## Series and other data related to the five-particle contribution χ(5) to the square lattice Ising model susceptibility.

First a bit of notation. The `natural' variables are s=sinh(2J/kT) and w=s/(2(1+s2))

## Some exact results for n-particle contributions to the susceptibility

 χ(1) = 2w/(1-4w) ODE for χ(3) in variable w. First column is the order k of the derivative, the second column is j and the third column is the j'th coefficient pk,j in the polynomial Pk(w) multiplying the k'th derivative. ODE for χ(5) in variable w mod the prime 32749. For χ(5) we express the solution in terms of a linear ODE with polynomial coefficients but using the diffential operator (wd/dw). The solution is of order n=56 with the degree of the polynomials equal 129. The data is organised as a list of lists with the first list being the coefficients of Pn(w), the second list being the coefficients of Pn-1(w) etc. etc. and the last list is the coefficients of P0(w)

## Some exact series mod various primes.

For the prime 32749 the series has some 10000 terms
while for the remaining primes the series has some 5600 terms.  Series for χ(3) in w mod the primes:     32749     32719     32717     32713 Series for χ(5) in w mod the primes:     32749     32719     32717     32713

## Some exact results for factors occuring in the order 29 differential operator L29 annihilating the series   Φ(5)=χ(5)-χ(3)/2+χ(1)/120

First we have the operator L11 in exact arithmetic. This operator has the factorization:

The multiplications are done as differential operators and the addition is the direct sum of operators.
In terms of commands from the Maple package DEtools we have:

Next we have the operator L5 mod 32749. This is the minimal order ODE for this operator.
A solution to this ODE is given by

where E and K denote the complete elliptic integrals

while P4-i,i are polynomials in w with coefficients known modulo the prime 32749 and of degree 200, 202, 204, 204 and 204, respectively. Click here to download these polynomials .

Here we have the operator L13 mod 32749. This is a non-minimal order ODE (order 19) for this operator.

In our paper J. Phys A 43 195205 (2010) we calculated the ODE in exact arithmetic. Below are some of main results.

Exact χ(5) to order 8000, Exact L29 operator, Exact L24 operator, Exact L12(right) operator, and sKE polynomials.

Exact L11 operator and `reduced' L13 polynomials Qk. The polynomials occuring in L13 are Pk=Qk*PAk, where PA is the apparent polynomial of L11

Since publication one of us (Bernie Nickel) has found several exact relations between the constants at the ferromagnetic point given in the arrays C0 and C1 of equation (B.4). These relations can be found here