Faber polynomials

In mathematics, the Faber polynomials P_m of a Laurent series

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

are the polynomials such that

[math]\displaystyle{ \displaystyle P_m(f)-z^{-m} }[/math]

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, doi:10.2307/2316567, ISSN 0002-9890 
• Faber, Georg (1903), "Über polynomische Entwickelungen", Mathematische Annalen (Springer Berlin / Heidelberg) 57: 389–408, doi:10.1007/BF01444293, ISSN 0025-5831, https://zenodo.org/record/2129496/files/article.pdf 
• Faber, G. (1919), "Über Tschebyscheffsche Polynome." (in German), Journal für die reine und angewandte Mathematik 150: 79–106, ISSN 0075-4102, http://resolver.sub.uni-goettingen.de/purl?GDZPPN00216874X 
• 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, doi:10.2307/2371913, ISSN 0002-9327 
• 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, https://books.google.com/books?id=uZZS0Eeh5RMC 
• Hazewinkel, Michiel, ed. (2001), "Faber polynomials", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page