Abstract
Stieltjes interlacing states that if \(\{p_{n}(z)\}_{n=0}^{\infty }\) is a sequence of orthogonal polynomials, then there is at least one zero of pn(z) in between any two consecutive zeros of pm(z), where m < n − 1. Stieltjes interlacing holds for the zeros of polynomials from different sequences of little q-Jacobi polynomials pn(z;a,b|q), 0 < aq < 1, bq < 1 and q-Laguerre polynomials \(L_{n}^{(\delta )}(z;q),\delta >-1\). We consider cases where the degree difference is 2 or 3 and, in each case, we derive the associated polynomials analogous to the de Boor-Saff polynomials whose zeros will complete the interlacing. We derive upper bounds for the smallest zeros of these polynomials and provide numerical examples to illustrate improvements on previously known bounds that have been obtained using different approaches.
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Funding
The research of PPK and PG is supported by OURIIP, Govt. of Odisha, India with Sanction Number 1040/69/OSHEC and KJ gratefully acknowledges the support of a Royal Society Newton Advanced Fellowship NAF∖R2∖180669.
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Kar, P.P., Jordaan, K. & Gochhayat, P. Stieltjes interlacing of zeros of little q-Jacobi and q-Laguerre polynomials from different sequences. Numer Algor 92, 723–746 (2023). https://doi.org/10.1007/s11075-022-01387-8
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DOI: https://doi.org/10.1007/s11075-022-01387-8