โ Scribed by Tom H Koornwinder
No coin nor oath required. For personal study only.
Let for a, j?, y > -1, a+y+3/2 > 0, /l+y+3/2 > 0 and n > k > 0 the orthogonal polynomials pE:fy( u, V) be defined as polynomials in u and o with "highest" term un-kvX which are obtained by orthogonalization of the sequence 1, u, V, uz, uw, ~2, us, u2u, . . . with respect to the weight function (1 -u+v)"( 1 +u+u)@(uz-4v)Y on the region bounded by the lines 1 -u+ v = 0 and 1 + u + w = 0 and by the parabola uz-4v=O.
Two explicit linear partial differential operators Dfp.7 and Dzp*Y of orders two and four, respectively, are obtained such that the polynomials pz;$Y(u, V) are eigenfunctions of [email protected] and DC@*?'. It is proved that if a differential operator D has the polynomials pz$Y( u, v as eigenfunctions then D can be expressed in ) one and only one way as a polynomial in D, a*fl*Y and D2p.y. The special case y= -+ can be reduced to Jacobi polynomials by the identity pz;f-*(x+y, xy)=oonst. (P~~'(x)P~~)(y)+P~~)(x)P~~)(y)).
F or certain values of oc, @, y and in terms of the coordinates s, t, where u=cos s+cos t, v=cos s cos t, the operator Dfp*Y is the radial part of the Lapllace-Beltrami operator on certain compact Riemannian symmetric spaces of rank two.
ERDI~LYI [3,$10.6]). KRALL and SCHEFFER [5] generalized this property to the case of orthogonal polynomials in two variables as follows.
๐ SIMILAR VOLUMES
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,