Classical symmetric orthogonal polynomials of a discrete variable

Abstract

Classification of polynomial solutions of second-order difference equation of hypergeometric type with real coefficients, orthogonal with respect to a positive symmetric weight function is presented.

A basic class of symmetric orthogonal polynomials of a discrete variable

“…If a = −2(b + 2c + 4d) in the main difference equation (20), after simplification of a common factor we get  x 2 (b + 2c + 4d) + x(c + 2d) + d  ∆∇φ n (x) − 2x(c + 2d) φ n (x) + n(−(b(n − 1) + 2(n − 2)(c + 2d)))φ n (x) = 0, which is the same equation as analyzed in [30], (âx 2 +bx +ĉ)∆∇y n (x) − 2bx y(x) + n(â(1 − n) + 2b)y(x) = 0, for d =ĉ, c =b − 2ĉ, b =â − 2b, a = −2â.…”

“…,b/2 − 1,b/2} when b ∈ N. Case 3. Ifâ = 1,b = −δ 1 − δ 2 andĉ = δ 1 δ 2 in (67), then σ (x) has two different real roots δ 1 and δ 2 and depending on the values δ 1 and δ 2 , the symmetric Hahn-Eberlein polynomials [30,2] are derived as S n  −2 1 + 2(δ 1 + δ 2 ) −δ 2 − δ 1 (1 + 2δ 2 ) δ 1 δ 2 x  = (δ 1 + δ 2 ) n (2δ 2 ) n (n + 2(δ 1 + δ 2 ) − 1) n 3 F 2  −n, n + 2(δ 1 + δ 2 ) − 1, δ 2 − x δ 1 + δ 2 , 2δ 2…”

“…For the two sub-cases (32), the difference equations corresponding to (30) respectively take the forms…”

“…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 [30] and is different from the two introduced hypergeometric sequences.…”

A characterization of Dunkl-classical d-symmetric d-orthogonal polynomials and its applications

“…The literature on these topics is extremely vast. We quote, for instance, [7][8][9]2,1,6,[10][11][12].…”