Using sums of squares to prove that certain entire functions have only real zeros

Abstract

It is shown how sums of squares of real valued functions can be used to give new proofs of the reality of the zeros of the Bessel functions $J_\alpha (z)$ when $\alpha \ge -1,$ confluent hypergeometric functions ${}_0F_1(c\/; z)$ when $c>0$ or $0>c>-1$, Laguerre polynomials $L_n^\alpha(z)$ when $\alpha \ge -2,$ and Jacobi polynomials $P_n^{(\alpha,\beta)}(z)$ when $\alpha \ge -1$ and $ \beta \ge -1.$ Besides yielding new inequalities for $|F(z)|^2,$ where $F(z)$ is one of these functions, the derived identities lead to inequalities for $\partial |F(z)|^2/\partial y$ and $\partial ^2 |F(z)|^2/\partial y^2,$ which also give new proofs of the reality of the zeros.