Orthogonality properties of some entire characteristic functions (Q1600622)

We need to prepare the background for Propositions 1 and 2 below which are proved in this paper. Background: An entire function of order 2 with a finite number of zeros has the form \(f(t)= w(t)\exp(A(t))\) where \(w\) and \(A\) are polynomials and \(A\) is of degree . If \(f\) is also a characteristic function (ch.f.), then \(w\) is of even degree (say \(2n\)) and has the following form (ignoring the trivial factors of the type \(\exp(i\beta t)\)): \[ f_{2n}(t)= [w_{2n}(t)]\exp(- t^2\sigma^2/2)= \Biggl[\sum^{2n}_{r=0} \lambda_r(it)^r\Biggr] \exp(-t^2\sigma^2/2),\tag{1} \] where \(\lambda_r\) are real and \(\lambda_0=0\). Conversely [the author and \textit{Plucinski}, J. Math. Sci. (to appear)], a necessary and sufficient condition for a function of the type (1) to be a ch.f. is: \[ f_{2n}(t)= \Biggl[\sum^n_{r,s=0} (b_r b_s+ d_r d_s) H_{r+s} (\sigma t)(i\sigma)^{r+ 2}\Biggr]\exp(- t^2 \sigma^2/2), \] where \(b_r\) and \(d_r\) are real and satisfy the condition \(\sum^n_{r,s=0} (b_r b_s+ d_r d_s)(r+ s-1)!!\sigma^{r+ s}= 1\). Here \((-1)!!=1\), and \(H_n\) are the usual Hermite polynomials: \[ H_n(x)= \exp(x^2/2)(d^n/dx^n) (\exp(-x^2/2)). \] . Examples of ch.f.'s of this type are: \[ f_{2n}(t)= (-1)^n \alpha_{2n}^{-1} H_{2n}(t)\exp(- t^2\sigma^2/2),\tag{2} \] \[ f_{2n+1}(x)= (-1)^n \alpha_{2n}^{-1}(1/t) H_{2n+ 1}(t) \exp(-t^2 \sigma^2/2),\tag{3} \] where \(\alpha_{2n}= (2n)!!/(n!2^n)\). Proposition 1: Let \(f_{2n}(t)\) be a ch.f. of the type (1). Then it is of the form (2) iff the sequence \(w_{2n}(t)\) is orthogonal with respect to the weight function \(\exp(-t^2\sigma^2/2)\). Proposition 2: Let \(f_{2n+1}(t)=(1/t) w_{2n+1}(t) \exp(-t^2\sigma^2/2)\) be a real ch.f. Then it is of the form (3) iff the sequence \(w_{2n+1}(t)\) is orthogonal with respect to the weight function \(\exp(-t^2\sigma^2/2)\). It is essential in these propositions for \(f_{2n}(t)\) and \(f_{2n+1}(t)\) to be ch.f.'s for the conclusions to be valid. The paper gives examples of \(w\) sequences which are orthogonal with respect to the weight function \(\exp(-t^2\sigma^2/2)\) but result in non-ch.f.'s of a form different than (2) and (3).