 b = a. This condition, however, implies that bq k = aq −(N −k) and that ρ vanishes at bq k for k = 0, 1, . . . , N , and, therefore, it must be rejected. The similar statement is true when a < b < 0, which can be obtained by a simple linear scaling transformation so that it does not represent an independent case.
(iii) Let a = 0 < b (or, a < b = 0). This case is much more involved. First of all, if a = 0 is a zero of σ 1 (x, q) then it is a zero of σ 2 (x, q) as well, both containing a factor x. Therefore, the r.h.s. of q-Pearson equation (1.5) can be simplified and PI(i)-(ii) are not valid anymore. In fact, in this case an OPS w.r.t. a measure supported on the set of points {bq k } k∈N 0 in (0, b] can be defined. PII. The relation in (1.6) suggest an alternative possibility to define an OPS on (a, b). Namely, if q −1 a and q −1 b are both zeros of σ 2 (x, q), by using (1.5) rewritten of the form it follows that ρ(x, q) vanishes at the points q k a and q k b for k ∈ N 0 . Then two different situations appear depending on whether a < 0 < b or 0 < a < b. In the first case, ρ(x, q) = 0 at the points q k a and q k b for k ∈ N 0 in [a, b], which is not interesting. In the second case, the q k b are in [a, b] whereas the q k a remain out of [a, b], so that we could have an OPS if there exists N such that q −N −1 a = q −1 b. However, since q −k a = q N −k b, ρ vanishes at the q −k a which are in [a, b] as well.
PIII. Let q −1 a and b be the roots of σ 2 and σ 1 , respectively. Then we see, from (2.9) and (2.10), that ρ = 0 at x = q −k b for k ∈ N and at x = q k a for k ∈ N 0 . That is, if a < 0 < b, ρ = 0 on x ∈ (a, 0) and, therefore, an OPS can not be constructed on (a, b) unless a → 0 − . In this limiting case of x ∈ (0, b], it can be possible to introduce a desired weight function supported on the set {bq k } k∈N 0 . If 0 < a < b, on the other hand, ρ vanishes for x < a and x > b. Thus there could be an OPS w.r.t. a measure supported on the finite set of points {q k b} N k=0 provided that q N +1 b = a for some finite N integer. Alternatively, we can define an equivalent OPS w.r.t. a measure supported on the equivalent finite set of points {q −k a} N k=0 provided now that q −N −1 a = b, where N is a finite integer. Note that in the limiting case of a → 0 + the set of points {q k b} k∈N 0 becomes infinity. PIV. Assume that a and q −1 b are the roots of σ 1 (x, q) and σ 2 (x, q), respectively. Then, from (2.9) and (2.10), it follows that ρ(x, q) vanishes at the points q −k a, k ∈ N and q k b, k ∈ N 0 . So, if a < 0 < b, it is not possible to find a weight function satisfying the BCs. Nevertheless, as in PIII, in the limiting case of b → 0 + an OPS w.r.t. a measure supported at the points q k a, k ∈ N 0 in [a, 0) can be constructed. If 0 < a < b, there is no possibility to introduce an OPS. Note that when a = 0 < b, an OPS also does not exist.

OPSs on infinite intervals
Assume now that (a, b) is an infinite interval. Without any loss of generality, let a be a finite number and b → ∞. In fact, the system on the infinite interval (−∞, b) is not independent which may be transformed into (a, ∞) on replacing x by −x. Obviously one BC in (2.5) reads as and there are additional cases for x = a.
PV. If x = a = 0 is root of σ 1 (x, q) then, from (2.9), ρ(x, q) vanishes at the points q −k a for k ∈ N which are interior points of (a, ∞) when a > 0. Therefore there is no OPS on (a, ∞) for a > 0. If a < 0 we can find a q-weight function in [a, 0) ∪ (0, ∞) supported on the union of the sets {q k a} k∈N 0 and {q ±k α} k∈N 0 for arbitrary α > 0 where α can be taken as unity. If a = 0, on the other hand, then a weight function in (0, ∞) can be defined at the points q ±k α for arbitrary α > 0 and k ∈ N 0 .
PVI. If x = q −1 a is a root of σ 2 (x, q), as we have already discussed, ρ is zero at q k a for k ∈ N 0 . Therefore, for a > 0 a q-weight function can exist in (q −1 a, ∞) supported on the set of points {q −k a} k∈N . An OPS does not exist if a < 0. Finally, if a = 0 it is possible to find a ρ on (0, ∞) supported at the points q ±k α for arbitrary α > 0 and k ∈ N 0 .
PVII. Finally, we consider the possibility of satisfying the BC lim a→−∞ σ 1 (a, q)ρ(a, q)a k = 0 in the limiting case as a → −∞. If this condition holds a weight function and, hence, an OPS w.r.t. a measure supported on the set of points {±q ±k α} k∈N 0 , for arbitrary α > 0, can be defined.
The aforementioned considerations are expressible as a theorem. Theorem 2.3 Let a 1 (q), b 1 (q) and a 2 (q), b 2 (q) denote the zeros of σ 1 (x, q) and σ 2 (x, q), respectively. Let ρ be a bounded and non-negative function satisfying the q-Pearson equation (1.5) as well as the BCs (2.5) or (2.7). Then ρ is a desired weight function for the polynomial solutions P n (x, q) of (1.1) only in the following cases: where the q-Jackson integral is of type (2.2).

