Symmetric orthogonal polynomials and the associated orthogonal 𝐿-polynomials

Abstract

Abstract. We show how symmetric orthogonal polynomials can be linked to polynomials associated with certain orthogonal L-polynomials. We provide some examples to illustrate the results obtained. Finally as an application, we derive information regarding the orthogonal polynomials associated with the weight function (1 + kx2)(l -x2)~1/2, k>0.

Lamé differential equations and electrostatics

“…Recently Sri Ranga [12] provided a ingenious way of obtaining sequences of orthogonal Laurent polynomials by a change of variables in a sequence {p n (t)} of orthogonal polynomials with respect to an even weight function on a symmetric interval (−d, d). Consider the classical Gegenbauer (ultraspherical) polynomials C In fact, Sri Ranga [12] proved that the zeros of B λ n (x) are 'symmetric' with respect to √ ab in the following sense: if x k is a zero, then ab/x k is also a zero.…”

“…Consider the classical Gegenbauer (ultraspherical) polynomials C In fact, Sri Ranga [12] proved that the zeros of B λ n (x) are 'symmetric' with respect to √ ab in the following sense: if x k is a zero, then ab/x k is also a zero.…”

Orthogonal Laurent polynomials corresponding to certain strong Stieltjes distributions with applications to numerical quadratures

“…On the other hand, sequences of polynomials {Q n } n≥0 satisfying (2.5) have been studied extensively by Ranga et al [26,27,28,29,31]. Assume now that the integral (2.6)…”

Monotonicity of zeros of orthogonal Laurent polynomials

“…The last example concerns largest zeros of the ultraspherical L-polynomials B λ n introduced in Example 2 in [13]. Since the orthogonal L-polynomials in [13] are obtained from symmetric orthogonal polynomials by a transformation which makes the ordinary central symmetry an inversion symmetry, it is clear that if the positive zeros of the symmetric orthogonal polynomials increase (decrease) then half of the zeros of the orthogonal L-polynomials will increase and the other half will decrease.…”

“…Since the orthogonal L-polynomials in [13] are obtained from symmetric orthogonal polynomials by a transformation which makes the ordinary central symmetry an inversion symmetry, it is clear that if the positive zeros of the symmetric orthogonal polynomials increase (decrease) then half of the zeros of the orthogonal L-polynomials will increase and the other half will decrease. It is shown in [13] that the ultraspherical polynomials are transformed to L-orthogonal polynomials which satisfy (1.2) with dψ(x) = x −λ (b − x) λ−1/2 (x − a) λ−1/2 dx, where √ b = √ α + β+ √ α, √ a = √ α + β− √ α, α, β > 0 and the coefficients in the recurrence relation (1.3) are given by α n = 1, β n = β and γ n = αn(n + 2λ − 1)/((n + λ)(n + λ)). By Theorem 2 all the zeros of B λ n are increasing functions of β. Furthemore, since ∂γ n /∂α > 0 and −nα(n + λ) 2 (n + λ − 1) 2 γ ′ n (λ) = 2λ 2 + (2n − 1)λ + 1, then the largest zeros of B λ n are increasing functions of α and decreasing functions of λ, λ ≥ 0.…”