On the zeroes of two families of polynomials arising from certain rational integrals (Q812498)

The value of the integral \(\int_0^\infty\frac{dx}{(x^4+2ax^2+1)^{m+1}}\), \(a>-1\) for positive integral \(m\) is expressed by terminating hypergeometric series \(_2F_1\), that is, by a polynomial, say \(P_m(a)\). The coefficients of these polynomials are expressed by means of two families of certain polynomials of variable \(m\). The author shows that these polynomials satisfy certain three-term recurence relations leading to the interlacing roots property like as for orthogonal polynomials. Then he proves the conjecure of Boros, Moll and Shallit saying that all roots of these polynomials lie on the line \(\Re(m)=-\frac{1}{2}\) in the complex plane.