Identity-Type Functions and Polynomials

For a noncommuting product of functions, similar to convolutions, an “identity-type function” leaving a specific function invariant is defined. It is evaluated for any choice of function on which it acts by solving a functional equation. A closed-form representation for the identity-type function of (1+t)−b(b>0) is obtained, which is a solution of a second-order linear differential equation with given boundary conditions. It yields orthogonal polynomials whose graphs are also given. The relevance for solution of boundary value problems by a series and convergence of the series are briefly discussed