We encountered polynomials defined by the recursive relations for the coefficients $b_k>0$ as defined below:
$$p_{n}(x)=\sum_{k=0}^{n}\binom{2n}{2k}b_k x^k$$
$$\frac{b_k^2}{b_{k-1}b_{k+1}}=1+\frac{\pi}{31(k+1/2)}=\frac{k+A}{k+B}>1$$
These polynomials showed up when we tried to find a polynomial approximation to Jensen polynomial associated with Riemann $\xi(z)$ function. For detailed background information on jensen polynomial and its relation to entire function like Riemann $\xi(z)$ function, see ref. 1 and ref. 2 below.
Here we provide some basic information from ref. 1 and ref. 2.
Riemann $\xi(z)$ function is defined as
$$ \xi (z/2)=8\int_0^{\infty}\Phi(t)\cos(zt)dt$$
where
$$\Phi(x)=\sum_{n=1}^{\infty}(2n^4\pi^2e^{9t}-3n^2\pi e^{5t})exp(-n^2\pi e^{4t})$$
The Riemann Hypothesis is equivalent to that all the infinite zeros of $\xi(z)$ are real.
Taking Taylor expansion on $\cos(zt)$, we obtain
$$\frac{1}{8}\xi(z/2)=\sum_{m=0}^\infty (-1)^m a_m\frac{z^{2m}}{(2m)!}$$
where
$$a_m=\int_0^{\infty}t^{2m} \Phi(t)dt$$
On setting $x=-z^2$ and $\xi_1(x)=\frac{1}{8}\xi(z)$, we obtain
$$\xi_1(x)=\sum_{m=0}^\infty a_{m}\frac{x^{m}}{(2m)!}$$
The function $\xi_1(x)$ is then an entire function of order 1/2.
The Jensen polynomial $g_n(x)$ associated with $\xi_1(x)$ is defined as
$$g_n(x)=\sum_{m=0}^n \binom{n}{m}a_{m}\frac{m!}{(2m)!}x^{m}$$
A Theorem due to Polya and Schur states that A real entire function $\phi(x)=\sum_{m=0}^\infty c_m \frac{x^m}{m!}$ to be in Laguerre-Plya class (i.e., all infinite zeros of $\phi(x)$ are real) if and only if the associated Jensen polynomials $g_n(x)=\sum_{m=0}^n \binom{n}{m}c^{m}x^{m}$ (n=1,2,3…) have only real zeros.
Therefore the Riemann Hypothesis is equivalent to that all the zeros of $g_n(x)$ are real.
Polya conjectured and Craven, Norfolk and Varga proved (cf ref. 1 and ref. 2) the following necessary condition (now also known as Turan inequality) for all the zeros of $g_n(x)$ to be real:
$$a_m^2\gt\frac{m-\frac{1}{2}}{m+\frac{1}{2}}a_{m-1}a_{m+1}$$
Since it is too hard to directly prove that all the zeros of $g_n(x)$ are real, we try to see if a polynomial (like $p_n(x)$ defined at the top) similar to $g_n(x)$ has all the real zeros.
Here by similar we mean that their coefficients obey similar recursive relations.
@gaoxinge found a closed-form solution of coefficients $b_k$ here
Let $\binom{2n}{2k}b_k=\gamma_k\binom{2n}{2k-2}b_{k-1}$. Then we have
$$\gamma_k=\frac{k-1+B}{k-1+A}\frac{(n-k+1)(2n-2k+1)}{k(2k-1)}\gamma_{k-1}$$
So we obtain
$$\gamma_k=\frac{(k-1+B)_k}{(k-1+A)_k}\frac{(n-k+1)_k(2n-2k+1)_k}{(k)_k(2k-1)_k}\gamma_{0}$$
where $(A)_k$ is the Pochhammer symbol and $\gamma_0=1$.
$$\binom{2n}{2k}b_k=\prod_{j=0}^k\gamma_j$$
$$p_{n}(x)=\sum_{k=0}^{n}(\prod_{j=0}^k\gamma_j) x^k$$
Numerical results showed that the roots for $p_{n}(x)$ with $ 1\leq n\leq 150$ are all real.
We are looking for a proof (or a reference on such proof) that all the zeros of $p_n(x)$ are real.
Any references on similar proofs will be helpful to us.
Best regards-
Mike
ref.1 G. Csordas, T. S. Norfolk and R. S. Varga, The Riemann Hypothesis and the Turán Inequalities, Transactions of the American Mathematical Society, Vol. 296, No. 2 (Aug., 1986), pp.521-541
ref.2 T. Craven, G. Csordas; Jensen polynomials and the Turan and Laguerre inequalities. Pacific J. Math., 136 (2) (1989), pp. 241–260
Best Answer
Just two pointers for this problem (sorry, no solution).
Hurwitz used a version of Sturm's theorem on numerators of continued fractions to study the zeros of the Bessel functions, and Watson's treatise on Bessel functions (1944) has this in section 9.7. The series in question can be written as $$\sum_{k=0}^\infty (-1)^k \frac{x^{2k}}{(2k)!} \prod_{j=1}^k \frac{2j-1}{2j+2p},\quad(p>-1).$$ (Hurwitz 1889 Ueber die Nullstellen der Bessel'schen Function)
There is a converse to Newton's inequalities for real polynomials with positive coefficients: if $a_i>0$ and $a_i^2 > 4 a_{i-1} a_{i+1}$, then all roots are real and distinct: A Sufficient Condition for All the Roots of a Polynomial To Be Real David C. Kurtz, The American Mathematical Monthly Vol. 99, No. 3 (Mar., 1992), pp. 259-263
J. Gélinas