Let
Then ρ is supported on the set of points {q k b} k∈N 0 in (0, b] and

12)
where the q-Jackson integral is of type (2.1).
3. Let 0 < a < b, where a = a 2 (q) and b = q −1 a 1 (q). Then ρ is supported on the finite set of points {q −k a} N k=0 when q −N −1 a = b and

13)
which is the finite sum of the form P n (q −k a 2 (q), q)P m (q −k a 2 (q), q)ρ(q −k a 2 (q), q).

4.
Let a = a 1 (q) < 0 and b → ∞. Then ρ is supported on the set {q k a} k∈N 0 {q ∓k α} k∈N 0 for arbitrary α > 0 and ∞ a 1 (q) where the first q-Jackson integral is of type (2.1) and the second one is of type (2.3), respectively.

5.
Let a = a 2 (q) > 0 and b → ∞. Then ρ is supported on the set of points {q −k a} k∈N 0 in [a, ∞) and ∞ a 2 (q) where the q −1 -Jackson integral is of type (2.4).
6. Let a = 0 and b → ∞. Then ρ is supported on the set of points {q ±k α} k∈N 0 for arbitrary α > 0 and where the q-Jackson integral is of type (2.3).

7.
Let a → −∞ and b → ∞. Then ρ is supported on the set of points {∓q ±k α} k∈N 0 for arbitrary α > 0 and where the bilateral q-Jackson integral is of type (2.3).

The q-weight function
In the following sections we will discuss the solutions of the q-Pearson equation (1.5) defined on the q-linear lattices enumerated in Remark 2.2. Since it is a linear difference equation on a given lattice, its solution can be uniquely determined by the equation (1.5) and the boundary conditions (for more details on the general theory of linear difference equations see e.g. [13, §1.2]). In fact, the explicit form of a q-weight function can be deduced by means of Theorem 3.1. are definite functions. Then f (x; q) admits the two q-integral representations provided that the integrals are convergent.
Proof Taking the logarithms of both sides of (3.1), multiplying by 1/(q − 1)t and then integrating from 0 to x, we have The l.h.s. is expressible as which completes the proof, on using the fact that f (q n+1 x, q) → f (0, q) as n → ∞ for 0 < q < 1. The second representation (3.3) can be proven in a similar way.
This theorem can be used to derive the q-weight functions for every σ 1 and σ 2 . However, here we take into account the quadratic coefficients leading to ∅-Jacobi/Jacobi and 0-Jacobi/Jacobi cases. The results, some of which can be found in [5], are stated by the next theorem.
(a −1 Proof We start proving the first expression in Table 1. Keeping in mind that q −1 σ 2 (0, q) = σ 1 (0, q) and that q −1 σ 2 (0, q)a 2 (q)b 2 (q) = σ 1 (0, q)a 1 (q)b 1 (q) we have from (1.5) on using (3.2). By definition (2.1) of the q-integral, we first obtain and, therefore, This implies that a 1 (q)q −1−k and b 1 (q)q −1−k for k ≥ 0 are zeros of ρ. Furthermore, a 2 (q)q −j and b 2 (q)q −j for j ≥ 0 stand for the simple poles of ρ. Note here that ρ(0, q) can be made unity, and a 1 (q), b 1 (q), a 2 (q) and b 2 (q) are non-zero constants. Therefore the solution in (3.5) is continuous everywhere except at the simple poles. To show 4., we rewrite the q-Pearson equation in the form and assume that ρ is a product of three functions ρ( respectively, then ρ = f gh is a solution of (3.6). A solution of (3.
, which can be verified by direct substitution. Here, the function with β = 0 was first defined in [5], and α = 0 is such that q α = a. The equation in (3.7) for g(x, q) can be solved in a way similar to that of (3.4). So we find that g(x, q) = g(0, q)(a −1 can be taken again as unity without loss of generality. Clearly h(x, q) is uniformly convergent in any compact subset of the complex plane that does not contain the point at the origin. Moreover, the product converges to an arbitrary constant c, which has been set to unity, as x → ∞. Thus, In order to obtain the expressions 2 and 3 in Table 1 for the weight function we use the same procedure as before, but starting from the q-Pearson equation written in the forms respectively. This completes the proof. , and different solution procedure in each case, therefore, it is not surprising that ρ has several equivalent representations displayed in Table 1. However, they all satisfy the same linear equation and, therefore, they differ only by a multiplicative constant.
For the sake of the completeness, let us obtain the analytic representations of q-weight functions satisfying (1.5) for the other cases.
Theorem 3.4 Let σ 1 and σ 2 be polynomials of at most 2nd degree in x as the form (2.18). Then a solution ρ(x, q) of q-Pearson equation (1.5) for each ∅-Jacobi/Laguerre, ∅-Jacobi/Hermite, ∅-Laguerre/Jacobi, ∅-Hermite/Jacobi, 0-Jacobi/Laguerre, 0-Bessel/Jacobi, 0-Bessel/Laguerre and 0-Laguerre/Jacobi case is expressible in the equivalent forms shown in Table 2. Table 2: Expressions for the q-weight function ρ(x, q) for the other cases Proof The proof is similar to the previous one. That is, to obtain the second formula for the ∅-Jacobi/Laguerre family we rewrite the q-Pearson equation (1.5) in the form and then apply the same procedure described in the proof of the previous theorem.
4 The orthogonality of q-polynomials: the Jacobi/Jacobi cases The rational function on the r.h.s. of the q-Pearson equation (1.5) has been examined in detail.
Since it is the ratio of two polynomials σ 1 and σ 2 of at most second degree, we deal with a definite rational function having at most two zeros and two poles. In the analysis of the unknown quantity ρ(qx, q)/ρ(x, q) on the l.h.s. of (1.5), we sketch roughly its graph by using every possible form of the definite rational function in question. In particular, we split the x-interval into subintervals according to whether ρ(qx, q)/ρ(x, q) < 1 or ρ(qx, q)/ρ(x, q) > 1, which yields valuable information about the monotonicity of ρ(x, q). Other significant properties of ρ are provided by the asymptotes, if there exist any, of ρ(qx, q)/ρ(x, q). A full analysis along these lines is sufficient for a complete characterization of the orthogonal q-polynomials. A similar characterization is made in a very recent book [20] based on the three-term recursion and the Favard theorem.
Here, in this section, we discuss only the cases in which both σ 1 and σ 2 are of second degree, i.e., the ∅-and 0-Jacobi/Jacobi cases.

