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SUMMARY:Kathy Driver (University of Cape Town)
DTSTART:20220215T140000Z
DTEND:20220215T150000Z
DTSTAMP:20260422T201236Z
UID:CAvid/65
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CAvid/65/">I
 nterlacing of zeros of Laguerre polynomials</a>\nby Kathy Driver (Universi
 ty of Cape Town) as part of CAvid: Complex Analysis video seminar\n\nLectu
 re held in N/A.\n\nAbstract\nThe sequence of Laguerre polynomials $\\{L_{n
 }^{(\\alpha)}(x)\\} _{n=0}^\\infty$ is orthogonal on $(0\, \\infty)$ with 
 respect to the weight function $e^{-x} x^{\\alpha}\,\\alpha > -1$ and the 
 real distinct positive zeros of $L_{n-1}^{(\\alpha)}(x)$ and $L_{n}^{(\\al
 pha)}(x)$ are interlacing for $\\alpha >-1\,  n \\geq 2.$  D-Muldoon (2015
 -2019) proved that for $\\alpha >-1\,$  the zeros of $L_{n-1}^{(\\alpha+t)
 }(x)$ and $L_{n}^{(\\alpha)}(x)$  are interlacing for $0 \\leq t \\leq 2\;
  $  the zeros of the equal degree Laguerre polynomials $L_{n}^{(\\alpha)}(
 x)$  and  $L_{n}^{(\\alpha+t)}(x)$ interlace for $0 < t \\leq 2$\, and the
  interval $0 \\leq t \\leq 2$ is sharp for interlacing to hold for every $
 n \\in \\mathbb{N}$. Further\, the zeros of $L_{n-k}^{(\\alpha+t)}(x)$ and
  $L_{n}^{(\\alpha)}(x)$  are interlacing (in the Stieltjes sense) for $0 \
 \leq t \\leq 2k$\, $1 < k < n$ and the interval $0 \\leq t \\leq 2k$ is sh
 arp. \nAt OPSFA 2019\, Alan Sokal: What happens to interlacing of roots if
  you increase parameter and increase degree of one polynomial relative to 
 the other? Simplest case: Are the zeros of $L_{n}^{(\\alpha)}(x)$  and  $L
 _{n+1}^{(\\alpha+1)}(x)$ interlacing for $\\alpha > -1$ and each $n \\in \
 \mathbb{N}$? We discuss this and related cases.\n
LOCATION:https://researchseminars.org/talk/CAvid/65/
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