Symmetrically coherent pairs of measures date back to their introduction by A. Iserles, P. E. Koch, S. P. Nørsett and J. M. Sanz-Serna [J. Approx. Theory 65, No. 2, 151–175 (1991; Zbl 0734.42016)] and have been studied extensively later on. The introduction of the paper under review contains a nice overview of the development relevant to the case of Gegenbauer-Sobolev orthogonality.
In a paper by W. G. M. Groenevelt [J. Approx. Theory 114, No. 1, 115–140 (2002; Zbl 1008.33006)] all possible types of coherent pairs in the Gegenbauer case have been studied and interlacing properties of zeros have been derived.
In the paper under review, the situation is addressed where the pair of measures is not coherent anymore, but the relation is relaxed to the situation of the existence of a recurrence relation connecting the ordinary Gegenbauer polynomials to the Gegenbauer-Sobolev orthogonal polynomials.
The following inner product is studied \[ \begin{split} \langle f,g\rangle = \int_{-1}^1\,f(x)g(x)(1-x^2)^{\lambda-1/2}dx\\ +\int_{-1}^1\,f'(x)g'(x)\left(\kappa_1+ {\kappa_2\over 1+qx^2}\right)(1-x^2)^{\lambda+1/2}dx \\ +\kappa_2M_q\left[f'\left({-1\over\sqrt{-q}}\right)g'\left({-1\over\sqrt{-q}}\right)+ f'\left({1\over\sqrt{-q}}\right)g'\left({-1\over\sqrt{-q}}\right)\right],\end{split}\tag{1} \] where \[ \lambda>-{1\over 2},\;q\geq -1;\;M_q\geq 0\text{ for }-1<q<0\text{ and }M_q=0\text{ for }q\geq 0 \] and \(q,\kappa_1,\kappa_2\) are chosen such that the integral forms above are positive definite inner products when studied separately.
When \(\kappa_1\not= 0\), the pair of measures (one consisting of the first integral and the other of the second integral together with the \(M_q\)-term) is no longer coherent, but nevertheless—triggered by numerical evidence—the authors succeeded in proving interlacing properties when the parameters satisfy \[ \lambda\geq 0,\;q\geq 1\text{ and }\kappa_2\geq\left[{2\lambda +3\over 2\lambda +2}+2(1+q)\right]\kappa_1>0. \eqno{(2)} \]
Finally, in section 5, several theorems on interlacing zeros are given in what the authors call ‘extremal cases’ arising from (1):
- monic polynomials from \(\lim_{\kappa_1\rightarrow\infty}\,{1\over\kappa_1}\,\langle f,g\rangle\),
- monic polynomials from \(\lim_{M_q\rightarrow\infty}\,\langle f,g\rangle\) where \(\kappa_1=0, \;\kappa_2={\kappa\over M_q}\).
A very interesting and well written paper, that shows an interlacing property when leaving the realm of coherent pairs, but (unfortunately?) at the cost of condition (2) on the parameters. You can’t win them all.