The non-zero case
Let the coefficients σ 1 and σ 2 be quadratic polynomials in x such that σ 1 (0, q)σ 2 (0, q) = 0. If σ 1 is written in terms of its roots, i.e., σ 1 (x, q) provided that the discriminant denoted by ∆ q , , of the quadratic polynomial in the nominator of f (x, q) in (4.1) is non-zero. Here x = a 2 and x = b 2 denote the zeros of f , and they are constant multiples of the roots of σ 2 (x, q). We see that the lines x = q −1 a 1 and x = q −1 b 1 stand for the vertical asymptotes of f (x, q) and the point y = 1 is always its y-intercept since σ 2 (0, q) = qσ 1 (0, q). Moreover, the locations of the zeros of f are determined by the straightforward lemma.
so that the line y = Λ q denotes the horizontal asymptote of f (x, q). Then we encounter the following cases for the roots of the equation f (x, q) = 0.
, f has two real and distinct roots with opposite signs.
Case 2. If Λ q > 0 and 0 < a 1 (q) < b 1 (q), there exist three possibilities (a) if ∆ q > 0, f has two real roots with the same signs f has a pair of complex conjugate roots.
Case 3. If Λ q < 0 and a 1 (q) < 0 < b 1 (q), there exist three possibilities (a) if ∆ q > 0, f has two real roots with the same signs Case 4. If Λ q < 0 and 0 < a 1 (q) < b 1 (q), f has two real distinct roots with opposite signs.
From (4.1) it is clear that we need to consider the cases Λ q > 1 and 0 < Λ q < 1 separately. Now, our strategy consists of sketching first the graphs of f (x, q) depending on all possible relative positions of the zeros of σ 1 and σ 2 . To obtain the behaviours of q-weight functions ρ from the graphs of f (x, q) = ρ(qx, q)/ρ(x, q), we divide the real line into subintervals in which ρ is either monotonic decreasing or increasing. We take into consideration only the subintervals where ρ > 0. Note that if ρ is initially positive then we have ρ > 0 everywhere in an interval where ρ(qx, q)/ρ(x, q) > 0. Then we find suitable intervals in cooperation with Theorem 2.3.
In Figure 1A, the intervals (q −1 a 1 , a 2 ) and (b 2 , q −1 b 1 ) are rejected immediately since f is negative there. The subinterval (a 2 , b 2 ) should also be rejected in which ρ = 0 by PII. For the same reason (q −1 b 1 , ∞) and (−∞, q −1 a 1 ), by symmetry, are not suitable by PV. Therefore, an OPS fails to exist.
Let us analyse the problem presented in Figure 1B. The positivity of ρ implies that the intervals  Figure 1: , then ρ is increasing on (b 2 , x 0 ) and decreasing on (x 0 , ∞). As is shown from the figure f has a finite limit as x → +∞ so that we could have the case ρ → 0 as x → ∞. Even if ρ → 0 as x → ∞, we must show also that σ 1 (x, q)ρ(x, q)x k → 0 as x → ∞ to satisfy the BC. In fact, instead of the usual q-Pearson equation we have to consider the equation in case of an infinite interval, what we call it here the extended q-Pearson equation to determine the behaviour of the quantity σ 1 (x, q)ρ(x, q)x k as x → ∞, which has been easily derived from (1.5). It is obvious that the extended q-Pearson equation is the difference equation not for the weight function ρ(x, q) but for σ 1 (x, q)ρ(x, q)x k . In Figure 2 we draw the graph of a typical g for some 0 < q < 1, where k is large enough. From this figure we see that g < 1 for x > b 2 so that σ 1 (x, q)ρ(x, q)x k does not vanish at ∞ since it is increasing as x increases. Thus we cannot find a weight function ρ on (b 2 , ∞).
From Figure 3A, we first eliminate the intervals (a 2 , q −1 a 1 ) and (q −1 b 1 , b 2 ) because of the positivity of ρ. The interval (b 2 , ∞) coincides again with the 5th case in Theorem 2.3. However, f (x, q) < 1 on this interval so that ρ is increasing on (b 2 , ∞) which implies that ρ can not vanish as x → ∞. Thus σ 1 (x, q)ρ(x, q)x k is never zero as x → ∞ for some k ∈ N 0 . The same is true for (−∞, a 2 ) by symmetry. For the last subinterval (q −1 a 1 , q −1 b 1 ), we face the 1th case in Theorem 2.3. Since ρ(qx, q)/ρ(x, q) = 1  Figure 3: As a result, the typical shape of ρ is shown in Figure 4 assuming a positive initial value of ρ in each subinterval. Then, an OPS with such a weight function in Figure 4 Figure 4: The graph of ρ(x, q) associated with the case in Figure 3A.
