* Bill Gosper <billgosper@gmail.com> [Feb 15. 2011 13:25]:

LambdaStar[r]== 4*eta[E^(-4*r*Pi)]^4/Sqrt[eta[E^(-r*Pi)]^8+16*eta[E^(-4*r*Pi)]^8]== 4*eta[E^(-4*r*Pi)]^8*eta[E^(-r*Pi)]^4/eta[E^(-2*r*Pi)]^12

where eta[q] := EllipticEta[q] := DedekindEta[Log[q]/2/I/Pi] --rwg

This is obtained from k^2+k'^2==1 and replacing k and k' in terms of eta(). There are other forms such as E^8(-q) - E^8(q) == 16*q*E^8(q^4) (where E(q) = prod(n>=1, 1-q^n), more given on p.609) Oh, wait: see (first formula in file eta07.gp of) http://eta.math.georgetown.edu/ for very many forms of this and no less than 5428 such eta-product identities (many with several equivalent forms such as in terms of theta functions or Ramanujan's functions).