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Jacobi-Stirling Numbers

The Jacobi-Stirling numbers of the second kind JS_n^j(z) are the polynomial sequence defined by

JS_n^j(z)=JS_(n-1)^(j-1)(z)+j(j+z)JS_(n-1)^j(z),

for n,j>=1, with JS_0^0(z)=1 and JS_n^0(z)=JS_0^j(z)=0 for positive integers n and j. The companion Jacobi-Stirling numbers of the first kind Jc_n^j(z) satisfy

Jc_n^j(z)=Jc_(n-1)^(j-1)(z)+(n-1)(n-1+z)Jc_(n-1)^j(z),

with the same boundary conditions. At z=1, the two families specialize to the Legendre-Stirling numbers of the second and first kinds, respectively; this simply sets z=1 in the two recurrences. The name "Jacobi" refers to the classical second-order Jacobi differential operator, whose polynomial eigenfunctions are the Jacobi polynomials. The Jacobi-Stirling numbers arise as connection coefficients for polynomial bases associated with powers of this operator.

See also

Jacobi Polynomial, Legendre-Stirling Number, 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. "Jacobi-Stirling Numbers." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Jacobi-StirlingNumbers.html

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