Frobenius Triangle Identities

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

$$C_{(L+1)/M}S_{(L-1)/M}-C_{L/(M+1)}S_{L/(M+1)} = C_{L/M}S_{L/M}$$ (1)

$$C_{L/(M+1)}S_{(L+1)/M}-C_{(L+1)/M}S_{L/(M+1)} = C_{(L+1)/(M+1)}xS_{L/M}$$ (2)

$$C_{(L+1)/M}S_{L/M}-C_{L/M}S_{(L+1)/M} = C_{(L+1)/(M+1)}xS_{L/(M-1)}$$ (3)

$$C_{L/(M+1)}S_{L/M}-C_{L/M}S_{L/(M+1)} = C_{(L+1)/(M+1)}xS_{(L-1)/M},$$ (4)

where

$$S_{L/M}=G(x)P_L(x)+H(x)Q_M(x)$$ (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.