A characterization of exponential and ordinary generating functions
This paper deals with the question why, of all possible generating functions, the ordinary and exponential generating functions are special. Let \(\Omega=(\omega_0,\omega_1,\dots)\) be a sequence of reals and \({\mathbf a}=(a_0,a_1,\dots)\) a sequence of complex numbers. Then \[ \Omega_{\mathbf a}(z)=\sum_{n=0}^{\infty}a_n \omega_n z^n \] is called the \(\Omega\)-generating function of \(\mathbf a\). Without loss of generality it may be assumed that \(\omega_0=\omega_1=1\). The two choices of \(\Omega\) corresponding to the ordinary and exponential generating function are \(\Omega=(1,1,1,1,\dots)=:O\) and \(\Omega=(1,1,1/2!,1/3!,\dots)=:E\), respectively. Given \(\Omega\) and \({\mathbf x}=(x_1,x_2,\dots)\) the \(n\)th Comtet polynomial \(C_n^{\Omega}({\mathbf x})\) is defined through \[ \sum_{n=0}^{\infty}C_n^{\Omega}({\mathbf x})\omega_n z^n =\sum_{n=0}^{\infty}\Bigl(\sum_{k=1}^{\infty} x_k\omega_k z^k\Bigr)^n \omega_n. \] Hence the ordinary and exponential Comtet polynomials are given by \[ \sum_{n=0}^{\infty}C_n^O({\mathbf x}) z^n ={1\over 1-\sum_{k=1}^{\infty} x_k z^k} \qquad \text{ and } \qquad \sum_{n=0}^{\infty}C_n^E({\mathbf x}) {z^n \over n!} =\exp\Bigl(\sum_{k=1}^{\infty} x_k {z^k\over k!}\Bigr). \] A sequence of polynomials \(P_n({\mathbf x})\) is called Hankel mean-independent if \(\det(P_{i+j}({\mathbf x}))_{i,j=0}^n\) does not depend on \(x_1\) for all \(n\geq 0\). The main result of the paper is that of the Comtet polynomials only the ordinary and exponential ones are Hankel mean-independent (up to normalization of \(\omega_2\)).
scientific article; zbMATH DE number 1760679