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A basic class of symmetric orthogonal polynomials of a discrete variable

By using a generalization of Sturm-Liouville problems in discrete spaces, a basic class of symmetric orthogonal polynomials of a discrete variable with four free parameters, which generalizes all classical discrete symmetric orthogonal polynomials, is introduced. The standard properties of these polynomials, such as a second order difference equation, an explicit form for the polynomials, a three term recurrence relation and an orthogonality relation are presented. It is shown that two hypergeometric orthogonal sequences with 20 different weight functions can be extracted from this class. Moreover, moments corresponding to these weight functions can be explicitly computed. Finally, a particular example containing all classical discrete symmetric orthogonal polynomials is studied in detail.

The difference equation (2), which is obtained by approximating the differential equation (1) on a non-uniform lattice, is of much importance [32] as its particular solutions have been applied in quantum mechanics, theory of group representations and especially computational mathematics, where one can point to the Clebsch-Gordan and Racah coefficients with wide applications in atomic and nuclear spectroscopy. There exist different approaches for the analysis of orthogonal polynomials of a discrete variable running from the classical references [8,17] to the recent monograph [16], which is a basic reference on orthogonal polynomials.
Also there exists a number of numerical and symbolic methods for solving hypergeometric equations of type (1) or (2), which are of interest in applications, particularly for cases containing symmetric solutions, such as resolution of the Gibbs phenomenon [10,11], Fourier-Kravchuk transform used in Optics [6], approximation of harmonic oscillator wave functions [4], tissue segmentation of human brain MRI through preprocessing [1], reconstructions for electromagnetic waves in the presence of a metal nanoparticle [27], efficient determination of the critical parameters and the statistical quantities for Klein-Gordon and sine-Gordon equations with a singular potential [7], image representation [33,14] and quantitative theory for the lateral momentum distribution after strong-field ionization [9].
The main aim of this paper is to introduce a basic class of symmetric orthogonal polynomials of a discrete variable with four free parameters, which is the polynomial solution of a symmetric generalization of equation (2) on the uniform lattice x(s) = s. Computational aspects of these new polynomials are described in detail giving their explicit representation as well as the three-term recurrence relation they satisfy. A full classification of weight functions and orthogonality supports is given together with computing the moments of the aforesaid weights. From this class all classical symmetric orthogonal polynomials of a discrete variable can be recovered (section 6), and its limit relation with the continuous type of generalized classical symmetric orthogonal polynomials is given (see remark 2).
A regular Sturm-Liouville problem of continuous type is a boundary value problem in the form (3) d dx k(x) dy n (x) dx + (λ n ̺(x) − q(x)) y n (x) = 0 (k(x) > 0, ̺(x) > 0), which is defined on an open interval (a, b), and has the boundary conditions (4) α 1 y(a) + β 1 y ′ (a) = 0, α 2 y(b) + β 2 y ′ (b) = 0, where α 1 , α 2 and β 1 , β 2 , are given constants and k(x), k ′ (x), q(x), and ̺(x) in (3) are to be assumed continuous for x ∈ [a, b]. If one of the boundary points a and b is singular (i.e. k(a) = 0 or k(b) = 0), the problem is transformed to a singular Sturm-Liouville problem. Let y n and y m two eigenfunctions of the operator D(k(x)D) − q(x)I, where D is the standard derivative operator. According to Sturm-Liouville theory [29], they are orthogonal with respect to the weight function ̺(x) under the given conditions (4) and satisfy the orthogonality relation Many of special functions are orthogonal solutions of a regular or singular Sturm-Liouville problem having the symmetry property (φ n (−x) = (−1) n φ n (x)) so that have found valuable applications in physics and engineering, as already mentioned.
Theorem 1. [20] Let φ n (−x) = (−1) n φ n (x) be a sequence of symmetric functions satisfying the equation and λ n is a sequence of constants. If A(x), (C(x) > 0), D(x) and E(x) are even functions and B(x) is odd then The weight function defined in (7) must be positive and even on [−ν, ν] and the function is an even function so A(−ν)K(−ν) = 0 automatically.
Orthogonal functions of a discrete variable can similarly be solutions of a regular or singular Sturm-Liouville problem of discrete type in the form [15] (8) ∆ (K * (x)∇y n (x)) + (λ n w(x) − q * (x)) y n (x) = 0 and (8) satisfies a set of discrete boundary conditions like (4). This means that if y n (x) and y m (x) are two eigenfunctions of difference equation (8), they are orthogonal with respect to the weight function ̺ * (x) on a discrete set [28].
Recently in [23] we have presented the following theorem by which one can generalize usual Sturm-Liouville problems with symmetric solutions in discrete spaces. As a very important consequence of this theorem, we can introduce a basic class of symmetric orthogonal polynomials of a discrete variable with four free parameters.