It should be noted that the difference between these conditions and those of Figure 3 comes from the fact that we have considered not only the conditions on ρ but also on σ 1 ρ in Theorem 4.2. Finally, the analysis of the case in Figure 3B does not yield an OPS.  Figure 5: The case in Figure 5B is inappropriate to define an OPS. On the other hand, in Figure 5A, the intervals (q −1 a 1 , a 2 ) and (q −1 b 1 , b 2 ) are rejected by the positivity of ρ. The intervals (−∞, q −1 a 1 ) and (b 2 , ∞) coincide with 4th, by symmetry, and 5th cases in Theorem 2.3. However, f (x, q) < 1 on (−∞, 0) and on (b 2 , ∞) so that ρ is decreasing on (−∞, 0) and increasing on (b 2 , ∞) which implies that ρ can not vanish as x → −∞ and x → ∞. Thus σ 1 (x, q)ρ(x, q)x k is never zero as x → −∞ and x → ∞ for some k ∈ N 0 . The only possible interval for the case in Figure 5A is (a 2 , q −1 b 1 ) which corresponds to the 3th case of Theorem 2.3. Note that ρ(qx, q)/ρ(x, q) = 1 at a 2 < x 0 = −τ (0, q)/τ (0, q) < q −1 b 1 and ρ is increasing on (a 2 , x 0 ) and decreasing on (x 0 , q −1 b 1 ). Furthermore, ρ(qa 2 , q) = 0 and ρ → 0 respectively. It is clear that the BCs are satisfied at x = a 2 and x = q −1 b 1 . Thus we can find a suitable ρ on [a 2 , q −1 b 1 ) supported at the points q −k a 2 , k = 0, 1, ..., N where q −N −1 a 2 = q −1 b 1 . Therefore, we state the following theorem.
Theorem 4.4 Consider the case where 0 < a 2 ≤ b 2 < a 1 ≤ b 1 and q 2 Λ q > 1. Let a = b 2 (q) and b = q −1 a 1 (q) be one of the zeros of σ 2 (x, q) and σ 1 (qx, q), respectively. Then there exists a finite family of polynomials {P n } orthogonal w.r.t. the weight function (see Eq. 2 in Table 1)  Figure 7: The graph of f (x, q) in Case 3(a) with Λ q < 0. In A, the zeros are in order q −1 a 1 < 0 < q −1 b 1 < a 2 < b 2 , and in B, q −1 a 1 < 0 < a 2 < b 2 < q −1 b 1 . 0 < a 2 ≤ b 2 < a 1 ≤ b 1 and q 2 Λ q > 1. The authors did not mention this different set of the parameters for the q-Hahn polynomials in [20]. However it is given in [21, page 76].
Theorem 4.6 Consider the case where a 1 < 0 < a 2 ≤ b 2 < b 1 and q 2 Λ q < 0. Let a = b 2 (q) and b = q −1 b 1 (q) be zeros of σ 2 (x, q) and σ 1 (qx, q), respectively. Then there exists a finite family of polynomials {P n } orthogonal w.r.t. the weight function (see Eq. 2 in Table 1)  1) and (1.3) given by (4.7), being The conditions a 1 < 0 < a 2 ≤ b 2 < b 1 and q 2 Λ q < 0 lead to the orthogonality relation for the q-Hahn polynomials that is valid in a larger set of the parameters, α < 0 and β ≥ q −N −1 . This new parameter set is not mentioned in [20].  Figure 8: We could not find an OPS in case of Figure 8A. In Figure 8B, the interval (q −1 a 1 , q −1 b 1 ) coincides with the 1st case in Theorem 2.3. Notice that ρ(qx, q)/ρ(x, q) = 1 at q −1 a 1 < x 0 = −τ (0, q)/τ (0, q) < q −1 b 1 , then ρ is increasing on (q −1 a 1 , x 0 ) and decreasing on ( Then there is a suitable ρ supported on the set {q k a 1 } k∈N 0 {q k b 1 } k∈N 0 . Therefore, the following theorem holds: Theorem 4.7 Consider the case a 1 < 0 < b 1 , a 2 , b 2 ∈ C and q 2 Λ q < 0. Let a = a 1 (q) and b = b 1 (q) be zeros of σ 1 (x, q). Then there exists a sequence of polynomials {P n } for n ∈ N 0 orthogonal w.r.t. the weight function (4.4), supported on {q k a} k∈N 0 {q k b} k∈N 0 , (see (2.11) of Theorem 2.3-1) with This case is included in the case VIIa1 in Chapter 10 of [20, pages 292 and 318] (with γ 2 = γ 1 ) but it is not mentioned there. In fact, this case is similar to the big q-Jacobi polynomials studied in (Cases 1 in Figure 3A and Case 3(a) in Figure 7A). The difference is that the roots a 2 (q) and b 2 (q) are complex.
In Figure 9, the only possible interval is [b 2 , q −1 b 1 ) which corresponds to the one described in Theorem 2.3-3. A similar analysis shows that there exists a q-weight function defined on the interval [b 2 , q −1 b 1 ) supported at the points q −k b 2 for k = 0, 1, ..., N where q −N −1 b 2 = q −1 b 1 which lead to the following theorem: Theorem 4.8 Consider the case where a 2 < 0 < a 1 < b 2 < b 1 and q 2 Λ q < 0. Let a = b 2 (q) and b = q −1 b 1 (q) be zeros of σ 2 (x, q) and σ 1 (qx, q), respectively. Then there exists a finite family of polynomials {P n } orthogonal w.r.t. the weight function (4.6) in Theorem  Figure 9: The graph of f (x, q) in Case 4 with Λ q < 0. The zeros are in order a 2 < 0 < q −1 a 1 < b 2 < q −1 b 1 . and b 2 = 1. The conditions a 2 < 0 < a 1 < b 2 < b 1 and q 2 Λ q < 0 lead to another new constrains 0 < α < q −1 and β < 0 on the parameters of the q-Hahn polynomials which extend the orthogonality relation for the q-Hahn polynomials and it has been not reported in [20].
For completeness, we have also examined the cases listed below for which an OPS fails to exist.

