d-Orthogonal Faber polynomials

Abstract

Douak and Maroni stated the following problem:

P1: For fixed non-negative integer r and positive integer d, find all d-orthogonal polynomial sets {P _n } _{n ≥ 0} which satisfy

being the r-th associated polynomial set of {P _n } _n≥0.

They solved it for the particular cases (r, d)=(1, 1) and (r, d)=(1, 2). In this paper, we show that the d-orthogonal Faber polynomials are solution of P1 for generals r and d. We also solve P1 for r=1, 2 and general d with the d-symmetry property as an additional condition. Some properties of the obtained polynomials are singled out.

Keywords:

• d-orthogonal polynomials
• Associated polynomials
• Faber polynomials
• Generatingfunctions
• d-symmetric polynomials
• d-orthogonal Chebyshev polynomials

Acknowledgements

The author thanks Professor Y. Ben Cheikh for many enlightening discussions, and the referee for his/her careful reading of the manuscript and corrections.

Additional information

Notes on contributors

N. Ben Romdhane

Email: [email protected]