  A quadrature formula is a formula computing a definite integration by
evaluation at finite points. The existence of certain quadrature formulas for
orthogonal polynomials is related to interesting problems such as Waring's
problem in number theory and spherical designs in algebraic combinatorics. Sawa
and Uchida proved the existence and the non-existence of certain rational
quadrature formulas for the weight functions of certain classical orthogonal
polynomials. Classical orthogonal polynomials belong to the Askey-scheme, which
is a hierarchy of hypergeometric orthogonal polynomials. Thus, it is natural to
extend the work of Sawa and Uchida to other polynomials in the Askey-scheme. In
this article, we extend the work of Sawa and Uchida to the weight function of
the Bessel polynomials. In the proofs, we use the Riesz--Shohat theorem and
Newton polygons. It is also of number theoretic interest that proofs of some
results are reduced to determining the sets of rational points on elliptic
curves.