The zero case
We make a similar analysis here with the same notations. Let the coefficients σ 1 and σ 2 be quadratic polynomials in x such that σ 1 (0, q) = σ 2 (0, q) = 0. If σ 1 is written as , x = 0 (4.9) provided that 1 + (1−q −1 )τ (0,q) . Let us point out that Λ q defined in (4.2) is also horizontal asymptote of f (x, q) in (4.9). Moreover, f intersects the y-axis at the point .
In the zero cases notice that one of the boundary of (a, b) interval could be zero. Therefore for such a case we need to know the behaviour of ρ at the origin.
In a similar fashion, we introduce the two additional cases which include all possibilities. In Figure 10A, we consider all possible intervals in which we can have a suitable q-weight function ρ. By the positivity of ρ, the interval (a 2 , q −1 a 1 ) is not suitable. The other intervals (0, a 2 ) and (q −1 a 1 , ∞) are both eliminated due to the PIV and PV, respectively. The interval (−∞, 0) is the one described in Theorem 2.3-6 by symmetry. Since ρ(qx, q)/ρ(x, q) = 1 at x 0 = −τ (0, q)/τ (0, q) < 0, ρ is increasing on (−∞, x 0 ) and decreasing on (x 0 , 0). Moreover, since 0 < y 0 < 1 ρ → 0 as x → 0 − according to Lemma 4.9. On the other hand, since ρ(qx, q)/ρ(x, q) has a finite limit as x → −∞, we may have ρ → 0 as x → −∞, but we should check that σ 1 (x, q)ρ(x, q)x k → 0 as x → −∞ by using the extended q-Pearson equation (4.3). However, the graph of g (4.3) looks like the one represented in Figure 10A together with the property that g(x, q) < 1 on (−∞, 0) for large k which leads to that σ 1 (x, q)ρ(x, q)x k is decreasing on (−∞, 0) with σ 1 (x, q)ρ(x, q)x k → 0 as x → −∞. Therefore, this case does not lead to any suitable ρ and, therefore, OPS. The same result is valid for the case in Figure  10B.

