On Certain Sums of Functions of Base B Expansions

arXiv: Number Theory, 2016

The balancing numbers $B_n$ ($n=0,1,\cdots$) are solutions of the binary recurrence $B_n=6B_{n-1}-B_{n-2}$ ($n\ge 2$) with $B_0=0$ and $B_1=1$. In this paper we show several relations about the sums of product of two balancing numbers of the type $\sum_{m=0}^n B_{k m+r}B_{k(n-m)+r}$ ($k>r\ge 0$) and the alternating sum of reciprocal of balancing numbers $\left\lfloor\left(\sum_{k=n}^\infty\frac{1}{B_{l k}}\right)^{-1}\right\rfloor$. Similar results are also obtained for Lucas-balancing numbers $C_n$ ($n=0,1,\cdots$), satisfying the binary recurrence $C_n=6C_{n-1}-C_{n-2}$ ($n\ge 2$) with $C_0=1$ and $C_1=3$. Some binomial sums involving these numbers are also explored.

BBP-type formulas are usually discovered experimentally, one at a time and in specific bases, through computer searches. In this paper, however, we derive explicit digit extraction BBP-type formulas in general binary bases b = 2 12p , for p ∈ Z + and mod (p, 2) = 1. As particular examples, new binary formulas are presented for π √ 3, π √ 3 log 2, √ 3 Cl 2 (π/3) and a couple of other polylogaritm constants. A variant of the formula for π √ 3 log 2 derived in this paper has been known for over ten years but was hitherto unproved. Binary BBP-type formulas for the logarithms of an infinite set of primes and binary BBP-type representations for the arctangents of an infinite set of rational numbers are also presented. Finally, new binary BBP-type zero relations are established.

CENTRAL EUROPEAN SYMPOSIUM ON THERMOPHYSICS 2019 (CEST)

In "Construction of some new families of Apostol-type numbers and polynomials via Dirichlet character and p-adic q-integrals Turk J Math (2018) 42: 557-577", we defined some new families of numbers and polynomials related to the Dirichlet character of a finite abelian group. In this paper, we investigate some properties and relations for these numbers and polynomials with their series representations and generating functions. By using functional equations of generating functions with their differential equations and series representations of these numbers and polynomials, we derive some formulas including recurrence relations, combinatorial numbers, combinatorial sums and special numbers and polynomials. By using p-adic integral method, we also derive p-adic series representations of these series representations of these numbers and polynomials.

Journal of Number Theory, 1990

Extensions and improvements of a recent paper, "On Digit Expansions with Respect to Linear Recurrences" by A. Petho and R. F. Tichy (J. Number Theory 33 (1989), 243-256) are established. Furthermore distribution properties mod 1 of the sequence @a(n)) are investigated, where so(n) denotes the sum-of-digits function with respect to the linear recurrence G. a

Mathematical Methods in the Applied Sciences, 2016

The aim of this article is to define some new families of the special numbers. These numbers provide some further motivation for computation of combinatorial sums involving binomial coefficients and the Euler kind numbers of negative order. We can show that these numbers are related to the well-known numbers and polynomials such as the Stirling numbers of the second kind and the central factorial numbers, the array polynomials, the rook numbers and polynomials, the Bernstein basis functions and others. In order to derive our new identities and relations for these numbers, we use a technique including the generating functions and functional equations. Finally, we not only give a computational algorithm for these numbers, but also some numerical values of these numbers and the Euler numbers of negative order with tables. We also give some combinatorial interpretations of our new numbers.

Journal of Number Theory, 2005

A new family of sequences is proposed. An example of sequence of this family is more accurately studied. This sequence is composed by the integers n for which the sum of binary digits is equal to the sum of binary digits of n 2 . Some structure and asymptotic properties are proved and a conjecture about its counting function is discussed.

We give two simple algorithms for the exact evaluation of the sum S(x) = 0≤n<x;n≡0(mod3) (−1) σ(n) , where σ(n) is the binary digit sum of n and obtain the sharp estimates for x −λ S(x), λ = ln 3 ln 4 .

Journal of the London Mathematical Society, 1998

We consider digital expansions with respect to complex integer bases. We derive precise information about the length of these expansions and the corresponding sum-of-digits function. Furthermore we give an asymptotic formula for the sum-of-digits function in large circles and prove that this function is uniformly distributed with respect to the argument. Finally the summatory function of the sum-of-digits function along the real axis is analyzed.

manuscripta mathematica, 2000

Let s(n) denote the sum of digits of the Zeckendorf representation of n and S q,i (N) = n<N,n≡i modq (−1) s(n). The aim of this paper is to discuss the behaviour of S q,i (N). First it is shown that that the values of S(N) = S 1,0 (N) admit a Gaussian limit law with bounded mean and variance of order log N. Conversely, for q > 1 S q,i (N) (mostly) has a periodic fractal structure. We also prove that S 3,0 (N) ≥ 0 which is an analogue to a well-known result by Newman [14] for binary digit expansions.

Applications of Fibonacci Numbers, 1993