Theorem 2. [23]
Let φ n (−x) = (−1) n φ n (x) be a sequence of symmetric functions that satisfying the difference equation where, as usual, ∆∇ = ∆ − ∇. If A(x) is a free real function and (C(x) > 0), D(x) and E(x) are even functions then and W (x) is the solution of the Pearson difference equation which is equivalent to Moreover, the weight function defined in (10) must be even over one of the four following symmetric counter sets i) S 1 = {−a − n, −a − n + 1, . . . , −a − 1, −a, a, a + 1, . . . , a + n}, a ∈ R, ii) S 2 = S 1 ∪ {0} (as any odd function is equal to zero at

2.
A basic class of symmetric orthogonal polynomials of a discrete variable using Theorem 2 As a special case of equation (5), the following differential equation is defined in [19]: (12) x 2 (p * x 2 +q * )Φ ′′ n (x)+x(r * x 2 +s * )Φ ′ n (x)−(n(r * +(n−1)p * )x 2 +σ n s * )Φ n (x) = 0. Here one of the basic solutions is the symmetric class of orthogonal polynomials denoted by By referring to theorem 1, we observe in (12) that A(x) = x 2 (p * x 2 + q * ) is a polynomial of degree at most four, C(x) = x 2 is a symmetric quadratic polynomial, D(x) = 0, and E(x) = s * . Since discrete orthogonal polynomials have a direct relationship with continuous polynomials, motivated by (12), we suppose in the main equation (9) that though there may exist some other possible cases. By noting the assumptions (13), we are now interested in obtaining a symmetric orthogonal polynomial solution. Hence, let (14) φ n (x) = x n + δ n x n−2 + · · · , satisfy a three term recurrence relation as From (9), (13) and (14), equating the coefficient in x n+2 gives provided that c 2 = 0 and also |a 3 | + |a 4 | = 0. By using the eigenvalue λ n in (16) and equating the coefficient in x n we obtain Also from (14) and (15) we have Therefore, in order that φ n (x) is a solution of (9), the following extra conditions must be considered for the initial values of n: As a conclusion, we get −a 4 (n − 1)((n − 5)n + 9))} / {24a 4 (3 − 2n) + 48a 3 } .
For simplicity, if we set a 4 = 2a, a 3 = a + 2b, a 2 = b + 2c, a 1 = c + 2d, then (18) changes to Hence the following difference equation appears Since the polynomial solution of equation (20) is symmetric, we use the notation for mathematical formulae and S n (a, b, c, d; x) in the sequel. This means that from now we deal with just one characteristic vector V = (a, b, c, d) for any given subcase.
If the polynomial sequence {S n (a, b, c, d; x)} n≥0 satisfies a three term recurrence relation of type (15) then by referring to (17) and (19) we obtain Proposition 1. The coefficient γ n in the three term recurrence relation (15) is a rational expression in n where the numerator is a polynomial of degree 4 and the denominator is a polynomial of degree 2, given by For n = 2m and n = 2m + 1, γ n in (21) becomes Remark 1. Once we have explicitly determined γ n in the recurrence relation (15), a discussion about the situation of this coefficient is extremely important. For instance, analyzing the location of the zeros of orthogonal polynomials would give rise to a positive definite case when γ n > 0 (∀n ∈ N), the quasi-definite case when γ n = 0, and weak orthogonality case when γ n = 0 for some values of n. However, this discussion completely depends on the four parameters a, b, c and d, because γ n is in general a rational expression in n. In section 4 we provide factorized representations of γ n in two hypergeometric sequences, which would allow us to easily determine the orthogonality situation based on the parameters.
Proof. By defining the symmetric basis the following straightforward properties are derived The expansion of S n (a, b, c, d; x) in terms of the above basis yields then by using the Navima algorithm [2,30,31] we can reach a solvable recurrence relation for the connection coefficients c j (n). So, by using the latter properties (24)- (26) and substituting (27) in (20) we obtain But since ϑ n (x) is linearly independent, the coefficients c j (n) would satisfy the relation which is explicitly solvable with the initial conditions c m (n) = 0 for m > [n/2], and c [n/2] (n) = 1, providing (22). (22), the corresponding polynomial satisfies a difference equation of type (20) that formally tends to the differential equation (12) as h → 0. Hence, the following limit relation can be directly deduced:

