Zeros of linear combinations of Laguerre polynomials from different sequences
نویسندگان
چکیده
منابع مشابه
Zeros of linear combinations of Laguerre polynomials from different sequences
We study interlacing properties of the zeros of two types of linear combinations of Laguerre polynomials with different parameters, namely Rn = L α n + aL α n and Sn = L α n + bL α n−1. Proofs and numerical counterexamples are given in situations where the zeros of Rn, and Sn, respectively, interlace (or do not in general) with the zeros of L α k , L ′ k , k = n or n− 1. The results we prove ho...
متن کاملStieltjes interlacing of zeros of Laguerre polynomials from different sequences
Stieltjes’ Theorem (cf. [11]) proves that if {pn}n=0 is an orthogonal sequence, then between any two consecutive zeros of pk there is at least one zero of pn for all positive integers k, k < n; a property called Stieltjes interlacing. We prove that Stieltjes interlacing extends across different sequences of Laguerre polynomials Ln, α > −1. In particular, we show that Stieltjes interlacing holds...
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We investigate the interlacing of the zeros of linear combinations pn+ aqm with the zeros of the components pn and qm, where {pn} ∞ n=0 and {qm} ∞ m=0 are different sequences of Jacobi polynomials. The results we prove hold when pn and qm are Jacobi polynomials P (α,β) n (x) and P (α,β) m (x) for certain values of α and β with m = n or m = n−1. Numerical counterexamples are given in situations ...
متن کاملStieltjes Interlacing of Zeros of Jacobi Polynomials from Different Sequences
A theorem of Stieltjes proves that, given any sequence {pn}n=0 of orthogonal polynomials, there is at least one zero of pn between any two consecutive zeros of pk if k < n, a property called Stieltjes interlacing. We show that Stieltjes interlacing extends, under certain conditions, to the zeros of Jacobi polynomials from different sequences. In particular, we prove that the zeros of P n+1 inte...
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Article history: Received 15 December 2008 Available online 3 March 2010 Submitted by D. Waterman
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ژورنال
عنوان ژورنال: Journal of Computational and Applied Mathematics
سال: 2009
ISSN: 0377-0427
DOI: 10.1016/j.cam.2009.02.091