Note on two extensions of the classical formula for sums of powers on arithmetic progressions

where Bm(x) is the mth Bernoulli polynomial. Since the time of James Bernoulli (), several methods have been developed to find such sums, trying in many occasions to obtain different generalizations. For instance, Kannappan and Zhang [] (see also the references therein) have used Cauchy’s equation to prove (), when the monomial function xm is replaced by a polynomial of degree m. Some q-analogues of formula () can be found in Guo and Zeng [] and the references therein. On the other hand, the sums in () can also be computed by means of the forward differences of the monomial function ψm(u) = um, u ∈R. We actually have (see, for instance,


Introduction
Let N be the set of nonnegative integers and N + = N \ {}. Throughout this note, we assume that m, n ∈ N, x ∈ R, and that f : R → R is an arbitrary function. The kth forward differences of f are recursively defined by  f (x) = f (x),  f (x) = f (x + )f (x), and The starting point of this note is the following classical formula for sums of powers: where m ∧ n = min(m, n). For x = , formula () can be written in terms of the Stirling numbers of the second kind S(m, k) defined as Computationally, formulas () and () are equivalent in the sense that the computation of a sum of n +  terms is reduced to the computation of a polynomial in n of degree m + . However, () can easily be derived from () as follows. Suppose that (P m (x)) m≥ is a sequence of polynomials satisfying for a certain constant c m only depending upon m. Then we have from (), (), and formula () below The Bernoulli polynomials satisfy () with c m = m + . However, one can construct other sequences of polynomials (P m (x)) m≥ fulfilling () (in this respect, see Luo et al. []). For this reason, we will extend formula () rather than (). This is done in Theorem . below by means of a simple identity involving binomial mixtures.

Main results
Let S n = (S n (t),  ≤ t ≤ ) be a stochastic process such that S n (t) has the binomial law with parameters n and t, i.e., and let T be a random variable taking values in [, ] and independent of S n . The random variable S n (T), obtained by randomizing the success parameter t by T, is called a binomial mixture with mixing random variable T (see [] and the references therein). As follows from (), the probability law of S n (T) is given by where E stands for mathematical expectation. Our first main result is the following.

Theorem . With the preceding notations, we have
Let U be a random variable having the uniform distribution on (, ). Observe that In the terminology of binomial transforms (see, for instance, Mu [] and the references therein), identity () means that (f (x + k)) k≥ is the binomial transform of ( k f (x)) k≥ . In this sense, Theorem . appears as a generalization of (). Every choice of the function f and the random variable T in Theorem . gives us a different binomial identity. Whenever the probability density of T includes the uniform density on (, ) as a particular case, we are able to obtain a different extension of formula (). In this respect, we give the following two corollaries of Theorem ..

Corollary . For any p >  and q > , we have
Finally, recall that the discrete Cesàro operator C is defined as We denote by C j the j iterate of C, j ∈ N + (see Galaz and Solís [] and Adell and Lekuona [] for the asymptotic behavior of such iterates, as j → ∞).
Corollary . For any j ∈ N + , we have Observe that both corollaries extend formula () by choosing f = ψ m and p = q =  in Corollary ., and f = ψ m and j =  in Corollary ..

The proofs
Proof of Theorem . Let t ∈ [, ]. We have from () and () Thus, it suffices to replace t by the random variable T and then to take expectations.
Proof of Corollary . Let T be a random variable having the beta density As in (), we have whenever r > -p and s > -q. Hence, applying Theorem ., we get The conclusion follows from () and the well-known formulas Proof of Corollary . Let j ∈ N + . The following formula for the j iterate of the discrete Cesàro operator C was shown by Hardy [], Section II., A probabilistic representation of () can be built as follows (see [] for more details). Let (U k ) k≥ be a sequence of independent identically distributed random variables having the uniform distribution on (, ), and denote T j = U  · · · U j . It turns out (cf. [], Lemma .) that the probability density of T j is given by On the other hand, we see that () Therefore, the conclusion follows by choosing T = T j in Theorem . and taking into account ()-().