Related papers: On a series of Ramanujan, dilogarithm values, and …
In this paper, we consider the degenerate multi-poly-Bernoulli numbers and polynomials which are defined by means of the multiple polylogarithms and degenerate versions of the multi-poly-Bernoulli numbers and polynomials. We investigate…
Studying degenerate versions of various special polynomials have become an active area of research and yielded many interesting arithmetic and combinatorial results. Here we introduce a degenerate version of polylogarithm function, called…
An identity by Ramanujan related to the multisection of Bernoulli numbers is revisited. Two alternative approaches are proposed, both relying on the multisection technique. A geometric approach reveals the role played by the symmetries of…
The study of Ramanujan-type congruences for functions specific to additive number theory has a long and rich history. Motivated by recent connections between divisor sums and overpartitions via congruences in arithmetic progressions, we…
A generalization of the classical Lipschitz summation formula is proposed. It involves new polylogarithmic rational functions constructed via the Fourier expansion of certain sequences of Bernoulli--type polynomials. Related families of…
This paper is an introduction to classical polylogarithms and is an expanded version of a talk given by the author at the Motives conference. Topics covered include, monodromy; the polylogarithm local systems; Bloch's constructions of…
We study orthogonal polynomials associated with a continued fraction due to Hirschhorn. Hirschhorn's continued fraction contains as special cases the famous Rogers--Ramanujan continued fraction and two of Ramanujan's generalizations. The…
As is well-known, poly-Bernoulli polynomials are defined in terms of polylogarithm functions. Recently, as degenerate version of such functions and polynomials, degenerate polylogarithm functions were introduced and degenertae…
We present a relationship between the generalized hyperharmonic numbers and the poly-Bernoulli polynomials, motivated from the connections between harmonic and Bernoulli numbers. This relationship yields numerous identities for the…
We evaluate binomial series with harmonic number coefficients, providing recursion relations, integral representations, and several examples. The results are of interest to analytic number theory, the analysis of algorithms, and…
In this lecture notes we try to familiarize the audience with the theory of Bernoulli polynomials; we study their properties, and we give, with proofs and references, some of the most relevant results related to them. Several applications…
We use sums over integer compositions analogous to generating functions in partition theory, to express certain partition enumeration functions as sums over compositions into parts that are $k$-gonal numbers; our proofs employ Ramanujan's…
Ramanujan derived the well known divergent-sum of integers in more than one way. We generalise the informal method to higher powers of the Riemann zeta function through a study of the Eulerian numbers in particular. Within the context of…
Let $\mathbb{A}=\mathbb{F}_{q}[T]$ be the polynomial ring over finite field $\mathbb{F}_{q}$, and $\mathbb{A}_{+}$ be the set of monic polynomials in $\mathbb{A}$. In this paper, we show that a large class of arithmetic functions in…
We give a simple proof and a multivariable generalization of an identity due to E. Alkan concerning a weighted average of the Ramanujan sums. We deduce identities for other weighted averages of the Ramanujan sums with weights concerning…
We construct polylogarithms on families of pointed Riemann surfaces of any genus which describe monodromies of meromorphic connections with simple poles. Furthermore, we show that the polylogaritms are computable as power series in…
In this paper, we give some recurrence formula and new and interesting identities for the poly-Bernoulli numbers and polynomials which are derived from umbral calculus.
We study the orthogonal projection of homogeneous polynomials onto the space of homogeneous polyharmonic polynomials. To do this we derive the decomposition of homogeneous polynomials in terms of the Kelvin transform of derivatives of the…
We show that certain weighted Fibonacci and Lucas series can always be expressed as linear combinations of polylogarithms. In some special cases we evaluate the series in terms of Bernoulli polynomials, making use of the connection between…
In this paper, we consider degenerate poly-Bernoulli numbers and polynomials associated with polylogarithmic function and p-adic invariant integral on Zp. By using umbral calculus, we derive some identities of those numbers and polynomials