I have recursive polynomials


$$Q\_{n}(t)=tQ\_{n-1}(t)+\frac{1-t^{2}}{n+1}Q'\_{n-1}(t)$$ 


and 


$$Q\_{0}(t)=1$$


Is there a theory for finding a factorisation of recursive polynomials?


It is possible to show that 
$$\sum\_{s=-\infty}^{\infty}sinc(\pi(x+s))^{p+1}=Q\_{p-1}[cos(\pi x)]$$
where $$sinc(x)=sin(x)/x$$


Maybe you have some ideas for my case?



roots of recursive polynomials
I have realised that your recurrent relation is exactly the one which appears in Eq. (2.4) in [[*Izvestiya: Mathematics* **66**:3 (2002) 489--542]](http://dx.doi.org/10.1070/IM2002v066n03ABEH000387) (see
also [here](http://wain.mi.ras.ru/PS/zete_main.pdf)). The properties of the corresponding polynomials are expressed in Lemma 2.2 and around (they seem to be exactly the ones you are trying to establish).


