SUMS OF THE POWERS OF RECIPROCALS OF POLYGONAL NUMBERS

Abstract: We address the question of expressing the sums of the powers of polygonal numbers in closed forms using some basic functions. We obtain explicit expressions for the closed form expressions for the sums of the squares of reciprocals of polygonal numbers, the sums of the cubes of reciprocals of polygonal numbers the sums of the fourth-powers of reciprocals of polygonal numbers. These closed form expressions are composed of digamma function, Riemann zeta function and the Hurwitz zeta function. It has been possible to obtain the general result for the sums of an arbitrary power of reciprocals of square numbers. An outline is given to extend the result to the general case of the sums of the powers of reciprocals of polygonal numbers.