An extended Laha-Lukacs characterization result based on a regression property (Q1372401)

The authors characterize the class of distributions for which \[ E\Biggl(X^2_1-aX_1X_2-bX_1- dX_1X_2X_3 \Biggl|\sum^n_{i=1}X_i\Biggr)= c\qquad\text{a.s.} \] for i.i.d. r.v.'s \(X_1,\dots,X_n\). This gives a characterization of the natural exponential families whose variances are cubic functions of the means.