A generalized Vandermonde matrix of two sequences 
and 
where 
is an increasing sequence of positive integers and 
is an increasing sequence of nonnegative integers of the same
length is the outer product of 
and 
with multiplication operation given by the power function.
The generalized Vandermonde matrix can be implemented in the Wolfram
Language as
Vandermonde[a_List?VectorQ, b_List?VectorQ] :=
Outer[Power, a, b] /; Equal @@ Length /@ {a, b}
A generalized Vandermonde matrix is a minor of a Vandermonde matrix. Alternatively, it has the same form as a Vandermonde
matrix 
, where 
is an increasing sequence of positive integers, except now
is any increasing sequence of nonnegative integers. In the special case of a Vandermonde
matrix, 
.
While there is no general formula for the determinant of a generalized Vandermonde matrix, its determinant is always positive. Since any minor of a generalized Vandermonde matrix is also a generalized Vandermonde matrix, they are always totally positive.
