Is the following polynomial positive: T_k(t)=(t/2)^p sum_(j=0)^k((−t^2/4)j Gamma(p+1))/(j! Gamma(p+j+1)).

Is the following polynomial positive:
${T}_{k}\left(t\right)={\left(\frac{t}{2}\right)}^{p}\sum _{j=0}^{k}\frac{{\left(-\frac{{t}^{2}}{4}\right)}^{j}\mathrm{\Gamma }\left(p+1\right)}{j!\mathrm{\Gamma }\left(p+j+1\right)}.$
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kuglatid4
Consider series
$T\left(t\right)={\left(\frac{t}{2}\right)}^{p}\sum _{j=0}^{\mathrm{\infty }}\frac{{\left(-\frac{{t}^{2}}{4}\right)}^{j}\mathrm{\Gamma }\left(p+1\right)}{j!\mathrm{\Gamma }\left(j+p+1\right)}$
This is related to the series representation of the Bessel function of order p of the first kind. Indeed
$T\left(t\right)=\mathrm{\Gamma }\left(p+1\right)\sum _{j=0}^{\mathrm{\infty }}\frac{\left(-1{\right)}^{j}}{j!\mathrm{\Gamma }\left(j+p+1\right)}{\left(\frac{t}{2}\right)}^{2j+p}=\mathrm{\Gamma }\left(p+1\right){J}_{p}\left(t\right)$
Since this series converges, then
$\begin{array}{}\text{(1)}& \underset{k\to \mathrm{\infty }}{lim}{T}_{k}\left(t\right)=\mathrm{\Gamma }\left(p+1\right){J}_{p}\left(t\right)\end{array}$
It is known that Bessel functions of the first kind take negative and positive values infinitely many times on $\left(0,+\mathrm{\infty }\right)$. Hence we may consider ${t}_{0}$ such that $\mathrm{\Gamma }\left(p+1\right){J}_{p}\left({t}_{0}\right)<0$. From (1) it follows that for some ${k}_{0}$ we would have