Remark 2. By substituting the characteristic vector
As the recurrence relation (15) is explicitly known, the complete form of the orthogonality relation is is the original weight function and W (x) satisfies the difference equation By noting that A(x) = (2x + 1)(ax 3 + bx 2 + cx + d) for |a| + |b| = 0, two cases can generally happen for the parameter a in (29), i.e. when a = 0 and b arbitrary or a = 0 and b = 0. In the first case, since any arbitrary polynomial of degree 3 has at least one real root, say x = p ∈ R, the aforementioned A(x) can be decomposed in three different forms, i.e.
Similarly, in the second case when a = 0 and b = 0, A(x) can be decomposed as For the two sub-cases A(x) = (2x+ 1) (x− p) a x 2 + u x + v in (30) and A(x) = (2x + 1) b x 2 + c x + d in (31), the difference equations corresponding to (29) respectively take the forms (32) W (x + 1) Since the denominators of the two fractions (32) and (33) are not decomposable in R, we shall deal with them in a separate work [under preparation]. However, it is important to note that the cases analyzed in this paper allow us to recover all classical symmetric orthogonal polynomials of a discrete variable (section 6). Hence, let us consider the rest of cases. For the second sub-case of (30), without loss of generality take a = 1 and then A( where p, q, r ∈ R, which indeed covers the third sub-case too. For the second sub-case of (31) we can similarly consider This means that there exist only two orthogonal sequences of S n (a, b, c, d; x) when A(x) is decomposable. Moreover, there exist 20 different weight functions on 32 supports for these two sequences, as the weight functions are not in general unique leading to an indeterminate moment problem. In summary, when the polynomial A(x) is of degree four, a three parametric family (with 14 subcases) appears and when A(x) is of degree three, a two parametric family (with 6 subcases) would appear. Finally, when A(x) is of degree 2, the well known classical symmetric discrete families appear.
However, this case is clear as the sum of any odd summand on a symmetric counter set is equal to zero. ii) If simultaneously A(−θ) = 0 and W (θ) = 0, then (36) is again equal to zero. This condition is in fact the key point for finding the orthogonality supports.

