An integral transformation relation

Abstract

with I (X+ 1)ao(x-1)a'I as weight function. We call (x-ao)ao ** (x-a,)aJP still the weight function of the generalized Jacobi polynomials of the order p. By introduction of the extended Heine-Stieltjes polynomials p(ao 'aP)(x) = (x-ao)aO * * (X-a,)cPP('O aP)(x) and by denoting fk+1(uvxt/(x-aj0) . . (x-ac)tP)dx, by Bkl, we obtain I BklI | 0.12 We turn finally to the special case, interesting among others for the theory of the transfinite diameter and the Fekete-diameters of a finite degree m, that ai=pi= 1, i=O, * * *, p. Then the generalized Vandermondean Vm(t, 42, * m; ei, e2, * * , em) becomes the usual Vandermondean Vm(~i, 62, , * m)= Ill?i<kgm (ti-ik) and, since Al(x) =A (x), there results B(x) =A'(x).

Locations

• Proceedings of the American Mathematical Society - View - PDF