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In mathematics, the Faber polynomials \( P_m \) of a Laurent series

\( \displaystyle f(z)=z^{-1}+a_0+a_1z+\cdots \)

are the polynomials such that

\( \displaystyle P_m(f)-z^{-m} \)

vanishes at z=0. They were introduced by Faber (1903, 1919) and studied by Grunsky (1939) and Schur (1945).
References

Curtiss, J. H. (1971), "Faber Polynomials and the Faber Series", The American Mathematical Monthly (Mathematical Association of America) 78 (6): 577–596, ISSN 0002-9890, JSTOR 2316567
Faber, Georg (1903), "Über polynomische Entwickelungen", Mathematische Annalen (Springer Berlin / Heidelberg) 57: 389–408, doi:10.1007/BF01444293, ISSN 0025-5831
Faber, G. (1919), "Über Tschebyscheffsche Polynome." (in German), Journal für die reine und angewandte Mathematik 150: 79–106, ISSN 0075-4102, JFM 47.0315.01
Grunsky, Helmut (1939), "Koeffizientenbedingungen für schlicht abbildende meromorphe Funktionen", Mathematische Zeitschrift 45 (1): 29–61, doi:10.1007/BF01580272, ISSN 0025-5874
Schur, Issai (1945), "On Faber polynomials", American Journal of Mathematics 67: 33–41, ISSN 0002-9327, JSTOR 2371913, MR 0011740
Suetin, P. K. (1998) [1984], Series of Faber polynomials, Analytical Methods and Special Functions, 1, New York: Gordon and Breach Science Publishers, ISBN 978-90-5699-058-9, MR 1676281
Suetin, P. K. (2001), "f/f038010", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104

Mathematics Encyclopedia

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