Theorem 4.11
Consider the case where a 2 < 0 < a 1 , 0 < qy 0 < 1 and q 2 Λ q < 0. Let a = 0 and b = a 1 (q) be zeros of σ 1 (x, q). Then there exists a sequence of polynomials {P n } for n ∈ N 0 orthogonal on (a, b] w.r.t. the weight function in Theorem 4.10, supported on {q k b} k∈N 0 (see (2.12) of Teorem 2.3-2).  Figure 12: The graph of f (x, q) in Case 6 with Λ q < 0 and a 2 < 0 < q −1 a 1 . In A, we have Case 6(a). In B, we have Case 6(b).
The OPS in Theorem 4.11 coincides with the case IVa4 in Chapter 10 of [20, pages 278 and 312]. An example of this family is again the little q-Jacobi polynomials with orthogonality on the set of points {..., q 2 , q, 1}. They satisfy the q-EHT with the coefficients in (4.11) where a 1 = 1 and a 2 = b −1 q −1 . These polynomials have the same orthogonality property with the same d 2 n as in (4.12) but the constrains 0 < a < q −1 and b < 0 on the parameters are different due to the conditions. This extends the orthogonality relation of the little q-Jacobi polynomials for 0 < a < q −1 and 0 < b < q −1 to a larger set of the parameters 0 < a < q −1 and b < 0. Notice that combining this with the previous Case 5(a) one can obtain the orthogonality relation of the little q-Jacobi polynomials for 0 < aq < 1, bq < 1.  Figure 13: The graph of f (x, q) in Case 6(c) with Λ q < 0. In A, the zeros are in order 0 < q −1 a 1 < a 2 . In B, the zeros are in order 0 < a 2 < q −1 a 1 .
Theorem 4.12 Consider the case where 0 < a 2 < a 1 , qy 0 < 0 and q 2 Λ q < 0. Let a = a 2 (q) and b = q −1 a 1 (q) be zeros of σ 2 (x, q) and σ 1 (qx, q), respectively. Then there exists a finite family of polynomials {P n } orthogonal w.r.t. the weight function (see Eq. 4 in Table 1) supported on the set of points {q −k a} N k=0 where q −N −1 a = b (see (2.13)

