On the zeros of some special functions: differential equations and Nicholson-type formulas.

*(English)*Zbl 0599.33013
Differential equations and their applications, Equadiff 6, Proc. 6th Int. Conf., Brno/Czech. 1985, Lect. Notes Math. 1192, 155-160 (1986).

The formulas in question, due to G. N. Watson (1922), G. H. Hardy (1925), A. L. Dixon and W. L. Ferrar (1930) and, more recently, L. Durand (1975) are typified by
\[
(*)J_{\mu}(z)J_{\nu}(z)+Y_{\mu}Y_{\nu}(z)=
\]

\[ \frac{4}{\pi^ 2}\int^{\infty}_{0}K_{\mu +\nu}(2z \sinh t)[e^{(\mu -\nu)t}\cos \nu \pi +e^{-(\mu -\nu)t}\quad \cos \mu \pi]dt, \] where Re \(z>0\) with the usual notations for Bessel functions. The formula (*), in the case \(\mu =\nu\), is useful in investigating the zeros of Bessel functions, and a related formula leads to an expression for the derivative of such zeros with respect to order. These formulas are generally proved by the methods of complex integration. Here we discuss an alternative approach, using a method pioneered by H. Bateman (1909), and extended by E. L. Ince (1927) for finding integral representations of solutions of linear differential equations. When the method is applied to the fourth-order differential equation satisfied by the left-hand side of (*), the kernel \(K_{\mu +\nu}(2z\) sinh t) is one of two that simplify the succeeding analysis and it is then a routine matter to find the other factor in the integrand. Some unsolved problems are mentioned.

\[ \frac{4}{\pi^ 2}\int^{\infty}_{0}K_{\mu +\nu}(2z \sinh t)[e^{(\mu -\nu)t}\cos \nu \pi +e^{-(\mu -\nu)t}\quad \cos \mu \pi]dt, \] where Re \(z>0\) with the usual notations for Bessel functions. The formula (*), in the case \(\mu =\nu\), is useful in investigating the zeros of Bessel functions, and a related formula leads to an expression for the derivative of such zeros with respect to order. These formulas are generally proved by the methods of complex integration. Here we discuss an alternative approach, using a method pioneered by H. Bateman (1909), and extended by E. L. Ince (1927) for finding integral representations of solutions of linear differential equations. When the method is applied to the fourth-order differential equation satisfied by the left-hand side of (*), the kernel \(K_{\mu +\nu}(2z\) sinh t) is one of two that simplify the succeeding analysis and it is then a routine matter to find the other factor in the integrand. Some unsolved problems are mentioned.