Quasi-orthogonality with applications to some families of classical orthogonal polynomials.
The authors present some results on the location of the zeros of quasi-orthogonal polynomials. A polynomial \(R_n\), \(\text{ deg} R_n=n\), is quasi-orthogonal of order \(r\) on \([a, b]\) with respect to a positive weight function \(w\) if \(n\geq r\) and \[ \int_a^b x^k R_n (x) w(x)\,dx \begin{cases} =0 &\text{for } k=0,\dots, n-1-r,\\ \neq 0 &\text{for } k=n-r.\end{cases} \] It is claimed that if \(R_n\) is quasi-orthogonal of order \(r\) on \([a, b]\), then \(R_n\) has at least \((n-r)\) distinct zeros in the interval \((a, b)\). This result is applied to the generalized Gegenbauer, Jacobi and Laguerre polynomials. In particular, if we take the generating function \[ (1-2xt+t^2)^{-\lambda}= \sum_{m=0}^\infty C_m^\lambda (x) t^m \] with \(\lambda\in (- {1 \over 2} -j, -{1 \over 2}-j-1)\), \(j\in \{1,\dots, \lfloor {{n_0}\over 2} \rfloor -1\}\), \(n_0\in {\mathbb{N}}\), then the authors show that the Gegenbauer polynomial \(C_{n_0}^\lambda\) is quasi-orthogonal of order \(2j\) with respect to the weight function \((1-x^2)^{\lambda+j- {1\over 2}}\) on \([-1, 1]\) and has at least \((n_0-2j)\) zeros in \((-1, 1)\). Recall that the usual Gegenbauer polynomials \(C_m^\lambda\) have the same generating function with \(\lambda >-{1\over 2}\) only.
scientific article; zbMATH DE number 2055526
scientific article