Sums of Powers of Integers

Abstract

and so on. Many different writers have worked on the problem of finding simple formulas for Skn, and many different methods have been used; see [3], [5], [6], [9], [10] for just a small sampling of recent articles. The formulas for Tk n are certainly less well known. In the present paper we use generating functions to find new recurrences for Skf7(a, d) and Tk n(a,d). We also show how Skn(a,d) and Tk n(a,d) can be determined from Sk_ln(a,d) and Tk_lr2(a, d), respectively, and we show how Skn(a, d) and Tkn(ay d) can be expressed in terms of Bernoulli numbers. One of the main results is a new "lacunary" recurrence formula for Skn with gaps of 6 (Theorem 3.1); that is, we can use the formula to find Smn for m = 0,1,..., 5; then, using only Smn, we can find S6+mt„; then, using only Smn and S6+nhn, we can find Si2+m3„, and so on. There is a similar recurrence for Tk^n. There are several motivations for this paper: (1) A recent article by Wiener [10] dealt with equation (1.1) and generalized some well-known properties of Skt„. We show how the formulas of [10] can be derived very quickly and how they can be extended. (2) In a recent article by Howard [4], formulas were found which connected Bernoulli numbers to Tkt„. Evidently, the properties of Tk^n are not well known, so the results of [4] are a stimulus to study Tk and Tkt„(a, d) in some detail. (3) The new lacunary recurrences mentioned above are useful and easy to use, and (in the writer's opinion) they are of considerable interest. In Section 3 we illustrate the formulas by computing S6n, Sl2}„, and^g^; in Section 6 we compute TAn,Tl0n, mdT16n. (4) Perhaps the main purpose of the paper is to show how generating functions provide a simple, unified approach to the study of sums of powers of integers. The many, and often repetitious