Associated Laguerre and Hermite polynomials

Abstract

SynopsisExplicit orthogonality relations are found for the associated Laguerre and Hermite polynomials. One consequence is the construction of the [n − 1/n] Padé approximation to Ψ(a + 1, b; x)/Ψ(a, b; x), where Ψ(a, b; x) is the second solution to the confluent hypergeometric differential equation that does not grow rapidly at infinity.

The associated Askey-Wilson polynomials

“…The associated polynomials are orthogonal if and only if the positivity condition (1.9) an+acn+a+x>0, « = 0,1,..., is satisfied, [5,7]. For recent work on associated polynomials we refer the reader to the articles [5,8,10,14,15,19,25,33], and their references.…”

The Associated Classical Orthogonal Polynomials

“…If we replace λ and x in (1.41) by 1 2 (α + 1) and −x/2 sin φ, respectively, and take the limit φ → 0, then (1.41) becomes the 3-term recurrence relation for the associated Laguerre polynomials, L α n (x; c), as was observed by Pollaczek in [40]. Askey and Wimp [5] took advantage of his property to obtain the weight function for the orthogonality of {L α n (x; c)} on 0 < x < ∞, namely,…”

“…Askey and Wimp [5] also found the weight function for the associated Hermite polynomials H n (x; c):…”

“…Askey and Wimp [5] found the measure of orthogonality for these associated Laguerre polynomials L α n (x; c), as well as their explicit polynomial form:…”

“…The resulting polynomials, called by Askey and Wimp [5] the associated MeixnerPollaczek polynomials, (Meixner [34] had also found them and their orthogonality relation when c = 0) are orthogonal on the infinite interval −∞ < x < ∞ wrt the same weight function (1.38) with appropriate replacement of the variables. If we replace λ and x in (1.41) by 1 2 (α + 1) and −x/2 sin φ, respectively, and take the limit φ → 0, then (1.41) becomes the 3-term recurrence relation for the associated Laguerre polynomials, L α n (x; c), as was observed by Pollaczek in [40].…”