Infinitely many explicit solutions of certain second-order differential equations with an apparent singularity of characteristic exponent −2 are constructed by adjusting the parameter of the multi-indexed Laguerre polynomials.

The multi-indexed Laguerre and Jacobi polynomials form a complete set of orthogonal polynomials. They satisfy second-order differential equations but not three term recurrence relations, because of the 'holes' in their ...