Two hypergeometric orthogonal sequences of S n (a, b, c, d; x)
In this section, we introduce two hypergeometric sequences of symmetric orthogonal polynomials, which are particular cases of S n (a, b, c, d; x) corresponding to two aforementioned cases a = 0 and a = 0, respectively and then obtain all possible weight functions together with orthogonality supports for these two sequences.
The latter representations would allow us to analyze the sign of γ n in terms of the values of p, q and r as explained in Remark 1. By noting the relations (38) and (39), the orthogonality relation of the first sequence is in which S n (x) = S n 1 −(p + q + r) pq + pr + qr −pqr x , and, finally denotes the original weight function corresponding to the hypergeometric polynomialdenotes the original weight function corresponding to the hypergeometric polynomial (37). As a consequence, we obtain Proposition 2. The function W (x) satisfies a particular case of the difference equation (29) (40) giving rise to the 16 symmetric solutions of (40) listed below (44) W 2,3 (x; p, q, r) = W 2,1 (x; r, q, p) (49) W 4,2 (x; p, q, r) = W 4,1 (x; q, p, r) (50) W 4,3 (x; p, q, r) = W 4,1 (x; r, q, p) (52) W 5,2 (x; p, q, r) = W 5,1 (x; p, r, q) (54) W 6 (x; p, q, r) As the original weight functions corresponding to all above 16 cases are as the two following identities are remarkable in this direction For example, for the last given case the original weight function becomes By noting section 3, the following table now shows the orthogonality supports of each weight function given in (41) Since some weight functions are symmetric with respect to the parameters p, q and r, e.g. W 4,2 (x; p, q, r) = W 4,1 (x; q, p, r) and W 4,3 (x; p, q, r) = W 4,1 (x; r, q, p), their orthogonality supports and parameter restrictions can be directly derived via the rows of table 1 by just interchanging the parameters. Also note that W 7 (x; p, q, r) and W 8 (x; p, q, r) have no valid orthogonality support. Therefore, there are totally 24 eligible orthogonality supports for the first sequence.
Remark 4. Let the weight functions corresponding to the first sequence be indicated as ̺ i (x) = (1/4 − x 2 )W i (x). Then they satisfy the difference equation which is equivalent to Thus, the hypergeometric polynomials (37) constitute a ∆-semiclassical sequence of class one [18], for which we have explicitly given their orthogonality condition, three term recurrence relation, hypergeometric representation and in the sequel in section 5 we will compute the factorial moments with respect to the basis ϑ n (x) defined in (23). Since p ∈ Z − , 1 − q ± p ∈ Z − and 1 − r ± p ∈ Z − , referring to Table 1 where ⌊x⌋ denotes the floor function, and for p, q ∈ R, is replaced in the difference equation (20), then has a basis solution, which can be written in terms of hypergeometric series The above polynomial is a limit case of the polynomial (37) when r → ∞. Moreover, it satisfies a recurrence relation of type (15) with which implies γ 2n = −n(−1 + n + p + q), and γ 2n+1 = −(n + p)(n + q). Hence, the orthogonality relation corresponding to the second sequence takes the form is the original weight function. Thus, we obtain Proposition 3. The function W (x) satisfies a particular case of the difference equation (29) as giving rise to 8 symmetric solutions for equation (59) respectively as follows: (60) W 9 (x; p, q) . Table 2 shows the orthogonality supports of each above-mentioned weights and their parameter restrictions.
As the main orthogonality relation (28) shows, an important part that one has to compute in norm square value is For example, for the first given weight function W 1 (x; p, q, r) on e.g. [−p, p] we have Remark 6. Let the weight functions corresponding to the second sequence be indicated as ̺ j (x) = (1/4 − x 2 )W j (x). Then they satisfy the difference equation which is equivalent to Since q ∈ Z − and p ± q ∈ Z − , referring to Table 2 shows that there are two possible weight functions for the selected parameters which are orthogonal with respect to the sequence § n (0, 1, −4, −140; x) on the support {−10, −9, . . . , 9, 10}, i.e.

Moments of the two introduced sequences
To compute the moments of a continuous distribution, different bases should be considered. For example, in the normal distribution the canonical basis {x j } j≥0 is used to get while for the Jacobi weight function (1 − x) α (1 + x) β as the shifted beta distribution on [−1, 1], using one of the two bases {(1 − x) j } j or {(1 + x) j } j is appropriate for this purpose. This matter similarly holds for the moments of discrete orthogonal polynomials. For instance, in the negative hypergeometric distribution corresponding to Hahn polynomials, it is more convenient to use the Pochhammer basis {(−x) n } n≥0 , instead of the canonical basis, to get Following this approach, for the weight functions ̺ i (x) appeared in the two introduced hypergeometric sequences, we shall compute the moments of the form where the basis ϑ n (x) is defined in (23).
Since ϑ 2n+1 (x) are odd polynomials, clearly all odd moments with respect to this basis are zero. Moreover, from (37) and (58) and also using the orthogonality property of the polynomials it can be proved by induction that the even moments corresponding to the first and second sequences respectively satisfy the following recurrence relations and (̺ j ) 2n = −(n + p − 1)(n + q − 1)(̺ j ) 2n−2 .
Therefore, if the aforesaid weight functions are normalized with the first moment equal to one, then we eventually obtain for the weight functions of the second sequence.

6.
A particular example of S n (a, b, c, d; x) generating all classical symmetric orthogonal polynomials of a discrete variable In this section, we present a particular interesting example of S n (a, b, c, d; x) that generates all classical symmetric orthogonal polynomials of a discrete variable studied and analyzed in [3] and is different from the two introduced hypergeometric sequences.