  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.
