Recurrence relations of the hypergeometric type functions on the quadratic-type lattices

The central idea of this article is to study the difference analogues of solutions of the second order linear difference equation of hypergeometric type defined on the quadratic-type lattices. We introduce various difference-recurrence relations for such solutions. Some applications to polynomials on the quadratic-type lattices are also considered.


Introduction
The q-special functions, particularly, q-polynomials has an increasing interest in the last years. In particular, q-polynomials on the q-linear lattices have been considered firstly, after the study by Hahn in 1949 (see e.g. [2,6] and references therein). In 1983 these functions were studied by Nikiforov and Uvarov who started from the second order linear difference equation of hypergeometric type that such kind of functions satisfy and then this theory is developed by several authors (see e.g. [4,8,12,17,18,19,20]). Moreover, qpolynomials on the q-quadratic lattices are of particular interest in the recent studies (see e.g. [9,10,11,17,18,19,20]) since they are the most general orthogonal families from which all the other hypergeometric orthogonal polynomials can be obtained. Such polynomials are the solutions of the second order difference equation of hypergeometric type defined on the q-quadratic lattices.
The main aim in this work is to introduce a constructive approach generating reccurence relations for the difference analogues of hypergeometric functions on the q-quadratic lattices x(s) = c 1 q s + c 2 q −s + c 3 which cover also the hypergeometric functions on the quadratic lattices x(s) = c 1 s 2 + c 2 s + c 3 as a limit case when q −→ 1 (see [1]).
The reccurence relations are quite useful for the evaluation of the functions rather than the direct method. Moreover, they are important not only theoretically but also numerically (see the references [7,14,15]).
In fact, in 1989 reccurence relations for the difference analogues of hypergeometric functions on the q-quadratic lattices are considered by Suslov in [19,20]. Here, we introduce a constructive approach for the reccurence relations for such functions which have more general form.
This paper is motivated after a very recent papers published by R.Álvarez-Nodarse and collaborators [3,5,6]. In fact, in [3], the author considered the continuous case and he obtained some reccurence relations for the Jacobi, Laguerre and Hermite polynomials and he applied the theory to the quantum mechanics. In [5] the authors studied difference analogues of hypergeometric functions on the linear lattices x(s) = s and they applied the theory to the Hahn, Meixner, Charlier and Kravchuk polynomials. and in [6], the authors studied the difference analogues of hypergeometric functions on the linear-type lattices where they applied to the q-polynomials on q-linear lattices x(s) = c 1 q s +c 2 , like big q-Jacobi, Alternative q-Charlier polynomials. Here we go further and consider the reccurence relations for the functions on the quadratic-type lattices and apply the theory to the q-Racah and dual Hahn polynomials.
Notice that since the lattice considered in this paper is not a linear-type, the general results of [6] may not be applied in general, therefore an appropriate method should be developed. In fact, the main aim of the present paper is to adapt the method presented in [6] for more general lattice. Therefore, this work is an generalization of the so-called papers [3,5,6] which contains more general form where the other cases can be obtained. Therefore, this article is important from two point of view. Firstly, it completes the study on the reccurrence relation using the similar idea of the papers [3,5,6] and secondly, it includes the most general reccurrence relations from where the others can be obtained as particular cases.
The structure of the paper is as follows: In section 2 the preliminary results are introduced. In section 3 and 4 the general theorems for reccurence relations are given. The last section concludes the paper with some examples.

