I am seeking solutions to the following difference equation: $$2c_k-c_{k-1}-c_{k+1}=\ln(k+A)-\ln(k+B)$$ where $A>B>0$.

This equation is related to a real polynomial (see here) which I want to prove that it has only real roots.

The related polynomials are defined by the recursive relations for the coefficients $b_k>0$ as defined below: $$p_{n}(x)=\sum_{k=0}^{n}\binom{2n}{2k}b_k x^k$$ $$\frac{b_k^2}{b_{k-1}b_{k+1}}=1+\frac{\pi}{31(k+1/2)}=\frac{k+A}{k+B}>1$$

So $$2{c_k}-{c_{k-1}}-{c_{k+1}}=2\ln{b_k}-\ln{b_{k-1}}-\ln{b_{k+1}}=\ln(k+A)-\ln(k+B)$$

These polynomials showed up when we tried to find a polynomial approximation to Jensen's polynomials associated with Riemann $\xi(z)$ function.

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

T. Craven, G. Csordas; Jensen polynomials and the Turan and Laguerre inequalities. Pacific J. Math., 136 (2) (1989), pp. 241–260

Thanks- Mike