# Zeroes of Laguerre polynomials

The simplest Laguerre polynomials are $$L_k(x)=(\frac{d}{dx}-1)^k\left(\frac{x^k}{k!}\right).$$ I would like to find a simple reference for proving or disproving the following assertions.

(1) All the $k$ zeroes of $L_k$ are simple and located on the positive half-line.

(2) The largest zero of $L_k$ is bounded above by $k^2$.

• (1) is granted, just because they are orthogonal polynomials on that domain. – Pietro Majer Oct 7 '16 at 16:33

## 2 Answers

Both assertions hold true, in fact the roots lie in the interval $$(0,k+(k-1)\sqrt{k}).$$

There are many books for references, one among which is

"Basic Hypergeometric Series", by George Gasper, Mizan Rahman, Encyclopedia of Mathematics and its applications.

hmm derivatives interlace zeros... however $a^+=1- d/dx$ behaves nicely on polynomials. It should preserve positivity of the roots and shift the leading zero by $+n$ with a small correction term where $n = \deg p$. here we have $x=0$ as a root with multiplicity $k$ and they should hop forward a bit