Orthogonality and asymptotics of Pseudo-Jacobi polynomials for non-classical parameters

Abstract

The family of general Jacobi polynomials P (α,β) n where α, β ∈ C can be characterised by complex (nonhermitian) orthogonality relations (cf. [15]). The special subclass of Jacobi polynomials P (α,β) n where α, β ∈ R are classical and the real orthogonality, quasi-orthogonality as well as related properties, such as the behaviour of the n real zeros, have been well studied. There is another special subclass of Jacobi polynomials P (α,β) n with α, β ∈ C, β = α which are known as Pseudo-Jacobi polynomials. The sequence of Pseudo-Jacobi polynomials {P α,α n } ∞ n=0 is the only other subclass in the general Jacobi family (beside the classical Jacobi polynomials) that has n real zeros for every n = 0, 1, 2, . . . for certain values of α ∈ C. For some parameter ranges Pseudo-Jacobi polynomials are fully orthogonal, for others there is only complex (non-Hermitian) orthogonality. We summarise the orthogonality and quasi-orthogonality properties and study the zeros of Pseudo-Jacobi polynomials, providing asymptotics, bounds and results on the monotonicity and convexity of the zeros.

Zeros of pseudo-ultraspherical polynomials

“…There is some overlap between our work (see, e.g., Sec. 3.4) and that in [21]. The results in [21] appear to be stronger than our results in Sec.…”

“…We show that simple methods also yield bounds for the endpoints of the interval in which all the zeros of C (λ) n must lie although the bounds obtained by these methods are not as sharp as those in [21]. Our first result is essentially an application of the Sturm comparison theorem.…”

“…While revising this article, we became aware of recent work of Jordaan and Tookos [21] on pseudo-Jacobi polynomials…”

Asymptotics of Pseudo‐Jacobi Polynomials with Varying Parameters

“…The definition of monic P‐J polynomials in terms of hypergeometric functions is given in , with parameters N and ν related to the parameters a and b in ; more precisely, with and , the monic P‐J polynomials are defined by By comparing the definitions of the P‐J polynomials and Jacobi polynomials , we immediately get the relation This relation has also been used as the definition of P‐J polynomials in [, p. 508] and . In view of the symmetry property of Jacobi polynomials, the P‐J polynomials satisfy see . Because the coefficients of P‐J polynomials are real, it also follows that …”

“…Probably due to their close relation with Jacobi polynomials, recently P‐J polynomials have attracted much attention . Orthogonality of the P‐J polynomials and properties of their zeros have been studied in for real parameter a in different ranges such as and . As a special case, the properties of the zeros of the pseudo‐ultraspherical polynomials have been independently investigated in almost at the same time.…”