Let the region S={(x, y)I,u(x+iy, x-iy) >0) be the interior of Steiner's hypocycloid, where ยต(z, z)=-z 222 +4z 3 +423-l8z2+27 . For each real a>-5/6 an orthogonal system of polynomials p.. n(z, z), m, n>0, can be defined on this region S such that pm,n (z, z)-zmzn has degree less than m+n and ff pr,,.,(z, z) q(z, z) (ยต(z, z))ยฐยฐdx dy=0 S for each polynomial q of degree less than m+n . If z = e i(8+t/YS)+ ei(-a+t/VS)+ e -Zit/vs then, in terms of 8 and t, the functions pm n and ylpm_ l . ,,_ 1 are the regular eigenfunctions of the operator ij2/a8 2 +a2/at2 which remain invariant or change sign, respectively, under the reflections in the edges of a certain equilateral triangle . Two explicit partial differential operators Di and D2 in z and z of orders two and three, respectively, are obtained such that the polynomials p',, are eigenfunctions of Di and D2 . The operators Di and D2 commute and are algebraically independent, and they generate the algebra of all differential operators for which the polynomials p"" ,. are eigenfunctions . If a=0, J, 3/2 or 7/2 then the operator Dl expressed in terms of 8 and t is the radial part of the Laplace-Beltrami operator on certain compact Riemannian symmetric spaces of rank two .
1 . INTRODUCTION
This paper deals with orthogonal polynomials in two variables on a region bounded by a closed three-cusped algebraic curve of fourth degree which is known as Steiner's hypocycloid . The weight function is some power of the fourth degree polynomial which vanishes on this curve . The main result in this paper is the construction of two algebraically independent partial differential operators of orders two and three, respectively, for which these orthogonal polynomials are eigenfunctions . Because of the existence of such operators these polynomials can be considered as a generalization of the classical orthogonal polynomials in one variable .