# What is the term analogous to “Wronskian” for difference equations?

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 2x2)

(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)

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To get your matrices right, you have to type four backslashes where you would ordinarily type two. Yes, four. Please edit your question for posterity's sake. – Harald Hanche-Olsen Nov 20 '09 at 21:23
done! thank you – Jean-Philippe Burelle Nov 21 '09 at 4:21
Good question, but lousy title. "What is the term analogous to "Wronskian" for difference equations?" is my suggestion. – Theo Johnson-Freyd Nov 21 '09 at 5:24

You mean "Casoratian"?

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Thanks a lot! that was fast, I find it really funny that it was only a typo in the notes of the teacher, and that the teacher himself didn't notice it! – Jean-Philippe Burelle Nov 20 '09 at 19:16

You can think of difference equations as "discretized differential equations," or alternately you can think of differential equations as "difference equations in the limit as the difference goes to zero." This shows that they're very closely related in their definitions alone. The integral version of this correspondence is made precise by the Euler-Maclaurin formula.

Another sense in which they're closely related is the nature of how they act on polynomials. The derivative $\frac{d}{dx}$ acts on the basis $\{ x^n \}$ for the space of polynomials by $\frac{d}{dx} x^n = n x^{n-1}$. It turns out there is a similar basis $\{ n! {x \choose n} \}$ for the space of polynomials such that the forward difference $f(x+1) - f(x)$ acts the same way: the forward difference of $n! {x \choose n}$ turns out to be $n(n-1)! {x \choose n-1}$. Like the derivative, the forward difference also has a notion of Taylor expansion in terms of this basis. This is generalized by the theory of Sheffer sequences and related topics.

Finally, solving some differential equations is equivalent to solving some difference equations. For example, solving a linear homogeneous differential equation is basically the same thing as solving a linear homogeneous recurrence for the coefficients of the Taylor series. (This is because the derivative acts as left shift on Taylor coefficients, which also implies that the Wronskian and the Casoratian are really the same thing.)

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I know this is a bit of a late response, but you may try taking a look at [1]. I believe Kelley and Peterson discuss casoratian in their text. Even still, this is a very nice text to reference when talking about difference equations. They do a very nice job presenting the material.

It is important to note, as Qiaochu Yuan has, that differential equations and difference equations parallel one another to an extent. They have the same goals, just on a different domain. If you have not heard of time scale calculus, I would recommend you take a look at that as well. It is the natural next step after seeing the connection between difference calculus and differential calculus. [2] is a good resource for time scale.

Let me know if I can be of any help to you!

[1] Kelley, W. & Peterson, A. (2001). Difference Equations: An Introduction with Applications (2nd Ed.). San Diego, CA: Academic Press.

[2] Bohner, M. & Peterson, A. (2001). Dynamic Equations on Time Scales: An Introduction with Applicaitons. Boston, MA: Birkhäuser.

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