Even Order Pascal Tensors are Positive Definite
Chunfeng Cui, Liqun Qi, Yannan Chen
TL;DR
The paper establishes that even-order Pascal tensors are positive definite and odd-order Pascal tensors are strongly completely positive, using an induction framework on tensor order and exploiting sub-tensors formed by fixing indices. It extends these results to broader families, notably generalized Pascal tensors and fractional Hadamard power tensors, via an Inherence Theorem that preserves PD for even orders and SCP for odd orders under suitable conditions. The determinant problem is tackled first in two dimensions, yielding $f(m,2)=\left[(m-1)\!\right]^m$ through Sylvester-Pascal matrices and LU decomposition, and then generalized to $n\ge3$ using Macaulay formula with several conjectures about the structure and factorization of $f(m,n)$. Together, these results illuminate the spectral and positivity structure of structured tensors and suggest a unifying framework for other completely positive tensor families. The work also outlines several open conjectures and directions for future research in determinant formulas and odd-order positivity concepts. $
Abstract
In this paper, we show that even order Pascal tensors are positive definite, and odd order Pascal tensors are strongly completely positive. The significance of these is that our induction proof method also holds for some other families of completely positives tensors, whose construction satisfies certain rules, such an inherence property holds. We show that for all tensors in such a family, even order tensors would be positive definite, and odd order tensors would be strongly completely positive, as long as the matrices in this family are positive definite. In particular, we show that even order generalized Pascal tensors would be positive definite, and odd order generalized Pascal tensors would be strongly completely positive, as long as generalized Pascal matrices are positive definite. We also investigate even order positive definiteness and odd order strongly completely positivity for fractional Hadamard power tensors. Furthermore, we study determinants of Pascal tensors. We prove that the determinant of the $m$th order two dimensional symmetric Pascal tensor is equal to the $m$th power of the factorial of $m-1$.