Approximate Closed-Form Formulas for the Zeros of the Bessel Polynomials

Source abstract

We find approximate expressions x ̃ ( k , n , a ) and y ̃ ( k , n , a ) for the real and imaginary parts of the k th zero z k = x k + i y k of the Bessel polynomial y n ( x ; a ) . To obtain these closed-form formulas we use the fact that the points of well-defined curves in the complex plane are limit points of the zeros of the normalized Bessel polynomials. Thus, these zeros are first computed numerically through an implementation of the electrostatic interpretation formulas and then, a fit to the real and imaginary parts as functions of k , n and a is obtained. It is shown that the resulting complex number x ̃ ( k , n , a ) + i y ̃ ( k , n , a ) is O ( 1 / n 2 ) -convergent to z k for fixed k .