Series and other data related to the five-particle contribution χ(6) 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)) and x=w2.
Some exact results for n-particle contributions to the susceptibility
|χ(2) = 4w4 2F 1[5/2,3/2;3;16w2]|
|ODE for χ(4) in variable x=w2.|
|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(6) in variable x mod the prime 32749.|
|For χ(6) 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(4) in w mod the primes: 32749 32719 32717 32713|
|Series for χ(6) in w mod the primes: 32749 32719 32717 32713|
Some exact results for factors occuring in the order 46 differential operator L46 annihilating the series Φ(6)=χ(6)-2/3χ(4)+2/45χ(2)
First we have the operator L17 in exact arithmetic.
This operator has the factorization:
Next we have the operator L6 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 .