Preliminaries
Here we include some useful information (see e.g. [1,18]) on the q-hypergeometric functions that we need in the rest of the paper.
The hypergeometric functions on the non-uniform lattices satisfy the following second order difference equation of hypergeometric type on the non-uniform lattices where Here, ∆y(s) = y(s + 1) − y(s), ∇y(s) = y(s) − y(s − 1) are the forward and backward difference operators, respectively, where ∆y(s) = ∇y(s + 1), 2 and the coefficients σ(x(s)) and τ (x(s)) are polynomials in x(s) of degree at most 2 and 1, respectively, and λ is a constant.
In this paper, we study on the quadratic-type lattices: the so-called quadratic lattice and q-quadratic lattice, with c 1 = 0, c 1 (q) = 0.
Remark 1. The quadratic-type lattices have the following properties: where Here [k] q is the symmetric q-number defined by The difference derivatives of a solution y(s) of (1), are defined by and satisfy a difference equation of the same type of (1). Furthermore, the solutions of the difference equation (1) have the following property.
Theorem 1. [18,19] The difference equation (1) has a particular solution of the form if the condition σ(s)ρ ν (s)∇x ν+1 (s) is satisfied and of the form if the condition is satisfied. Here, C is a contour in the complex plane, C ν is a constant.
Notice here that Here, ν ∈ C is the root of the equation where [x] q and α k are defined by (9) and (8), respectively. In the following we will use the function σ ν (s) defined as, Notice that by (17) and (19).
of the lattices (4) and (5) are defined as follows: • For the quadratic lattice of the form (4) • For the q-quadratic lattice of the form (5) where C = log(c 2 (q)/c 1 (q)) log q , η = c 2 (q) c 1 (q) and classical q-Gamma function, Γ q , is related to the infinite q-product [13] by formula Here, the infinite q-product [13] is defined by (a; q) ∞ = ∞
Proposition 2. [6,19] Let ν be complex number and m, k be positive integers with m ≥ k.
For the q-quadratic lattice of the form (5), we have The proof is straightforward by using (22) and (23) and we omit it here. The generalization of above expressions can be written by the following lemma. Lemma 3. Let µ i and ν i , i = 1, 2, 3 be complex numbers such that where ν 0 is the ν i , i = 1, 2, 3, with the largest real part, µ 0 is the µ i , i = 1, 2, 3, with the largest real part and the differences ν i − ν j and µ i − µ j i, j = 1, 2, 3 are integers. Then the ratio of the generalized power can be calculated by the following formulas: Proof. Here we only sketch the proof for more general case, i.e., the 3rd case. The others can be done in an analogous way. One can write the ratio of the generalized power in the 3rd case by Then, from the hypothesis and the formulas (24), (25) of the Proposition 2 the results in the lemma follow.

Reccurence relation on the quadratic-type lattices
Here, we are going to obtain the general reccurence relation on the quadratic-type lattices defined by (4) and (5). In order to get the relation we generalize the idea used for the lineartype lattices in the recent papers [3,5,6]. Firstly, we define the following functions, We remark that the functions y ν and Φ ν,µ are related as Lemma 4. Let the function Φ ν,µ (z) be defined by (28) and (29). Then, the following relation holds where [s] q is the symmetric q-number defined by (9).
Proof. The proof is similar to the one in [6].
We next prove the following lemma for the quadratic-type lattices which is the generalization of the linear-type lattices considered in Lemma 3.2. of [6, page 4].
Proof. We consider the proof for the function Φ ν i ,µ i defined by (28). For (29), the proof is similar. By substituting the function Φ ν i ,µ i defined by (28) in the sum, we have by the Pearson equation (15). Thus, we have where the ratio of the generalized power can be computed by using the Lemma 3.
We need to show that there exists a polynomial Q(s) whose basis is {1, z −1 , z, z −2 , z 2 , z −3 , z 3 , ...}, where z = q s , such that ρ ν * (s) 8 If such polynomial Q(s) exists, then by taking the sum over s = a to b − 1, and using the boundary condition (33), we get the relation (32). First of all, we will show in the following that Π(s) is a polynomial whose basis is {1, z −1 , z, z −2 , z 2 , z −3 , z 3 , ...}, where z = q s . By substituting the q-quadratic lattice x(s) = c 1 (q)q s + c 2 (q)q −s + c 3 (q) in each factors of Π(s) in (36) one can rewrite it as a polynomial in z = q s and 1/z = q −s , which is a special class of Laurent polynomials, [16] Λ 2n = {R ∈ Λ −n,n | the coefficent of z n is nonzero} The so-called Laurent polynomial is a function of a real variable x of the following form Λ m,n denotes the linear space of Laurent polynomials, i.e., which is a subspace with dim(Λ m,n ) = n−m+1 of the linear space of all Laurent polynomials. Notice that if P k denotes the linear space of polynomials of degree at most k, then P k = Λ 0,k . A Laurent polynomial is called L-degree m if it belongs to class Λ m , m ∈ N 0 , [16]. Now if we return to prove the existence of the polynomial Q(s), we rewrite the right hand side of (37) σ(s + 1)ρ ν * (s + 1) By using the Pearson equation (15) and the formulas (26) and (25) of the Proposition 2, respectively, one gets ρ ν * (s) Therefore, where, Φ ν * (s) = σ(s) + τ ν * (s)∇x ν * +1 (s).
Recall that Π(s) is a Laurent polynomial on the basis {1, z −1 , z, z −2 , z 2 , z −3 , z 3 , ...}, where z = q s . Then, note that σ(s) is a polynomial of degree at most two in x(s) and also a Laurent polynomial of L-degree at most four, whose basis is {1, z −1 , z, z −2 , z 2 }, where z = q s . Moreover, τ ν * (s) is polynomial of degree one in x ν * (s) and also a Laurent polynomial of L-degree two, whose basis is {1, z −1 , z}, where z = q s . In addition, x k (s) is a Laurent polynomial of L-degree two, whose basis is {1, z −1 , z}, where z = q s . Therefore, by substituting the q-quadratic lattice (5) and taking into account the property (10), one can see that Q(s) is also a Laurent polynomial whose L-degree is at least six less than the L-degree of Π(s).
Note that two Laurent polynomials are equal if their coefficients are equal likewise the ordinary polynomials. Since both parts of (38) are Laurent polynomials, one can use the equality of the coefficents of the Laurent polynomials in order to find A i (z). This completes the proof.
In the limit case as q → 1, one can also get the results of Lemma 5 for the quadratic lattice x(s) = c 1 s 2 + c 2 s + c 3 .

