Related papers: Digit sums and generating functions

Functional digraphs are unlabelled finite digraphs where each vertex has exactly one out-neighbor. They are isomorphic classes of finite discrete-time dynamical systems. Endowed with the direct sum and product, functional digraphs form a…

The floor and ceiling functions appear often in mathematics and manipulating sums involving floors and ceilings is a subtle game. Fortunately, the well-known textbook Concrete Mathematics provides a nice introduction with a number of…

We prove that the greedy sum of a direct product of two numeric arrays of complex numbers is equal to the product of the greedy sums of the factors provided that all the mentioned sums exist.

We prove a folklore conjecture concerning the sum-of-digits functions in bases two and three: there are infinitely many positive integers $n$ such that the sum of the binary digits of $n$ equals the sum of the ternary digits of $n$.

We find general solutions to the generating-function equation sum c_q^{(X)} z^q = F(z)^X, where X is a complex number and F(z) is a convergent power series with |F(0)| >0. We then use these results to derive finite expressions containing…

We study a generating function for the sum over fatgraphs with specified valences of vertices and faces, inversely weighted by the order of their symmetry group. A compact expression is found for general (i.e. non necessarily connected)…

We provide algorithms computing power series solutions of a large class of differential or $q$-differential equations or systems. Their number of arithmetic operations grows linearly with the precision, up to logarithmic terms.

A number of constructions in function field arithmetic involve extensions from linear objects using digit expansions. This technique is described here as a method of constructing orthonormal bases in spaces of continuous functions. We…

We give the q-analogue of the sums of the n-th powers of positive integers up to k-1.

In this paper, we study a Dirichlet series generated by powers of harmonic numbers. As an application of these functions, we derive certain series involving harmonic numbers. We also study the analytic properties of these Dirichlet series…

Many combinatorial generating functions can be expressed as combinations of symmetric functions, or extracted as sub-series and specializations from such combinations. Gessel has outlined a large class of symmetric functions for which the…

In this paper, we study the sum of the divisor function over sets with digit restrictions.

We derive generating functions for the ranks of pre-modular categories associated to quantum groups at roots of unity.

The Landau-Selberg-Delange method gives precise asymptotic formulas for the partial sums $\sum_{n \le x} \, a_n$ of a Dirichlet series $\sum_n \, a_n/n^s$ that behaves like a complex power of the Riemann zeta function. However, situations…

We connect Dedekind sums and some formulas for numerical semigroups.

We review some probabilistic properties of the sum-of-digits function of random integers. New asymptotic approximations to the total variation distance and its refinements are also derived. Four different approaches are used: a classical…

This paper focuses on greedy expansions, one possible representation of numbers, and on arithmetical operations with them. Performing addition or multiplication some additional digits can appear. We study bounds on the number of such digits…

The main object of this paper is to give the generalized von mangoldt function using the L-additive function which can help us to make it possible to calculate The Dirichlet series of the arithmetic derivative $\delta$ and Dirichlet series…

A classical result of Dirichlet shows that certain elementary character sums compute class numbers of quadratic imaginary number fields. We obtain analogous relations between class numbers and a weighted character sum associated to a…

Motivated by earlier work of P.~A.~MacMahon and recent contributions of T.~Amdeberhan, G.~E.~Andrews, K.~Ono, A.~Singh, and R.~Tauraso on higher-order partition enumerants, we study a class of $q$-series arising from nested divisor…