Related papers: On Powers of Some Power Series
Using appropriate power series evaluations, we determine all moments of arbitrary positive powers of the arcsine. As consequences we evaluate several doubly infinite classes of power series involving central binomial coefficients and…
In this paper we compute the sum of the $k$-th powers over any finite commutative unital rings, thus generalizing known results for finite fields, the rings of integers modulo $n$ or the ring of Gaussian integers modulo $n$. As an…
Some class of sums which naturally include the sums of powers of integers is considered. A number of conjectures concerning a representation of these sums is made.
We construct an explicit filtration of the ring of algebraic power series by finite dimensional constructible sets, measuring the complexity of these series. As an application, we give a bound on the dimension of the set of algebraic power…
We investigate compositions of a positive integer with a fixed number of parts, when there are several types of each natural number. These compositions produce new relationships among binomial coefficients, Catalan numbers, and numbers of…
The aim of the present article is to explore the possibilities of representing positive integers as sums of other positive integers and highlight certain fundamental connections between their multiplicative and additive properties. In…
We improve a result of Bennett concerning certain sequences involving sums of powers of positive integers.
Power nonnegative matrices are defined as complex matrices having at least one nonnegative integer power. We exploit the possibility of deriving a Perron Frobenius-like theory for these matrices, obtaining three main results and drawing…
Using a pointwise version of Fej\'{e}r's theorem about Fourier series, we obtain two formulae related to the series representations of positive integral powers of $\pi$. We also check the correctness of our formulae by the applications of…
This paper gives new explicit formulas for sums of powers of integers and their reciprocals.
A family of formal power series, such that its coefficients satisfy a recursion formula, is characterized in terms of the summability, in the sense of J. P. Ramis, of its elements along certain well chosen directions. We describe a set of…
For a fixed integer base $b\geq2$, we consider the number of compositions of $1$ into a given number of powers of $b$ and, related, the maximum number of representations a positive integer can have as an ordered sum of powers of $b$. We…
A general integral expression to transform power series is applied to $\arcsin{x}$ and its positive integer powers. We concentrate on the first to the fourth powers and obtain infinite classes of new power series involving central binomial…
We consider the set of power functions defined on the set of positive real number, and their linear combinations. After recalling some properties of the gamma function, we give two general definitions of derivatives of positive and negative…
We consider the linear combinations of elements of two sequences: the first one a priory given nonnegative sequence and the second random sequence from the unit interval. We investigate the expected value of the smallest natural number such…
We present a variation and generalization of a determinant evaluation of Wilf (math.CO/9809120). His result concerns a matrix whose entries are the coefficients of powers of a given power series; we replace the powers by repeated…
This paper presents expressions for sums of powers of sine and cosine in terms of the basis for the field extension obtained by adjoining the sine or cosine to the field of rational numbers.
We compute two parametric determinants in which rows and columns are indexed by compositions, where in one determinant the entries are products of binomial coefficients, while in the other the entries are products of powers. These results…
In this note, we give examples of formal power series satisfying certain conditions that cannot be realized as Hilbert series of finitely generated modules. This answers to the negative a question raised in a recent article by the second…
Formal power series come up in several areas such as formal language theory , algebraic and enumerative combinatorics, semigroup theory, number theory etc. This paper focuses on the set x R[[x]] consisting of formal power series with zero…