scholarly journals On the Non-Holonomic Character of Logarithms, Powers, and the $n$th Prime Function

10.37236/1894 ◽  
2005 ◽  
Vol 11 (2) ◽  
Author(s):  
Philippe Flajolet ◽  
Stefan Gerhold ◽  
Bruno Salvy
Keyword(s):  
Differential Equations ◽  
Complex Analysis ◽  
Structure Theorem ◽  
Prime Numbers ◽  
Fractional Powers ◽  
Open Problems ◽  
Naturally Occurring ◽  
Abelian Theorem ◽  
Linear Differential ◽  
Prime Function

We establish that the sequences formed by logarithms and by "fractional" powers of integers, as well as the sequence of prime numbers, are non-holonomic, thereby answering three open problems of Gerhold [El. J. Comb. 11 (2004), R87]. Our proofs depend on basic complex analysis, namely a conjunction of the Structure Theorem for singularities of solutions to linear differential equations and of an Abelian theorem. A brief discussion is offered regarding the scope of singularity-based methods and several naturally occurring sequences are proved to be non-holonomic.

scholarly journals DIFFERENCE ALGEBRAIC RELATIONS AMONG SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS

2015 ◽  
Vol 16 (1) ◽  
pp. 59-119 ◽  
Author(s):  
Lucia Di Vizio ◽  
Charlotte Hardouin ◽  
Michael Wibmer
Keyword(s):  
Differential Equations ◽  
Characteristic Zero ◽  
Galois Theory ◽  
Structure Theorem ◽  
Special Functions ◽  
Algebraic Groups ◽  
Galois Groups ◽  
Linear Differential Equations ◽  
Linear Differential

We extend and apply the Galois theory of linear differential equations equipped with the action of an endomorphism. The Galois groups in this Galois theory are difference algebraic groups, and we use structure theorems for these groups to characterize the possible difference algebraic relations among solutions of linear differential equations. This yields tools to show that certain special functions are difference transcendent. One of our main results is a characterization of discrete integrability of linear differential equations with almost simple usual Galois group, based on a structure theorem for the Zariski dense difference algebraic subgroups of almost simple algebraic groups, which is a schematic version, in characteristic zero, of a result due to Z. Chatzidakis, E. Hrushovski, and Y. Peterzil.

scholarly journals Dual exponential polynomials and a problem of Ozawa

2021 ◽  
pp. 1-19
Author(s):  
Janne Heittokangas ◽  
Katsuya Ishizaki ◽  
Kazuya Tohge ◽  
Zhi-Tao Wen
Keyword(s):  
Differential Equations ◽  
Polynomial Solution ◽  
Linear Differential Equations ◽  
The Other ◽  
Exponential Polynomial ◽  
Exponential Polynomials ◽  
Open Problems ◽  
Duality Property ◽  
Complex Linear ◽  
Linear Differential

Complex linear differential equations with entire coefficients are studied in the situation where one of the coefficients is an exponential polynomial and dominates the growth of all the other coefficients. If such an equation has an exponential polynomial solution $f$ , then the order of $f$ and of the dominant coefficient are equal, and the two functions possess a certain duality property. The results presented in this paper improve earlier results by some of the present authors, and the paper adjoins with two open problems.

Topics in Quaternion Linear Algebra

2014 ◽  
Author(s):  
Leiba Rodman
Keyword(s):  
Linear Algebra ◽  
Complex Analysis ◽  
Dimensional Space ◽  
Applied Mathematics ◽  
Three Dimensional ◽  
Number System ◽  
Graduate Course ◽  
Open Problems ◽  
Unpublished Research ◽  
Reference Tool

Quaternions are a number system that has become increasingly useful for representing the rotations of objects in three-dimensional space and has important applications in theoretical and applied mathematics, physics, computer science, and engineering. This is the first book to provide a systematic, accessible, and self-contained exposition of quaternion linear algebra. It features previously unpublished research results with complete proofs and many open problems at various levels, as well as more than 200 exercises to facilitate use by students and instructors. Applications presented in the book include numerical ranges, invariant semidefinite subspaces, differential equations with symmetries, and matrix equations. Designed for researchers and students across a variety of disciplines, the book can be read by anyone with a background in linear algebra, rudimentary complex analysis, and some multivariable calculus. Instructors will find it useful as a complementary text for undergraduate linear algebra courses or as a basis for a graduate course in linear algebra. The open problems can serve as research projects for undergraduates, topics for graduate students, or problems to be tackled by professional research mathematicians. The book is also an invaluable reference tool for researchers in fields where techniques based on quaternion analysis are used.

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