On conjectures by Csordas, Charalambides and Waleffe (Q2790920)

In the present paper the authors studied two conjectures stated by \textit{G. Csordas} et al. [Proc. Am. Math. Soc. 133, No. 12, 3551--3560 (2005; Zbl 1078.33005)] regarding the interlacing property of zeros of special polynomials \( \phi_{n}^{(\alpha,\beta)}(\mu)\) defined by NEWLINE\[NEWLINE \phi_{n}^{(\alpha,\beta)}(\mu)=\! \sum_{k=0}^{\left[ n/2 \right]}\! \left. \frac{d^{2k}}{dx^{2k}} P_{n}^{(\alpha,\beta)}(x) \right|_{x=1} \!\!\!\!\mu^k =\frac{(\alpha\!+\!1)_n}{n!}\! \sum_{k=0}^{\left[ n/2 \right(t]} \frac{(-n)_{2k}(n\!+\!\alpha\!+\!\beta\!+\!1)_{2k}}{(\alpha+1)_{2k}}\! \left(\frac{\mu}{4}\right)^k\!\!, NEWLINE\]NEWLINE \(n=1,2,\dots,\) where the notation~\([a]\) stands for the integer part of the number~\(a\) and \(P_{n}^{(\alpha,\beta)}(x)\) are the Jacobi polynomials.