20. Let $P _ { 1 }$ and $P _ { 2 }$ be two $N$-conductive polynomials. Show that if $P _ { 1 }$ is of positive type in dimension $N$, then $P _ { 1 } P _ { 2 }$ is $N$-conductive. We fix an integer $N \geqslant 4$ and an integer $n \geqslant 2$. We admit that the polynomial $a _ { n } ^ { \left( \frac { N } { 2 } - 1 \right) }$ has $n$ simple real roots $r _ { 1 } > r _ { 2 } > \cdots > r _ { n }$ in $] - 1,1 [$. Let $f : [ - 1,1 ] \rightarrow \mathbb { R }$ be an absolutely monotone function.
20. Let $P _ { 1 }$ and $P _ { 2 }$ be two $N$-conductive polynomials. Show that if $P _ { 1 }$ is of positive type in dimension $N$, then $P _ { 1 } P _ { 2 }$ is $N$-conductive.\\
We fix an integer $N \geqslant 4$ and an integer $n \geqslant 2$. We admit that the polynomial $a _ { n } ^ { \left( \frac { N } { 2 } - 1 \right) }$ has $n$ simple real roots $r _ { 1 } > r _ { 2 } > \cdots > r _ { n }$ in $] - 1,1 [$. Let $f : [ - 1,1 ] \rightarrow \mathbb { R }$ be an absolutely monotone function.\\