Summation of Series of Fractions Depending Upon the Roots of the Airy Function

Abstract

The following lines bring an extension of a method indicated by G. N. Watson¹ to the evaluation of sums connected with the roots of the so-called ‘Airy function’ (defined by Watson²). The need for the extension arose in connection with investigations on certain integrals depending upon functions derived from the Airy function. These integrals had been found to give information on statistical problems referring to sets of solutions of the non-linear, one-dimensional diffusion or heat flow equation. The statistical relations and the integrals will be left aside here, and the following note is restricted to the evaluation of certain sums.