of Teorem 2.3-3).
An example of this family is the q-Kravchuk polynomials K n (x; p, N ; q) satisfying the q-EHT with the coefficients where a 1 = q −N and a 2 = 1. The conditions q 2 Λ q < 0, qy 0 < 0 and 0 < a 2 < a 1 lead to the condition p > 0 on the parameter of K n (x; p, N ; q) with orthogonality on {1, q −1 , q −2 , ..., q −N } in the sense (2.13) where In the literature, this relation is usually written as a finite sum [21,Page 98]. This case is not mentioned in [20] for 0 < q < 1. However the q-Kravchuk polynomials with this set of parameters are described in [21, page 98].
In the following independent cases we fail to define an OPS.

The orthogonality of the q-polynomials: other cases
This section includes the main analysis of the other families by taking into account the rational function on the r.h.s. of the q-Pearson equation (1.5) along the same lines with the ∅-Jacobi/Jacobi and 0-Jacobi/Jacobi cases handled in the previous section.

q-Classical ∅-Jacobi/Laguerre Polynomials
Let the coefficients σ 2 and σ 1 be quadratic and linear polynomials in x, respectively, such that σ 1 (0, q)σ 2 (0, q) = 0. If σ 1 is written in terms of its root, i.e., where τ (0, q) = 0 by hypothesis. Then the q-Pearson equation (1.5) takes the form provided that the discriminant denoted by ∆ q , of the quadratic polynomial in the nominator of f in (5.1) is non-zero. Note that here x = a 2 and x = b 2 are roots of f which are constant multiplies of the roots of σ 2 . Moreover, x = q −1 a 1 is the vertical asymptote of f and y = 1 is its y-intercept since σ 2 (0, q) = qσ 1 (0, q). On the other hand, the locations of the zeros of f are introduced by the following straightforward lemma.
Lemma 5.1 Let Λ q = τ (0,q) σ 1 (0,q) = 0. Then, we have the following cases for the roots of the equation f (x, q) = 0. Case 1. If Λ q and a 1 (q) have opposite signs, then there are two real distinct roots with opposite signs.
Case 2. If Λ q and a 1 (q) have same signs, then there exist three possibilities (a) if ∆ q > 0, f has two real roots with same signs   Figure 14: The graph of f (x, q). In A, we have Case 1 with Λ q < 0 and a 2 < 0 < q −1 a 1 < b 2 , and in B, we have Case 2(a) with Λ q < 0 and a 2 < b 2 < q −1 a 1 < 0.
In Figure 14A, we first start with positivity condition of q-weight function which allows us to exclude the intervals (−∞, a 2 ) and (q −1 a 1 , b 2 ). Moreover, due to PIII (a 2 , q −1 a 1 ) can not be used. On the other hand, the interval (b 2 , ∞) coincides with the 5th case of Theorem 2.3. Notice that since ρ(qx, q)/ρ(x, q) = 1 at x 0 = −τ (0, q)/τ (0, q) > b 2 , ρ is decreasing on (x 0 , ∞). Moreover, Since ρ(qx, q)/ρ(x, q) has an infinite limit as x → +∞, we have ρ → 0 as x → ∞. However, since it is infinite interval, we should check that σ 1 (x, q)ρ(x, q)x k → 0 as x → ∞ by using extended q-Pearson equation (4.3). The graph of the function g defined in (4.3) looks like the one for f . Then the analysis of the extended q-Pearson equation leads to σ 1 (x, q)ρ(x, q)x k → 0 as x → ∞. Therefore, there exists a suitable ρ supported on the set of points {q −k b 2 } k∈N 0 . Thus, we have the following theorem.
In Figure 14B, the only possible interval is (q −1 a 1 , ∞) which is the one identified in Theorem 2.3-4. Notice that ρ(qx, q)/ρ(x, q) = 1 at x 0 = −τ (0, q)/τ (0, q) > q −1 a 1 , then ρ is increasing on (q −1 a 1 , x 0 ) and decreasing on (x 0 , ∞) which leads to ρ → 0 as x → ∞ since ρ(qx, q)/ρ(x, q) → ∞. But we still need to show σ 1 (x, q)ρ(x, q)x k → 0 as x → ∞ by using the extended q-Pearson equation (4.3). By applying the same procedure to the extended q-Pearson equation (4.3) whose graph looks like the one for f , we get σ 1 (x, q)ρ(x, q)x k → 0 as x → ∞. Consequently, we have a suitable ρ on the interval [a 1 , ∞) supported on the set of points 1 be a zero of σ 1 (x, q) and b → ∞. Then, there exists a sequence of polynomials (P n ) n for n ∈ N 0 orthogonal w.r.t. the weight function (see the 1st expression of the ∅-Jacobi/Laguerre case in Table 2) supported on the set of points {q k a} k∈N 0 {q ±k α} k∈N 0 for arbitrary α > 0 in the sense (2.14) of Theorem 2.3-4 with  Figure 15: The graph of f (x, q) in Case 2(a). In A, we have Λ q < 0 and q −1 a 1 < 0 < a 2 < b 2 and in B, Λ q > 0 and 0 < a 2 < b 2 < q −1 a 1 .
In Figure 15A, the only possible interval is (b 2 , ∞). An analogous analysis as the one that has been done for the case in Figure 14A yields ρ → 0 as x → ∞. Moreover, since from (4.3) σ 1 (x, q)ρ(x, q)x k → 0 as x → ∞ for k ∈ N 0 , then there exists a q-weight function on [b 2 , ∞) supported at the points q −k b 2 for k ∈ N 0 . Thus we have the following result.
A typical example of this family is again the q-Meixner polynomials orthogonal on {1, q −1 , q −2 , ...}. They satisfy the q-EHT with the coefficients (5.3) where a 1 = bq, a 2 = −bc and b 2 = 1. This set of q-Meixner polynomials corresponds to the case IIa2 in Chapter 11 of [20, pages 337 and 358] and their orthogonality relation is valid in a larger set of parameters. In fact the conditions a 1 < 0 < a 2 ≤ b 2 and Λ q < 0 yield c > 0, b < 0 and 0 < −bc ≤ 1. This was not reported in [20].
Case 2. If Λ q < 0, there exist three possibilities (a) if ∆ q > 0, f has two real roots with same signs   Figure 17: The graph of f (x, q). In A, we have Case 1 with Λ q > 0 and a 2 < 0 < b 2 , and in B, Case 2(a) with Λ q < 0 and 0 < a 2 < b 2 .
The next step is sketching roughly all graphs of f by taking into account all possible relative positions of the zeros of f in question. As a result of analysis of the graphs of f , we determine a suitable ρ > 0 satisfying the q-Pearson equation (1.5) with BCs (2.5), (2.7).
In Figure 17A, let us consider the possible intervals in which we can have a suitable weight function ρ which are defined by the zeros of the polynomials σ 1 and σ 2 . First of all, notice that since ρ should be a positive weight function and f is negative in the intervals (−∞, a 2 ) and (b 2 , ∞), they are not suitable. On the other hand, the interval (a 2 , b 2 ) is also eliminated in which ρ = 0 due to PII.As a result, an OPS fails to exist.
