In my graduate discrete math course I talked about real-rootedness of some combinatorial polynomials, and as a homework asked for a proof of the real-rootedness of the matching polynomial (matching generating function) of the complete graph, 
$$
p\_n(x) = \sum\_{k \geq 0} m\_{n,k} x^k
$$ 
where $m\_{n,k}$ is the number of matchings of $K\_n$ with $k$ edges. (These are essentially the Hermite polynomials.)


I was thinking of the recurrence 
$$
p\_n(x) = p\_{n-1}(x) + (n-1)xp\_{n-2}x,
$$
with $p\_0(x)=p\_1(x)=1$, from which it is an easy induction that the roots of $p\_n(x)$ and $p\_{n-1}(x)$ are both all real, negative and distinct, that they interlace, and that the largest (least negative) root of $p\_n(x)$ is closer to $0$ than the largest root of $p\_n(x)$. (This is essentially Heilman and Lieb's proof that the matching generating function of any simple graph is real-rooted).


One of my students found instead the recurrence 
$$
p\_n(x) = 1 + \binom{n}{2}\int\_0^x p\_{n-2}(t)~dt~~~~~(\*)
$$ 
(or $p\_n'(x)=\binom{n}{2}p\_{n-2}(x)$), but neither he nor I were able to use this recurrence in an inductive proof that the roots of $p\_n$ and $p\_{n-2}$ are real, distinct, negative and interlace. (And this is not a fact I had previously known).


**Question**: Can the recurrence (\*) be used to prove that the roots of $p\_n(x)$ and $p\_{n-2}(x)$ are real and interlace? 


It's not generally true that if $a(x)$ is a polynomial with only real roots, and $b(x) = C\_1 + C\_2\int\_0^x a(t)~dt$, then $b(x)$ has all real roots; take e.g. $C\_1=C\_2=1$ and $a(x)=1+2x+x^2$. 



Matching polynomial of complete graph
I am not sure about how to use this recurrence to show that the roots are real and distinct, but I think if we assume we know this via other methods, I can show that the roots of $p\_n$ and $p\_{n-2}$ are interlaced.


Indeed, let $r\_1,\dots,r\_k$ be the roots of $p\_{n-2}$, and consider the intervals
$(\infty,r\_1],(r\_1,r\_2],\dots,(r\_k,\infty)$. Then $p\_n$ is monotone on each of these intervals by $(\ast)$. Thus $p\_n$ can have at most one root in each interval. Since we are assuming the roots of $p\_n$ are real and distinct, and we know the degree of $p\_n$ is one greater than the degree of $p\_{n-2}$, we know that there is exactly one root in each interval by pigeonhole. Thus the roots are interlaced.


