In this paper D-optimal designs for the weighted polynomial regression mode
l of degree p with efficiency function (1 + x(2))(-n) are presented. Intere
st in these designs stems from the fact that they are equivalent to locally
D-optimal designs for inverse quadratic polynomial models. For the unrestr
icted design space R and p < n, the D-optimal designs put equal masses on p
+ 1 points which coincide with the zeros of an ultraspherical polynomial,
while for p = n they are equivalent to D-optimal designs for certain trigon
ometric regression models and exhibit all the curious and interesting featu
res of those designs. For the restricted design space [-1, 1] sufficient, b
ut not necessary, conditions for the D-optimal designs to be based on p + 1
paints are developed. In this case the problem of constructing (p + 1)-poi
nt D-optimal designs is equivalent to an eigenvalue problem and the designs
can be found numerically. For n = 1 and 2, the problem is solved analytica
lly and, specifically, the D-optimal designs put equal masses at the paints
+/- 1 and at the p - 1 zeros of a sum of n + 1 ultraspherical polynomials.
A conjecture which extends these analytical results ta cases with n an int
eger greater than 2 is given and is examined empirically.