Varying Jacobi-Krall orthogonal polynomials: local asymptotic behaviour and zeros
The author considers polynomials \(P_n^{\alpha,\beta,M_n}\;(n\geq 0)\), orthogonal with respect to the varying inner product \[ (f,g)_n =\int_{-1}^1\,f(x)g(x)(1-x)^{\alpha}(1+x)^{\beta}dx+M_nf(1)g(1),\;\alpha,\beta>-1, \] with varying size of the delta function type mass at \(x=1\) (added to the standard Jacobi measure on \((-1,1)\)). These polynomials are referred to as varying Jacobi-Krall polynomials (cf. \textit{H. L. Krall} [Pennsylvania State College Studies, No. 6, 24 p. (1940; Zbl 0060.19210)] and \textit{A. M. Krall} [Proc. R. Soc. Edinb., Sect. A 87, 271--288 (1981; Zbl 0453.33006)]). After the Introduction (\S1), the layout of the paper is as follows. \S 2: Introduction of the basic properties of the varying Jacobi-Krall orthogonal polynomials for a sequence \(\{M_n\}_n\) of non-negative numbers with \[ \lim_{n\rightarrow\infty}\,M_n\gamma^n=M>0,\;\gamma\in\mathbb R. \eqno{(\ast)} \] \S3: Local asymptotics around the point \(x=1\), consequences for the zeros (including similar results for Gegenbauer-Krall polynomials. \S4: Numerical experiments illustrating the analytical results of the previous section. Using the known formula \[ P_n^{\alpha,\beta,M_n}(x)=P_n^{\alpha,\beta}(x)+a_{n-1}P_{n-1}^{\alpha+1,\beta}(x), \] with explicitly given \(a_{n-1}\) and \(P_n^{\alpha,\beta}\) the Jacobi polynomials with weight \((1-x)^{\alpha}(1+x)^{\beta}\) (cf. for instance [\textit{P. G. Nevai}, ``Orthogonal polynomials'', Mem. Am. Math. Soc. 213, 185 p. (1979; Zbl 0405.33009)]), the main result of the paper is Theorem 1. Let \(z_{\alpha}(x)=x^{-\alpha}J_{\alpha}(x)\) (where \(J_{\alpha}\) is the Bessel function of the first kind), then the Mehle-Heine type asymptotic formula for \(P_n^{\alpha,\beta,M_n}\) with \(M_n\) as in \((\ast)\), is given by \[ \lim_{n\rightarrow\infty}\,{2^nP_n^{\alpha,\beta,M_n}(\cos{(x/n)})\over n^{\alpha+1/2}}= \begin{cases} -{\sqrt{\pi}\over 2^{\beta}}x^2z_{\alpha+2}(x), &\text{if }\gamma<2\alpha+2,\\ {\sqrt{\pi}\over 2^{\beta}}(z_{\alpha+2}(x)+a_{\alpha,\beta,M}z_{\alpha+1}(x)), &\text{if }\gamma=2\alpha+2,\\ {\sqrt{\pi}\over 2^{\beta}}z_{\alpha+2}(x), &\text{if }\gamma>2\alpha+2, \end{cases} \] uniformly on compact subsets of \(\mathbb C\), where \[ a_{\alpha,\beta,M}=-{2(\alpha+1)M\over M+2^{\alpha+\beta+1}\Gamma(\alpha+2)\Gamma(\alpha+1)}. \] As a corollary to this result, information is obtained about the asymptotic properties of the \(i\)th positive zero of the varying Jacobi-Krall polynomial for the three cases of values for \(\gamma\) as above (\(\gamma<2\alpha+2\): limit \(0\) for \(i=1\) and \(j_{\alpha+2,i-1}\) for \(i\geq 2\); for \(\gamma>2\alpha+2\): limit \(j_{\alpha,i}\) for all \(i\) - \(j_{\alpha,i}\) being the \(i\)th positive zero of \(J_{\alpha}\)).
scientific article; zbMATH DE number 6084538