Two-Series Approach to Partial Differential Approximants: Three-Dimensional Ising Models

A two-series approach to partial differential approximant analysis of power series is presented. Instead of double series, $f(x,y)=\Sigma c_{\mathrm{ij}}{x}^i{y}^j$, our approach uses two one-variable series in $x$, $f$ and $\frac{\partial f}{\partial y}$, and has the efficiency and stability of one-variable methods. 21-term high-temperature series are analyzed for the susceptibility and correlation length squared for double-Gaussian Ising models on the bcc lattice. Critical exponents are $\gamma =1.2378\mathrm{}\left(6\right)\mathrm{}$, $2\nu =1.2623\mathrm{}\left(6\right)\mathrm{}$, and $\eta =0.0375\mathrm{}\left(5\right)\mathrm{}$; correction-to-scaling exponents are ${\theta }_{\chi }=0.52\mathrm{}\left(3\right)\mathrm{}$ and ${\theta }_{{\xi }^2}=0.49\mathrm{}\left(4\right)\mathrm{}$; and the subdominant critical amplitude ratio is $\frac{a_{\xi }}{a_{\chi }}=0.83\mathrm{}\left(5\right)\mathrm{}$.