Perturbations of Jacobi polynomials

Yu. A. Neretin

Sbornik. Mathematics, Tome 197 (2006) no. 11, pp. 1607-1633

Résumé

A family of non-complete orthogonal systems of functions on the ray $[0,\infty]$ depending on three real parameters $\alpha$, $\beta$, $\theta$ is constructed. The elements of this system are piecewise hypergeometric functions with singularity at $x=1$. For $\theta=0$ these functions vanish on $[1,\infty)$ and the system is reduced to the Jacobi polynomials $P_n^{\alpha,\beta}$ on the interval $[0,1]$. In the general case the functions constructed can be regarded as an interpretation of the expressions $P_{n+\theta}^{\alpha,\beta}$. They are eigenfunctions of an exotic Sturm–Liouville boundary-value problem for the hypergeometric differential operator. The spectral measure for this problem is found. Bibliography: 27 titles.