Generalized Vandermonde tensors.

We extend Vandermonde matrices to generalized Vandermonde tensors. We call an mth order n-dimensional real tensor $$A = (A_{i_1 i_2 } \cdots _{i_m } )$$ a type-1 generalized Vandermonde (GV) tensor, or GV tensor, if there exists a vector v = ( v, v,..., v) such that $${A_{{i_1},{i_2} \ldots {i_m}}} = v_{{i_1}}^{{i_2} + {i_3} + \ldots + {i_m} - m + 1},$$ and call A a type-2 ( mth order n dimensional) GV tensor, or GV tensor, if there exists an ( m − 1)th order tensor $$B = \left( {{B_{{i_1}{i_2} \ldots {i_{m - 1}}}}} \right)$$ such that $${A_{{i_1},{i_2} \ldots {i_m}}} = B_{{i_1}{i_{2 \ldots {i_{m - 1}}}}}^{{i_m} - 1},$$. In this paper, we mainly investigate the type-1 GV tensors including their products, their spectra, and their positivities. Applications of GV tensors are also introduced. [ABSTRACT FROM AUTHOR]