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Sums of Products Involving Power Sums of Integers

DOI: 10.1155/2014/158351

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Abstract:

A sequence of rational numbers as a generalization of the sequence of Bernoulli numbers is introduced. Sums of products involving the terms of this generalized sequence are then obtained using an application of Faà di Bruno's formula. These sums of products are analogous to the higher order Bernoulli numbers and are used to develop the closed form expressions for the sums of products involving the power sums which are defined via the M?bius function μ and the usual power sum of a real or complex variable . The power sum is expressible in terms of the well-known Bernoulli polynomials by . 1. Introduction Singh [1] introduced the power sum of real or complex variable and positive integer defined by the generating function from which he derived the following closed form formula for these power sums: for all where are the Bernoulli numbers and runs over all prime divisors of . In particular, gives the sum of th power of those positive integers which are less than and relatively prime to . We will call as M?bius-Bernoulli power sums. Present work is aimed at describing sums of products of the power sums via introducing yet another sequence of rational numbers which we will call as the sequence of M?bius-Bernoulli numbers. The rational sequence that appears in (2) is defined via the generating function , , and was known to Faulhaber and Bernoulli. Many explicit formulas for the Bernoulli numbers are also well known in the literature. One such formula is as follows [2]: The rest of the paper is organized as follows. M?bius Bernoulli numbers are introduced in Section 2 and their sums of products are discussed via Faà di Bruno’s formula. In Section 3 sums of products of power sums of integers are obtained in closed form using sums of products of M?bius Bernoulli numbers. 2. M?bius-Bernoulli Numbers Definition 1. We define M?bius-Bernoulli numbers , , via the generating function We immediately notice from (4) that the M?bius-Bernoulli numbers are given by Note that, for a fixed , the M?bius Bernoulli number is a multiplicative function of . Singh [1] has obtained the following identity relating the function to the M?bius-Bernoulli numbers. , from which we observe that , , and for all , where is Euler’s totient. Use of M?bius Bernoulli numbers is inherent in studies recently done by Alkan [3] on averages of Ramanujan sums which are defined for any complex number and integer by , where . M?bius Bernoulli numbers are also related to Jordan’s totient (a generalization of Euler’s totient) by , where is the square free part of . The notion of Bernoulli polynomials

References

[1]  J. Singh, “Defining power sums of n and φ(n) integers,” International Journal of Number Theory, vol. 5, no. 1, pp. 41–53, 2009.
[2]  H. W. Gould, “Explicit formulas for Bernoulli numbers,” The American Mathematical Monthly, vol. 79, no. 1, pp. 44–51, 1972.
[3]  E. Alkan, “Distribution of averages of Ramanujan sums,” The Ramanujan Journal, vol. 29, pp. 385–408, 2012.
[4]  H. M. Srivastava and P. G. Todorov, “An explicit formula for the generalized Bernoulli polynomials,” Journal of Mathematical Analysis and Applications, vol. 130, no. 2, pp. 509–513, 1988.
[5]  W. P. Johnson, “The curious history of Faà di Bruno's formula,” American Mathematical Monthly, vol. 109, no. 3, pp. 217–234, 2002.

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