Using sums of squares to prove that certain entire functions have only real zeros
Author: George Gasper
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.
This article published in
Fourier Analysis: Analytic and Geometric Aspects, W.O. Bray, P.S.
Milojevic and C.V. Stanojevic, eds., Marcel Dekker, 1994, pp. 171-186.
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