I am currently following a course on differential equations and difference equations (recurrence relations).
The teacher tries to make parallels between the two concepts, because the methods for solving both of these kinds of equations are essentially the same(sub-question : is there a deeper reason of why this is so?).
For example, to find solutions for linear differential equations or linear difference equations when the coefficients are constants, you find the roots of the caracteristic polynomial.
The teacher introduced the Wronskian
$$W(f,g)=
\begin{vmatrix}
f& g\\\\ f'&g'\\\\
\end{vmatrix}$$
That is used to know if two solutions of a differential equation are linearly independent (it is zero if they are dependent, and everywhere non-zero if not)
He told us that the equivalent of the Wronskian for difference equations was the "Carosatian" (my course is in french, the exact term was "Carosatien") which is the determinant:
$$C_n(x,y)=
\begin{vmatrix}
x_n & y_n \\\\ x_{n+1} & y_{n+1}\\\\
\end{vmatrix} $$
for two sequences x, y. It works in the same way, in that this Carosatian is 0 only if x and y are linearly dependent.
When I search google for carosatian or carosatien, I get exactly 0 results, which is very surprising usually for things that actually exist. I was wondering if there was a more popular name for this concept?
(edit : I didn't get the matrices to work, but they're supposed to be 2×2)
(edit 2 : I got a very fast answer, but I would still be very happy to get an answer as to why are the two kinds of equations solvable with the same methods)
Best Answer
You mean "Casoratian"?