       MORE ON RSOLVE--Discrete Painleve Equations

• To: mathgroup at smc.vnet.net
• Subject: [mg43806] MORE ON RSOLVE--Discrete Painleve Equations
• From: Peter Szabo <peterszabo20022003 at yahoo.co.uk>
• Date: Mon, 6 Oct 2003 02:07:57 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```Dear Colleagues,

I sent a message today about my failure to use RSolve
to solve a set of discrete/difference equations:

y[n+1]+(a_1 -b*n*n -2)*y[n] + y[n-1]==0.
y[m+1]+(a_2 -b*m*m -2)*y[m] + y[m-1]==0.

The problem I faced when using RSolve was either it
said that the "Out" was the same as the "In" and hence
no operation was performed, or that it said that the
expression was not a (discrete) equation.  This
happened for both a single equation and for the system
given above.

However, the package worked for the ALL test examples
in the Mathematica book.  Thus, the possibility of a

To recapitulate, these are ordinary difference
representations (lattice equations) for the 2-D time
independent Schroedinger equation with harmonic
potential.

The condition is a_1 + a_2=a, which is the coefficient
of the partial difference equation (combined case).
Also, "n" and "m" are the iteration indices
(independent variables or lattice variables) for the 2
dimensions respectively.

I tried RSolve with a simple form of the first
discrete Painleve equation:

y[n+1]+y[n]+y[n-1]+((a*n +b)/(1+y[n])) +mu==0.

Here, "a", "b" and "mu" are constants.  As you very
well know, this is a simple modification of the
example given by Eq. (3.3.1) in B. Grammaticos, F. W.
Nijhoff and A. Ramani, "Discrete Pailleve Equations",
Lecture Notes for the Cargese School, (1996).

The only modifications were that the third constant
"gamma" is set to zero and the translation
y[n]->y[n+1] is done in the denominator of the fourth
term above, that is associated with the constants.
This is done fo the case of siplification.

Here too, RSolve gives NO  ANSWER!.  The same also
occurs for the original form of the first discrete
Painleve eqation given in the above citation.

Most Respectfully Yours

Peter Szabo

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```

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