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Faber Polynomial

Let

f(z)=z+a_1+a_2z^(-1)+a_3z^(-2)+...
(1)
=zsum_(n=0)^(infty)a_nz^(-n)
(2)
=zg(1/z)
(3)

be a Laurent polynomial with a_0=1. Then the Faber polynomial P_m(f) in f(z) of degree m is defined such that

P_m(f)=z^m+c_(m1)z^(-1)+c_(m2)z^(-2)+...=z^m+G_m(1/z),
(4)

where

G_m(x)=sum_(n=1)^inftyc_(mn)x^n
(5)

(Schur 1945). Writing

[g(x)]^m=sum_(k=0)^inftya_(mk)x^l
(6)

for m=1, 2, ... gives the relationship

a_(m,m+n)=c_(mn)+a_(m1)c_(m-1,n)+a_(m2)c_(m-2,n) +...+a_(m,m-1)c_(1n).
(7)

connecting a_(mn) and c_(mn).

This polynomial can be used to calculate the number of lattice paths from a point (r,0) to a point (a,b) that remain below the line y=cx.

See also

Lattice Path

Explore with Wolfram|Alpha

References

Gessel, I. M. Ree, S. "Lattice Paths and Faber Polynomials." In Advances in Combinatorial Methods and Applications to Probability and Statistics (Ed. N. Balakrishnan). Boston, MA: Birkhäuser, 1997.Pommerenke, C. "Über die Faberschen Polynome schlichter Funktionen." Math. Z. 85, 197-208, 1964.Schiffer, M. "Faber Polynomials in the Theory of Univalent Functions." Bull. Amer. Math. Soc. 54, 503-517, 1948.Schur, I. "On Faber Polynomials." Amer. J. Math. 67, 33-41, 1945.

Referenced on Wolfram|Alpha

Faber Polynomial

Cite this as:

Weisstein, Eric W. "Faber Polynomial." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FaberPolynomial.html

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