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Using integrals of squares of certain real-valued special functions to prove that the P\'olya \Xi^*(z) function, the functions K_{iz}(a), a>0, and some other entire functions have only real zeros

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 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, Jacobi polynomials P_n^{(\alpha,\beta)}(z) when \alpha \ge -1 and \beta \ge -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. Milojevi\'c and C. V. Stanojevi\'c, 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 P\'olya \Xi^*(z) function, the K_{iz}(a) functions when a>0, and some other entire functions.


Introduction
It is well-known [21] that the Riemann Hypothesis is equivalent to the statement that all of the zeros of the Riemann Ξ(z) function are real. Ξ(z) is an even entire function of z with the integral representations In a 1926 paper Pólya [18] observed that Φ(u) ∼ 8π 2 cosh( 9 2 u) e −2π cosh(2u) as u → ±∞ (1.3) and, in view of this asymptotic equivalence to Φ(u), considered the problem of determining whether or not the entire function Ξ * (z) = 16π 2 ∞ 0 cosh( 9 2 u) e −2π cosh(2u) cos(zu) du (1.4) has only real zeros. Here, as is now customary, the capital letter Ξ is used instead of the original lower case ξ. Pólya was able to prove that Ξ * (z) has only real zeros by using (in a different notation) a difference equation in z for the modified Bessel function of the third kind [7], [22] K z (a) = ∞ 0 e −a cosh u cosh(zu) du, a > 0, (1.5) to prove for each a > 0 that K iz (a) has only real zeros, and then applying the identity and the special case G(z) = K iz/2 (2π), c = 9 4 , of the lemma (derived via an infinite product representation for G(z)): Lemma. If −∞ < c < ∞ and G(z) is an entire function of genus 0 or 1 that assumes real values for real z, has only real zeros and has at least one real zero, then the function also has only real zeros. More generally, in a subsequent paper Pólya [19] pointed out that from the case G(z) = K iz/2 (a) of this lemma it follows that each of the entire functions has only real zeros. He also used a differential equation in the variable a to give a proof of the reality of the zeros of K iz (a), a > 0, that was simpler than his previous proof. Our main aim in this paper is to show how integrals of squares of certain real-valued special functions can be used to give new proofs of the reality of the zeros of the above Ξ * (z), K iz (a), and F a,c (z) functions. This paper is a sequel to the author's 1994 paper [14] in which he showed how sums of squares of certain real-valued special functions could be used to prove the reality of the zeros of the Bessel functions J α (z) when α ≥ −1, confluent hypergeometric functions 0 F 1 (c; z) when c > 0 or 0 > c > −1, Laguerre polynomials L α n (z) when α ≥ −2, Jacobi polynomials P (α,β) n (z) when α ≥ −1 and β ≥ −1, and some other entire functions. Also see the applications of squares of real-valued special functions in [2], [3], [6], [8], [9], [10], [11], [12], [13], and [15].
where the path of integration is along the upwardly oriented vertical line ℜ(s) = c. Next note that, by Gauss' summation formula [15, ( where (y) k = Γ(y + k)/Γ(y) and the series is absolutely convergent. Hence, a 2s ds.
(2.3) From (2.2) it follows that in order to prove that K iz (a) has only real zeros it suffices to prove that L a (x, y) > 0, −∞ < x, y < ∞.
by Fubini's Theorem and (2.1), which gives for −∞ < x, y < ∞. Equations (2.2) and (2.5) can be combined to give the formula (2.6) from which it follows that K iz (a) has only real zeros when a > 0, since the integrand is clearly nonnegative.
The reality of the zeros of K iz (a), a > 0, can also be proved by taking the first and second partial derivatives of the formula (2.6) with respect to y to obtain the formulas and In [4] Askey and the author utilized the reality of the zeros of the continuous dual Hahn polynomials [1, p. 331] to show that f t (y) (and a generalization of it) is an (even) absolutely monotonic function (one whose power series coefficients are nonnegative) of y when 0 < t < 1, which shows that y ∂ ∂y f t (y) ≥ 0, −∞ < y < ∞, 0 < t < 1, (2.10) and ∂ 2 ∂y 2 f t (y) ≥ 0, −∞ < y < ∞, 0 < t < 1. (2.11) and, hence, that the integrals in (2.7) and (2.8) are positive when y = 0. Thus, it follows from (2.7) that |K iz (a)| 2 is an increasing (decreasing) even function of y when y > 0 (y < 0), and it follows from (2.8) that |K iz (a)| 2 is a convex even non-constant function of y, each of which implies that for any x the function |K iz (a)| 2 assumes its minimum value when y = 0 and proves that K iz (a) has only real zeros. Corresponding to the above proofs via formulas (2.7) and (2.8), it should be noted that in 1913 Jensen [17] showed that each of the inequalities and is necessary and sufficient for a real entire function F (z) ≡ 0 of genus 0 or 1 to have only real zeros (see [5,Chapter 2] and the necessary and sufficient conditions in [20]).

3.
Reality of the zeros of the functions Ξ * (z) and F a,c (z) Because of Ξ * (z) = 4π 2 F 2π,9/4 (z/2), it suffices to show how formula (2.6) can be used to prove that F a,c (z) has only real zeros when a, c > 0. Fix a, c > 0 and suppose that z 0 = x 0 + iy 0 is a zero of F a,c (z). Then K i(x 0 +i(y 0 +c)) (a) = −K i(x 0 +i(y 0 −c)) (a) by (1.7) and hence 0 = |K i(x 0 +i(y 0 +c)) (a)| 2 − |K i(x 0 +i(y 0 −c)) (a)| 2 by (2.6) and (2.9). Since f t (y) is an even convex non-constant function of y when 0 < t < 1, and it follows from (3.1) that y 0 = 0 and, thus, the function F a,c (z) has only real zeros when a, c > 0. One can also try to give another proof of the reality of the zeros of F a,c (z) via a formula for the function that contains integrals of nonnegative functions as in (2.6). By using the right-hand side of (3.2), [7, Eq. 5.6(66)], [7,Eq. 5.3(1)], Gauss' summation formula and the beta integral, it can be shown that with g t,c (y) = y(y + 2c)t y+c 2 F 1 y + 1, y + 2c + 1 2 The function g t,c (y) is clearly an even function of c such that g t,0 (y) = 0 and g t,c (0) = 0. It can be shown that g t,c (y) is also an even function of y by applying the Euler transformation formula [15, (1.4.2)] to the hypergeometric functions in (3.4). Extensive analysis of g t,c (y) via Mathematica and generating functions of the form [4, (5)] strongly suggest that g t,c (y) is nonnegative, convex, and an absolutely monotonic function of y when 0 < t < 1 and −∞ < c < ∞. If the nonnegativity, convexity, or absolute monotonicity of g t,c (y) could be proved, then (3.3) and its partial derivatives with respect to y would give additional proofs of the reality of the zeros of F a,c (z). Pólya [19] derived a theorem concerning the zeros of Fourier transforms and universal factors, and then used it to prove that his [18, p. 317] Ξ * * (z) = 8π