A formula involving sums of the form _ d(n)f(n) and _ d(n)g(n) is derived, where d(n) is the number of divisors of n, andf(x), g(x) are Hankel transforms of each other. Many forms of such a formula, generally known as Voronoi's summation formula, are known, but we give a more symmetrical formula. Also, the reciprocal relation between f(x) and g(x) is expressed in terms of an elementary kernel, the cosine kernel, by introducing a function of the class L2(0, cc). We use L2-theory of Mellin and Fourier-Watson transformations. Introduction. In 1904 Voronoi [10] published the following general formula: If 7(n) is an arithmetic function andf(x) is continuous and has a finite number of maxima and minima in a a n2~_,na a = a One of the better known special cases of this formula is when 7(n)=d(n), the number of divisors of n, and a(x) = (2/i7)Ko(4i7x1/2) Yo(4i7x1/2), 8(x) = log x + 2y, y being Euler's constant and Y0, Ko denote Bessel functions of second and third kinds respectively, of order zero. This special case is generally known as Voronoi's summation formula. Later, this formula received considerable attention as a result of which many modifications were put forth by A. L. Dixon and W. L. Ferrar [2], J. R. Wilton [13], A. P. Guinand [3] and others. Most of the authors used complex analysis and in all the new forms of the Voronoi formula, the kernel used was a combination of the Bessel functions Yo(x) and Ko(x). Our object in this paper is to obtain a more symmetric and simplified form of Voronoi's formula, which holds under simple conditions. We state below the main result. First, a definition, due to Miller [6] and Guinand [4]. Received by the editors August 29, 1968 and, in revised form, February 18, 1969. AMS 1970 subject classifications. Primary 1OA20; Secondary 1OA20.