Nonlinear Integral Equation Formulation of Orthogonal Polynomials

The nonlinear integral equation P (x) = ∫ β α dy w(y)P (y)P (x + y) is investigated. It is shown that for a given function w(x) the equation admits an infinite set of polynomial solutions P (x). For polynomial solutions, this nonlinear integral equation reduces to a finite set of coupled linear algebraic equations for the coefficients of the polynomials. Interestingly, the set of polynomial solutions is orthogonal with respect to the measure xw(x). The nonlinear integral equation can be used to specify all orthogonal polynomials in a simple and compact way. This integral equation provides a natural vehicle for extending the theory of orthogonal polynomials into the complex domain. Generalizations of the integral equation are discussed. PACS numbers: 2.30.Rz, 2.10.Yn, 2.10.Ud There are many ways to specify uniquely a set of orthogonal polynomials. One can specify the domain (α, β) and the measure with respect to which the polynomials are orthogonal and then use the cumbersome Gramm-Schmidt orthogonalization procedure to construct the polynomials. For example, for the domain (−1, 1) and measure (1 − x), the Gramm-Schmidt procedure yields the Chebyshev polynomials Tn(x). Alternatively, one can specify a recursion relation. The linear recursion relation Tn+1(x) = 2xTn(x)− Tn−1(x) along with the initial conditions T0(x) = 1 and T1(x) = x again produces the Chebyshev polynomials. Another approach is to give the differentialequation eigenvalue problem satisfied by the polynomials. The Chebyshev polynomials Tn(x) satisfy the eigenvalue equation (1 − x )y(x) − xy(x) + ny(x) = 0. Stating the generating function is yet another way to specify a set of polynomials. For the Chebyshev polynomials the generating function 1 − xt 1 − 2xt + t2 = ∞