On an Arithmetic Convolution
نویسنده
چکیده
The Cauchy-type product of two arithmetic functions f and g on nonnegative integers is defined by (f • g)(k) := ∑k m=0 ( k m ) f(m)g(k −m). We explore some algebraic properties of the aforementioned convolution, which is a fundamental characteristic of the identities involving the Bernoulli numbers, the Bernoulli polynomials, the power sums, the sums of products, and so forth.
منابع مشابه
Arithmetic Convolution Rings
Arithmetic convolution rings provide a general and unified treatment of many rings that have been called arithmetic; the best known examples are rings of complex valued functions with domain in the set of non-negative integers and multiplication the Cauchy product or the Dirichlet product. The emphasis here is on factorization and related properties of such rings which necessitates prior result...
متن کامل18.781 (Spring 2016): Floor and arithmetic functions
2. Arithmetic functions 19 2.1. Arithmetic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2. Multiplicative functions . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3. The Dirichlet convolution . . . . . . . . . . . . . . . . . . . . . . . . 27 2.4. Examples of Dirichlet convolutions . . . . . . . . . . . . . . . . . . . 34 2.5. Möbius inversion . . . . . . . . . ....
متن کاملVector convolution in O(n) steps and matrix multiplication in O(n^2) steps : -)
We observe that if we allow for the use of division and the floor function besides multiplication, addition and subtraction then we can compute the arithmetic convolution of two n-dimensional integer vectors in O(n) steps and perform the arithmetic matrix multiplication of two integer n×n matrices in O(n) steps. Hence, any method for proving superlinear (in the input size) lower bounds for the ...
متن کاملThe Convolution Ring of Arithmetic Functions and Symmetric Polynomials
Inspired by Rearick (1968), we introduce two new operators, LOG and EXP. The LOG operates on generalized Fibonacci polynomials giving generalized Lucas polynomials. The EXP is the inverse of LOG. In particular, LOG takes a convolution product of generalized Fibonacci polynomials to a sum of generalized Lucas polynomials and EXP takes the sum to the convolution product. We use this structure to ...
متن کاملfür Mathematik in den Naturwissenschaften Leipzig Renormalization : A number theoretical model
We analyse the Dirichlet convolution ring of arithmetic number theoretic functions. It turns out to fail to be a Hopf algebra on the diagonal, due to the lack of complete multiplicativity of the product and coproduct. A related Hopf algebra can be established, which however overcounts the diagonal. We argue that the mechanism of renormalization in quantum field theory is modelled after the same...
متن کامل