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

10more comments