Using integrals of squares of certain real-valued special functions to prove that the Pólya Ξ*(z) function, the functions Kiz(a), a > 0, and some other entire functions have only real zeros

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Analogous to the use of sums of squares of certain real-valued special functions to prove the reality of the zeros of the Bessel functions Jα(z) when α≥ -1, confluent hypergeometric functions ₀F₁(c; z) when c > 0 or 0 > c > -1, Laguerre polynomials Lₙα(z) when α≥ -2, Jacobi polynomials Pₙ⁽α,β⁾(z) when α≥ -1 and β≥ -1, and some other entire special functions considered in G. Gasper [Using sums of squares to prove that certain entire functions have only real zeros, in Fourier Analysis: Analytic and Geometric Aspects, W. O. Bray, P. S. Milojević and C. V. Stanojević, eds., Marcel Dekker, Inc., 1994, 171--186.], integrals of squares of certain real-valued special functions are used to prove the reality of the zeros of the Pólya Ξ*(z) function, the Kiz(a) functions when a > 0, and some other entire functions.