On totally positive sequences and functions
A doubly infinite sequence \((b_n)^{ \infty}_{ n=- \infty}\) is called totally positive if for every \(n \in \mathbb N\) and integers \(m_1 < \dots < m_n\), \(k_1 < \dots < k_n\) we have \(\text{det}(b_{m_{i} -k_{j}})_{1 \leq i , j \leq n} \geq 0\). A sequence \((b_n)^{ \infty}_{ n=- \infty}\) is said to be admissible if it does not coincide with a sequence of the form \((A \lambda ^n )^{ \infty}_{ n=- \infty}\) where \(A \geq 0\) and \(\lambda > 0\). A function \(f : \mathbb R \rightarrow \mathbb R\) is called totally positive if for every \(n \in \mathbb N\) and reals \(u_1 < \dots < u_n\), \(v_1 < \dots < v_n\) we have \(\text{det}(f (u_i - v_j ))_{1 \leq i , j \leq n} \geq 0\). Let \(a > 0\) be fixed. A function \(f : \mathbb R \rightarrow \mathbb R\) is called \(a\)-shift-generating if for every \(x \in \mathbb R\), \(( f (an - x))^{ \infty}_{ n=- \infty}\) is an admissible totally positive sequence. The generating function of \(f\) is \(F (x, z) = \sum ^{ \infty}_{ n=- \infty} f (an - x)z^n\). The author obtains a characterization of \(a\)-shift-generating functions in terms of their generating functions. It is also obtained that for every continuous function \(f : \mathbb R \rightarrow \mathbb R\) which is not of the form \(e^{A x + B }\), \(f\) is totally positive if and only if for every \(a > 0\), \(f\) is \(a\)-shift-generating. Finally the author presents several examples of \(a\)-shift-generating functions, and shows that for every \(0 < q < 1\), \(x \in \mathbb R\) and \( 0< |z| < \infty\), \[ \sum ^{ \infty}_{ n=- \infty}q^{(n-x)^{ 2}} z^{n} = C_q z^{[ x +1/2]} q^{(\langle x+1/2\rangle-1/2) ^{2}} \prod _{i =1} ^{\infty} (1 + q ^{2i -2\langle x+1/2\rangle} z) ( 1 + q ^{2i +2\langle x+1/2\rangle-2} z^{-1}), \] where \(C_{q} = \prod _{i =1} ^{\infty} (1-q^{2i})\).
scientific article; zbMATH DE number 5629201