Sums of products involving power sums of $\varphi(n)$ integers

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 the Fa\`a 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 $\displaystyle \Psi_k(x,n):=\sum_{d|n}\mu(d)d^k S_k(\frac{x}{d}), n\in\mathbb{Z}^+$ which are defined via the M\"obius function $\mu$ and the usual power sum $S_k(x)$ of a real or complex variable $x.$ The power sum $S_k(x)$ is expressible in terms of the well known Bernoulli polynomials by $\displaystyle S_k(x):=\frac{B_{k+1}(x+1)-B_{k+1}(0)}{k+1}.$


Introduction
Singh [1] introduced the power sum Ψ k (x, n) of real or complex variable x and positive integer n defined by the generating function from which he derived the following closed form formula for these power sums: (1 − p 2m−1 ), k = 0, 1, ...
(1.2) where B m are the Bernoulli numbers and p runs over all prime divisors of n. In particular Ψ k (n, n) gives the sum of k−th power of those positive integers which are less then n and relatively prime to n. We will call Ψ k (x, n) as Möbius-Bernoulli power sums. Present work is aimed at describing sums of products of the power sums Ψ k (x, n) via introducing yet another sequence of rational numbers which we shall call as the sequence of Möbius-Bernoulli numbers. The rational sequence {B k } that appears in Eq.(1.2) 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 literature. One such formula is the following [2]:

Möbius-Bernoulli numbers
Definition: We define Möbius-Bernoulli numbers M k (n), k = 0, 1, ... via the generating function We immediately notice from the Eq. (2.1) that the Möbius-Bernoulli numbers are given by Singh [1] has obtained the following identity relating the function Ψ k (x, n) to the Möbius-Bernoulli numbers.
Definition: Let n be a positive integer and k a nonnegative integer. Define higher order Möbius-Bernoulli numbers by which are described by the generating function Note that M 1 k (n) = M k (n) ∀ k = 0, 1, ... Some of the first few higher order Möbius-Bernoulli numbers are given by the following: In this regard, a formula for the higher order Möbius-Bernoulli numbers can be obtained from the following version of the well known Faà di Bruno's formula [3]. Lemma 2.1. Let N be a positive integer and f : R → R be a function of class C k , k ≥ 1. Then is the multiplicity of occurrence of k i in the partition {k 1 , ..., k j } of n of length j and λ(k i )! contributes only once in the above product.
Proof. We use induction on k in proving the result. For k = 1, we see that . This proves that the result is true for k = 1. Let us assume that the formula (2.4) holds for all positive integers ≤ k. Now assume that f is of class C k+1 and consider At this point observe that any partition π ′ of k + 1 can be obtained from a partition π of k by adjoining 1 and let us denote the set of all such partitions of k + 1 by S. Denote by T the set of remaining all partitions of k + 1 where each π ′ is obtained simply by adding 1 to exactly one member of π. In each of these cases one has k + 1 choices of doing so for a fixed π. In the former case for each π ′ ∈ S, |π ′ | = |π| + 1 which happens in the first summation above in (2.5) which reduces to the following In the latter case for each π ′ ∈ T, |π ′ | = |π| and the terms after first summation in (2.5) reduce to  Proof. Observe from Eq.(2.3) that for a positive integer N, the following holds: for all n > 1 where the arithmetic function d|n µ(d) = δ 1n is the Kronecker delta. We have proved that H is an even function of t for n > 1. Thus the coefficient of t 2k−1 in the RHS of Eq.(2.3) (which is precisely M N 2k−1 (n)) vanishes for each k = 1, 2, ...

Remark:
If we extend the definition of higher Möbius-Bernoulli numbers to complex N = 0, the formula (2.9) for M N k (n) is still valid just on replacing N ! (N −j)! by N(N − 1) · · · (N − j + 1) in it. In this regard we note that M N 2k−1 (n > 1) = 0 holds for all k = 1, 2, ... and N ∈ C.

Remark:
The formula (2.9) is not suitable for explicit evaluation of M N k for large k. Because number of partitions of k increases at a faster rate than k. For example number of partitions of 10 is 42, which is the number of terms in the expression for M N 20 . In this regards, it will be good to see a formula for the higher order Möbius Bernoulli numbers which can describe them better than the one we have given above! Remark: If n = p s for some positive integer s and prime p, then the simplest possible formula (2.9) for the higher order Möbius-Bernoulli numbers can be found as follows: where we have utilized the Leibniz product rule for higher order derivatives and B m j is the higher order Bernoulli number given by (see for more details Srivastava and Todorov [4]) Similarly if we take n = p s 1 1 p s 2 2 for some positive integers s 1 , s 2 and distinct primes p 1 , p 2 then These formulas involve products of higher order Bernoulli numbers. So in general, the formulas for M N k (n) involve sums containing product of several higher order Bernoulli numbers and such a formula in the above sense would be complicated and will take the following form:

Sums of products of Möbius-Bernoulli power sums
Having developed the expressions for the Möbius Bernoulli numbers in the previous section, we will now use them in expressing the sums of products of the Möbius-Bernoulli power sums Ψ k (x, n).
Definition: We define sums of products of the Möbius Bernoulli power sums as integers k and N which are described by the generating function The next result evaluates the sums of products Ψ N k (x, n).
Theorem 3.1. For a positive integer N and nonnegative integer k, for all n = 2, 3, ... where S(ℓ, m) are the Stirling numbers of second kind.
Proof. Observe from the generating function for Ψ N k (x) that