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Frobenius Triangle Identities

Frobenius triangle identities are identities for Padé approximants. 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

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References

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

Referenced on Wolfram|Alpha

Frobenius Triangle Identities

Cite this as:

Weisstein, Eric W. "Frobenius Triangle Identities." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FrobeniusTriangleIdentities.html

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