Differential operators and the Gegenbauer polynomials: The limit circle case (Q2731572)

The authors consider the Gegenbauer differential equation NEWLINE\[NEWLINE-\bigl[(1-x^2)^{\nu +1/2} y'\bigr]'+ \nu^2(1-x^2)^{\nu- 1/2}y= \lambda(1- x^2)^{\nu -1/2}y,NEWLINE\]NEWLINE where \(x\in(-1,1)\) and \(\nu\in [1/2,3/2)\) while \(\lambda\) stands for a complex-valued parameter, and show that the set of its polynomial solutions (i.e.,the Gegenbauer polynomials) is complete in a certain Hilbert space \(L_w(-1,2)\). To this end, they dwell on the Titchmarsh-Weyl theory of singular differential equations.