scholarly journals Behavior of Digital Sequences Through Exotic Numeration Systems

10.37236/6581 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Julien Leroy ◽  
Michel Rigo ◽  
Manon Stipulanti
Keyword(s):  
Number Theory ◽  
Linear Algebra ◽  
Analytic Number Theory ◽  
Binomial Coefficients ◽  
Periodic Fluctuation ◽  
Sum Of Digits ◽  
Analytic Number ◽  
Summatory Function ◽  
Sum Of Digits Function ◽  
Numeration Systems

Many digital functions studied in the literature, e.g., the summatory function of the base-$k$ sum-of-digits function, have a behavior showing some periodic fluctuation. Such functions are usually studied using techniques from analytic number theory or linear algebra. In this paper we develop a method based on exotic numeration systems and we apply it on two examples motivated by the study of generalized Pascal triangles and binomial coefficients of words.

Titchmarsh's theorem of Hankel transform

2019 ◽  
pp. 11-24
Author(s):  
Mohamed-Ahmed Boudref
Keyword(s):  
Number Theory ◽  
Data Analysis ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Lipschitz Condition ◽  
Analytic Number Theory ◽  
Hankel Transform ◽  
Partial Differential ◽  
Analytic Number ◽  
Bessel Transform

Hankel transform (or Fourier-Bessel transform) is a fundamental tool in many areas of mathematics and engineering, including analysis, partial differential equations, probability, analytic number theory, data analysis, etc. In this article, we prove an analog of Titchmarsh's theorem for the Hankel transform of functions satisfying the Hankel-Lipschitz condition.

scholarly journals A Survey of Some Recent Developments on Higher Transcendental Functions of Analytic Number Theory and Applied Mathematics

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2294
Author(s):  
Hari Mohan Srivastava
Keyword(s):  
Number Theory ◽  
Special Functions ◽  
Applied Mathematics ◽  
Analytic Number Theory ◽  
Review Article ◽  
Mathematical Functions ◽  
Analytic Number ◽  
Recent Developments ◽  
Transcendental Functions ◽  
Mathematical Sciences

Often referred to as special functions or mathematical functions, the origin of many members of the remarkably vast family of higher transcendental functions can be traced back to such widespread areas as (for example) mathematical physics, analytic number theory and applied mathematical sciences. Here, in this survey-cum-expository review article, we aim at presenting a brief introductory overview and survey of some of the recent developments in the theory of several extensively studied higher transcendental functions and their potential applications. For further reading and researching by those who are interested in pursuing this subject, we have chosen to provide references to various useful monographs and textbooks on the theory and applications of higher transcendental functions. Some operators of fractional calculus, which are associated with higher transcendental functions, together with their applications, have also been considered. Many of the higher transcendental functions, especially those of the hypergeometric type, which we have investigated in this survey-cum-expository review article, are known to display a kind of symmetry in the sense that they remain invariant when the order of the numerator parameters or when the order of the denominator parameters is arbitrarily changed.

scholarly journals Output Sum of Transducers: Limiting Distribution and Periodic Fluctuation

10.37236/5026 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Clemens Heuberger ◽  
Sara Kropf ◽  
Helmut Prodinger
Keyword(s):  
Periodic Function ◽  
Error Term ◽  
Fourier Coefficients ◽  
Higher Dimensions ◽  
Expected Value ◽  
Hölder Continuous ◽  
Periodic Fluctuation ◽  
Sum Of Digits ◽  
Sum Of Digits Function ◽  
Random Integer

As a generalization of the sum of digits function and other digital sequences, sequences defined as the sum of the output of a transducer are asymptotically analyzed. The input of the transducer is a random integer in $[0, N)$. Analogues in higher dimensions are also considered. Sequences defined by a certain class of recursions can be written in this framework.Depending on properties of the transducer, the main term, the periodic fluctuation and an error term of the expected value and the variance of this sequence are established. The periodic fluctuation of the expected value is Hölder continuous and, in many cases, nowhere differentiable. A general formula for the Fourier coefficients of this periodic function is derived. Furthermore, it turns out that the sequence is asymptotically normally distributed for many transducers. As an example, the abelian complexity function of the paperfolding sequence is analyzed. This sequence has recently been studied by Madill and Rampersad.

Sign in / Sign up

Export Citation Format

Share Document