On $d$-orthogonal Tchebychev polynomials, II

Khalfa Douak, Pascal Maroni
1997 Methods and Applications of Analysis  
In Part I, we considered the following problem: Find all d-orthogonal polynomial sequences {Pn}n>0 such that P^ ^ = P n , n = 0,1,2,..., where {Pn }n>0 is the associated polynomial sequence of {Pn}n>0-The resulting polynomials are an extension of the classical Tchebychev polynomials of the second kind. A detailed study was made in the particular case d = 2. The purpose of this paper is to make similar investigations by considering the analogous problem: Find all d-orthogonal polynomial
more » ... {Pn}n>0 such that pW = Q n , n = 0,1,2,..., where Qn(x) := (n + l)~1P4 +1 (a;), n > 0. Here, we show that the resulting polynomials are a natural extension of the classical Tchebychev polynomials of the first kind. The recurrence coefficients and generating function are determined explicitly. As in Part I, a detailed study will be carried out for the case d = 2. A third-order differential equation and a differential-recurrence relation are given. Again, in the particular case, integral representations are obtained of the linear functionals with respect to which the resulting polynomials are 2-orthogonal.
doi:10.4310/maa.1997.v4.n4.a3 fatcat:p5ilb5w3p5dsncjhhvu53j4ghu