BASIC SETS OF POLYNOMIALS FOR GENERALIZED BELTRAMI AND EULER-POISSON-DARBOUX EQUATIONS AND THEIR ITERATES1 E. P. MILES, JR. AND EUTIQUIO C. YOUNG 1. Introduction. This paper concerns basic sets of polynomial solu- tions for the class of partial differential equations in m variables, m^2, (1) Lj(u) = lDm + (-1) £ DA u = 0, j = 0,1; k = 1,2, • • • , where 2 2 Di = d /dxi + (ai/xi)(d/dxi) with a< 3; 0; i = 1, • • • , m. The iterated operators L* are defined by the relations l7 (u) = Lj[Lj(u)], s = 1, • • • , k - 1. When ax= ■ ■ ■ =aro_i = 0 and am>0, Lj(u)=0 is known as the Beltrami or the Euler-Poisson-Darboux (EPD) equation according as.7 = 0 or j= 1. If am = 0 too, then L0(u) =0 and Lx(u) =0 become the Laplace and wave equations, respectively. Basic sets of polynomial solutions for the Laplace and wave equations have been given in a number of papers [l]-[5]. In [6] Miles and Williams obtained basic sets of polynomials for the Beltrami and EPD equations from their result in [3]. In [7] the result of [3] was extended to form basic sets for the iterated Laplace and wave equations. Here we derive basic sets for (1) from the basic sets given in [7]. The Miles and Williams basic set of homogeneous polynomials of degree n for the £-fold iterated Laplace equation Aku = 0 (A= XXi d2/dx2) may be represented by [(>■-««.)/21 . /,• 4- \a /2l\ r2;+°™ (2)Hai....,am= 2w (-1)1 r , )A(x! ■ • -xm-x) , i=o \ [am/2\ / (2j + am)l where ax, ■ ■ ■ , am are nonnegative integers such that 2^™iat = w and am^2k— 1. In particular, when am = 0, H21,...,am-x,o is harmonic, Presented to the Society, January 24, 1967 under the title Basic sets of polynomials for iterated generalized Beltrami and EPD equations; received by the editors September 6, 1966. 1 This work was supported by NSF research grant GP-817. 981