Recursive Computation of Logarithmic Derivatives, Ratios, and Products of Spheroidal Harmonics and Modified Bessel Functions and Applications

Abstract

Spheroidal harmonics and modified Bessel functions have wide applications in scientific and engineering computing. Recursive methods are developed to compute the logarithmic derivatives, ratios, and products of the prolate spheroidal harmonics ( $$P_n^m(x)$$ , $$Q_n^m(x)$$ , $$n\ge m\ge 0$$ , $$x>1$$ ), the oblate spheroidal harmonics ( $$P_n^m(ix)$$ , $$Q_n^m(ix)$$ , $$n\ge m\ge 0$$ , $$x>0$$ ), and the modified Bessel functions ( $$I_n(x)$$ , $$K_n(x)$$ , $$n\ge 0$$ , $$x>0$$ ) in order to avoid direct evaluation of these functions that may easily cause overflow/underflow for high degree/order and for extreme argument. Stability analysis shows the proposed recursive methods are stable for realistic degree/order and argument values. Physical examples in electrostatics are given to validate the recursive methods.