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 $k^2$.
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
