Question: How can I show Jensen's Polynomial for $k$ and $k+1$ as it relates to $f(x)$ and $g(x)$ if possible? I need someone to show me.
I just started to know about Jensen Polynomials so here is it :
Let $f(x)$ be a real entire function of the form
$f(x)=$ $\sum_{k=0}^\infty$ $y_k$ ${x^k\over k!}$
where the $y_k$ are positive and satisfies Turan's inequalities
for $k=1,2,...$
The Jensen polynomial $g(t)$ associated with $f(x)$ is then given by
$g_n (t)=$ $\sum_{k=0}^\infty$ ${n \choose k}$ $y_k$ $t^k$
where ${a \choose b}$ is a binomial coefficient
Here is a link to Jensen polynomial below
[mathworld.wolfram.com/JensenPolynomial.html][1]
[1]: http://mathworld.wolfram.com/JensenPolynomial.html
Update: Here is something I am trying to see:
$g_1(x)=$ $\binom{1}{0}t^0 +$ $\binom{1}{1}t^1+$ $\cdots$
where the 1st two terms are $1 + t$ and so forth if we continue this. My problem is, if I put $k$ and $k+1$ here, I do not understand fully what to do and what it will the end result be.