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Fasi-Higham Positivity Conjecture

The Fasi-Higham positivity conjecture concerns the reciprocal denominator of a Padé approximant to the exponential function. For integers k>=0 and m>=1, put

q_(k,m)(z)=sum_(j=0)^m(-1)^j(m; j)((k+m-j)!)/((k+m)!)z^j.

This is the denominator, normalized by q_(k,m)(0)=1, of the [k/m] approximant at zero. The conjecture states that every coefficient c_n(k,m) in 1/q_(k,m)(z)=sum_(n>=0)c_n(k,m)z^n is positive when m is odd (Fasi and Higham 2019).

Scardino (2026) disproved the unrestricted conjecture with exact rational counterexamples, including c_(38)(66,5)<0, c_(37)(44,7)<0, and c_(42)(45,9)<0. There are infinitely many counterexamples with m=5. The diagonal case k=m remains open, and the counterexamples do not imply failure of the Fasi-Higham matrix exponential algorithm.

See also

Matrix Exponential, Padé Approximant, Scaling and Squaring Method

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References

Fasi, M. and Higham, N. J. "An Arbitrary Precision Scaling and Squaring Algorithm for the Matrix Exponential." SIAM J. Matrix Anal. Appl. 40, 1233-1256, 2019. https://doi.org/10.1137/18M1228876.Scardino, F. "On a Positivity Conjecture of Fasi and Higham for Padé Approximants to the Matrix Exponential." 28 Oct 2026. https://arxiv.org/abs/2610.39051.

Cite this as:

Weisstein, Eric W. "Fasi-Higham Positivity Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Fasi-HighamPositivityConjecture.html

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