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Legendre-Stirling Number

The Legendre-Stirling numbers are the specialization at z=1 of the Jacobi-Stirling numbers. The numbers of the second kind LS_n^j satisfy

LS_n^j=LS_(n-1)^(j-1)+j(j+1)LS_(n-1)^j,

while the numbers of the first kind Lc_n^j satisfy

Lc_n^j=Lc_(n-1)^(j-1)+n(n-1)Lc_(n-1)^j.

Both use the boundary values LS_0^0=Lc_0^0=1 and zero when exactly one index is zero. They are connection coefficients for polynomial bases associated with the Legendre differential operator, whose polynomial eigenfunctions are the Legendre polynomials.

See also

Jacobi-Stirling Numbers, Legendre Polynomial, Stirling Number of the First Kind, Stirling Number of the Second Kind

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References

Andrews, G. E.; Egge, E. S.; Gawronski, W.; and Littlejohn, L. L. "The Jacobi-Stirling Numbers." J. Combin. Theory Ser. A 120, 288-303, 2013. https://doi.org/10.1016/j.jcta.2012.08.006.

Cite this as:

Weisstein, Eric W. "Legendre-Stirling Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Legendre-StirlingNumber.html

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