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Frobenius Triangle Identities

Let $C_{L,M}$ be a Padé Approximant. Then


$\displaystyle C_{(L+1)/M}S_{(L-1)/M}-C_{L/(M+1)}S_{L/(M+1)}$ $\textstyle =$ $\displaystyle C_{L/M}S_{L/M}$ (1)
$\displaystyle C_{L/(M+1)}S_{(L+1)/M}-C_{(L+1)/M}S_{L/(M+1)}$ $\textstyle =$ $\displaystyle C_{(L+1)/(M+1)}xS_{L/M}$ (2)
$\displaystyle C_{(L+1)/M}S_{L/M}-C_{L/M}S_{(L+1)/M}$ $\textstyle =$ $\displaystyle C_{(L+1)/(M+1)}xS_{L/(M-1)}$ (3)
$\displaystyle C_{L/(M+1)}S_{L/M}-C_{L/M}S_{L/(M+1)}$ $\textstyle =$ $\displaystyle C_{(L+1)/(M+1)}xS_{(L-1)/M},$ (4)

where
\begin{displaymath}
S_{L/M}=G(x)P_L(x)+H(x)Q_M(x)
\end{displaymath} (5)

and $C$ is the C-Determinant.

See also C-Determinant, Padé Approximant


References

Baker, G. A. Jr. Essentials of Padé Approximants in Theoretical Physics. New York: Academic Press, p. 31, 1975.




© 1996-9 Eric W. Weisstein
1999-05-26