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Let $p\geq 2$, and $p$ is not a half odd integer. $t\in R$.

Is the following polynomial positive:

$$ T_k(t)=\left(\frac t2\right)^p\sum_{j=0}^k\frac{\left(-\frac{t^2}{4}\right)^j\Gamma(p+1)}{j!\Gamma(p+j+1)}. $$

Thank you for your help

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Why the gamma notation? It's not a polynomial unless $p$ is a natural number. – John Bentin Jul 6 '12 at 16:02
Sorry, I did not mentioned that $p$ is not half odd integer. – Michael Jul 6 '12 at 17:45
$p$ is not a half-integer $\left(p \neq \frac{5}{2},\frac{7}{2},\frac{9}{2},\dots\right)$? – Vandermonde Jul 6 '12 at 18:07
@Vandermonde: Yes,exactly like you wrote. – Michael Jul 6 '12 at 21:31
up vote 5 down vote accepted

Consider series $$ T(t)=\left(\frac{t}{2}\right)^p\sum\limits_{j=0}^\infty\frac{\left(-\frac{t^2}{4}\right)^j\Gamma(p+1)}{j!\Gamma(j+p+1)} $$ This is related to the series representation of the Bessel function of order $p$ of the first kind. Indeed $$ T(t)=\Gamma(p+1)\sum\limits_{j=0}^\infty\frac{(-1)^j}{j!\Gamma(j+p+1)}\left(\frac{t}{2}\right)^{2j+p}=\Gamma(p+1)J_p(t) $$ Since this series converges, then $$ \lim\limits_{k\to\infty} T_k(t)=\Gamma(p+1)J_p(t)\tag{1} $$ It is known that Bessel functions of the first kind take negative and positive values infinitely many times on $(0,+\infty)$. Hence we may consider $t_0$ such that $\Gamma(p+1)J_p(t_0)<0$. From $(1)$ it follows that for some $k_0$ we would have $$ T_k(t_0)<0\quad\text{ for all}\quad k>k_0. $$

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