Some representative examples
Example 6. The following relation holds where the coefficients A 1 (z), A 2 (z) and A 3 (z) are the functions defined by where α ν , β ν and γ ν are defined by (8) and are the Taylor polynomial expansion of the functions σ ν (s) and τ ν (s) defined by (19) and (16), respectively.
Example 7. For the function Φ ν i ,µ i , following relation holds where the coefficients A 1 (z), A 2 (z) and A 3 (z) are the functions defined by where γ ν is defined by (8).
Example 8. The following relation holds where the coefficients A 1 (z), A 2 (z) and A 3 (z) are the functions defined by where γ ν is defined by (8).
Example 9. The following relation holds where the coefficients A 1 (z), A 2 (z) and A 3 (z) are the functions defined by and γ ν is defined by (8).

Example 10. The following relation holds
where the coefficients A 1 (z), A 2 (z) and A 3 (z) are the functions defined by where γ ν is defined by (8).
Example 11. The following relation holds where the coefficients A 1 (z), A 2 (z) and A 3 (z) are the functions defined by where γ ν is defined by (8).

Reccurence relations including the solutions y ν and their difference derivatives
Here we include the reccurence relations related with solutions y ν and their difference derivatives which are defined in [19,20] by where The following theorem has been proved for the lineer-type lattices in [3,5,6] and is also valid for the quadratic-type lattices.
Theorem 12. In the same conditions as in the Lemma 5 any three functions y where B i (s), i = 1, 2, 3 are some functions on s.
Proof. By the Lemma 5 there exist the functions A i (z), i = 1, 2, 3, such that the following linear relation holds, Therefore, by the definition of the difference derivative (46), we have which can be rewritten as the following by dividing the equality with ρ k * (s), where k * = min{k 1 , k 2 , k 3 } and then using the expression (35), which completes the proof.
Corollary 13. The following three-term recurence relation holds provided that the conditions in the Lemma 5 exist. Here, the coefficients A i (s), i = 1, 2, 3 are the functions.
Theorem 14. The following ∆-ladder-type reccurence relation holds provided that the conditions in the Lemma 5 are satisfied. Here, B i (s), i = 1, 2, 3 are functions.
We remark here that the cases m = ∓1 in (49) lead to the raising and lowering operators respectively, where B i (s) and B i (s), i = 1, 2, 3, are the functions.
Theorem 15. The following ∇-ladder-type reccurence relation holds provided that the conditions in the Lemma 5 are satisfied. Here, B i (s), i = 1, 2, 3 are functions.
Proof. By applying the ∇/∇x(s) operator to y ν (s) defined by (30), we have It is sufficient to substitute the above relation into (52), and then using the relation (31) where µ = ν.
By the formula defined in (46), the examples 6, 7, 8 and 9 lead to the following relations Notice that the last two relations are the so-called raising and lowering operators, respectively.

Application to polynomials on the quadratic-type lattices
Here we include the application of the method to the q-Racah and dual Hahn polynomials which are defined by (13) with ν = n. The q-Racah and dual Hahn polynomials are the general polynomials which are defined on the q-quadratic lattices of the form x(s) = q −s + δγq s+1 and the quadratic lattices of the form x(s) = s(s + 1), respectively. The other polynomials can be obtained by the limit cases.
One can find a detailed study on these polynomials in [1,17,18]. Since the q-Racah and dual Hahn polynomials are defined by (13) where ν = n and the contour C is closed, then the condition (14) is satisfied, therefore, the Lemma 5 and hence the Theorem 12 hold for such polynomials.
Notice that the above differentiation formulas valid for the q-polynomials on the q-quadratic lattices. In order to get such formulas for the polynomials on the quadratic lattices, one can consider the limit case when q → 1.
By considering the TTRR, we have following system of equations