Let us analyse the case in Figure 17B. The positivity of ρ implies that the interval (a 2 , b 2 ) should be eliminated. On the other hand, (−∞, a 2 ) is not suitable since ρ = 0 in (0, a 2 ) (this situation is similar to the one described in PVI. The interval (b 2 , ∞) coincides with the 5th case of Theorem 2.3. Notice that ρ(qx, q)/ρ(x, q) = 1 at x 0 = −τ (0, q)/τ (0, q) > b 2 , then ρ is decreasing on (x 0 , ∞). Since f has infinite limit as x → +∞, ρ → 0 as x → ∞. As a result, the typical shape of ρ is constructed in Figure 18 assuming a positive initial value of ρ in each subinterval.  Figure 18: The graph of ρ(x, q) associated with the case in Figure 17B.
However, it is not enough to assure that ρ satisfies the BC at +∞. In fact, even if ρ → 0 as x → ∞ we should check that σ 1 (x, q)ρ(x, q)x k → 0 as x → ∞ by using the extended q-Pearson equation (4.3), which is represented in Figure 19 for some 0 < q < 1, where k is large enough.
If we now provide a similar anaysis for g in (4.3), we see from Figure 19 that, g has the same property with f . Therefore, σ 1 (x, q)ρ(x, q)x k → 0 as x → ∞. That is, an OPS, to be stated in Theorem 5.8, exists on the supporting set of points {b 2 q −k } k∈N 0 .  Figure 19: The graph of g(x, q) corresponding to Figure 17B.
In the literature, this relation is usually written as an infinite sum [20, page 357].  Figure 20: The graph of f (x, q) in Case2(c) with Λ q < 0 and a 2 (q), b 2 (q) ∈ C.
Theorem 5.9 Let Λ q < 0 and a 2 , b 2 ∈ C. Let a → −∞ and b → ∞. Then, there exists a sequence of polynomials (P n ) n for n ∈ N 0 orthogonal w.r.t. the weight function (see expression 1 for the ∅-Jacobi/Hermite in Table 2) The OPS in Theorem 5.9 corresponds to the case Ia1 in Chapter 11 and case Va2 in chapter 10 of [20, pages 335, 355-356, 283 and 314-315]. An example of this family is the discrete q-Hermite II polynomials h n (x; q) satisfying the q-EHT with the coefficients
Theorem 5.11 Let 0 < a 1 < a 2 < b 1 and Λ q > 0. Let a = a 2 be the zero of σ 2 (x, q) and b = q −1 b 1 of σ 1 (qx, q). Then, there exists a finite family of polynomials (P n ) n orthogonal w.r.t. the weight function (see the 2nd expression of the ∅-Laguerre/Jacobi case in Table 2) The OPS in Theorem 5.11 coincides with the case IIIb3 in Chapter 11 of [20, pages 343 and 363]. An example of this family is the affine q-Kravchuk polynomials K Af f n (x; p, N ; q) satisfying the q-EHT with the coefficients where a 1 = pq, b 1 = q −N and a 2 = 1. The conditions Λ q > 0 and 0 < a 1 < a 2 < b 1 give the constrain 0 < p < q −1 on the parameter of K Af f n (x; p, N ; q) with orthogonality {1, q −1 , q −2 , ..., q −N } in the sense (2.13) where In the literature, this relation is usually written as a finite sum [20, page 364].
The following four cases listed below fail to define an OPS.
Notice that the point y = 1 is y-intercept of f . In a similar fashion as before, we introduce the following two cases.
Theorem 5.12 Let a 1 < 0 < b 1 . Let a = a 1 and b = b 1 be the zeros of σ 1 (x, q). Then, there exists a sequence of polynomials (P n ) n for n ∈ N 0 orthogonal w.r.t. the weight function (see the ∅-Hermite/Jacobi case in Table 2) The OPS in Theorem 5.12 coincides with the case VIIa1 in Chapter 10 of [20, pages 292 and 318-320] An example of this family is Al-Salam-Carlitz I polynomials U (a) n (x; q) satisfying the q-EHT with the coefficients where a 1 = a and b 1 = 1. The condition a 1 < 0 < b 1 gives the restriction a < 0 on the parameter of U (1 − q)(q; q) n (q, aq, a −1 q; q) ∞ .
Another example of this family is the discrete q-Hermite I polynomials which are special case of Al-Salam-Carlitz I polynomials (see [20, page 320] for further details). Finally, let us mention that the case represented in Figure 22B is inappropriate to define an OPS.
For this case the q-Pearson equation reads where −(1 − q −1 ) τ (0,q) σ 1 (0,q) a 2 (q) = 1 + (1−q −1 )τ (0,q) . Let us point out that f intersects the y-axis at the point Notice that for the zero cases one of the boundary of (a, b) interval could be zero. This requires to find the behaviour of ρ at the origin.
Proof: From (5.6) it follows that from where the result follows.
Then, there exists a sequence of polynomials (P n ) n for n ∈ N 0 orthogonal w.r.t. the weight function (see the 2nd expression of the 0-Jacobi/Laguerre case in Table 2) supported on the set of points {q −k a 2 } k∈N 0 (see (2.15) of Theorem 2.3-5).
In the literature, this relation is usually written as an infinite sum [20, page 360].
In the literature, this relation can be found as an infinite sum [20, page 314].
The Case 1 represented in Figure 26A as well as the Case 2 do not yield any OPS. From Figure  26B, it follows that the only possible interval is (0, q −1 a 1 ) which coincides with the 2nd case of Theorem 2.3. A completely similar analysis as the one done in the previous case allows us to conclude that in (0, a 1 ] an OPS can be defined which is orthogonal w.r.t. a suitable ρ supported on the set of points {q k a 1 } k∈N 0 . Then, we have the following Theorem. Theorem 5.18 Let a 1 > 0 and 0 < qy 0 < 1. Let a = 0 and b = a 1 be the zeros of σ 1 (x, q). Then, there exists a sequence of polynomials (P n ) n for n ∈ N 0 orthogonal w.r.t. the weight function (see the 0-Laguerre/Jacobi case in Table 2) ρ(x, q) = |x| α (b −1 qx; q) ∞ , q α = − q −2 1 2 σ 2 (0, q) ]. An example of this family is the little q-Laguerre (Wall) polynomials P n (x; α|q) satisfying the q-EHT with the coefficients σ 1 (x, q) = q −2 x(a 1 − x), σ 2 (x, q) = ax, τ (x, q) = − 1 (1 − q)q x + 1 − aq (1 − q)q and λ n (q) = q −n 1 − q [n] q where a 1 = 1. The conditions 0 < qy 0 < 1 and a 1 > 0 give the restriction 0 < a < q −1 on the parameter of P n (x; α|q) with orthogonality on {..., q 2 , q, 1} in the sense (2.12) where d 2 n = a n q n 2 (q; q) ∞ (aq; q) ∞ (q, aq; q) n .
In the literature, this relation can be found as an infinite sum [20, page 312].

Concluding remarks
The q-polynomials of the Hahn class have been revisited by use of a direct and very simple geometrical approach based on the qualitative analysis of solutions of the q-Pearson (1.5) and the extended q-Pearson (4.3) equations. By this way, it is shown that it is possible to introduce in a unified manner all orthogonal polynomial solutions of the q-EHT, which are orthogonal w.r.t. a measure supported on some set of points in certain intervals. In this review article we are able to extend the well known orthogonality relations for the big q-Jacobi polynomials (see Theorem 4.5 and Theorem 4.7), q-Hahn polynomials (see Theorem 4.6 and Theorem 4.8), and for the q-Meixner polynomials (see Theorem 5.4) to a larger set of their parameters.