Orthogonal Polynomials through Moment Generating Functionals

It is shown that if the linear functional w generates moments $\{ \mu _i \} _{i = 0}^\infty $ through the formula $\mu _i = \langle {w,x^i } \rangle $, $i = 0,1, \cdots $, then the Chebyshev polynomials $\{ p_i (x)\} _{i = 0}^\infty $ are orthogonal in the sense that $\langle {w,p_m p_n } \rangle = 0$ when $m \ne n$. In particular the Cauchy representations of the functionals associated with the Legendre, Jacobi and Bessel polynomials have this property when their action upon these polynomials is defined by a contour integral of sufficiently large radius.