SINGULAR INTEGRAL-OPERATORS WITH INTEGRAL EIGENVALUES AND POLYNOMIAL EIGENFUNCTIONS

The singular integral operator A defined by the formula[InlineEquation not available: see fulltext.] is investigated. In particular it is shown that, if and only if a(x)=1-x 2)1/2[1+α/(1-x)+β/(1+x)], the eigenfunctions of A are polynomials (of degree n=0, 1, 2, 3 ...) and the corresponding eigenvalues are just the nonnegative integers n. Some other related results are also reported, including certain neat integral relationships satisfied by the Tchebichef polynomials of the second kind. © 1979 Società Italiana di Fisica.