Related papers: Auxiliary functions in transcendence proofs
Problems in additive number theory related to sum and difference sets, more general binary linear forms, and representation functions of additive bases for the integers and nonnegative integers.
We consider the dynamics associated with an arbitrary semigroup of transcendental entire functions. Fatou-Julia theory is used to investigate the dynamics of these semigroups. Several results of the dynamics associated with iteration of a…
In this paper we prove a functional transcendence statement for the j-function which is an analogue of the Ax-Schanuel theorem for the exponential function. It asserts, roughly, that atypical algebraic relations among functions and their…
The results of difference sequences theory are applied to analytic function theory and Diophantine equations. As a result we have the equation which connects the $n$-th derivative of a function with the difference sequence for the values of…
We derive new reduction formulas for the incomplete beta function and the Lerch transcendent in terms of elementary functions. As an application, we calculate some new integrals. Also, we use these reduction formulas to test the performance…
We prove that entire transcendental holomorphic functions with an omitted value have infinite entropy. A proof for general transcendental entire functions will be given in an upcoming paper.
In this sequel to arXiv:0905.3327, we continue to study the congruence properties of the alternating version of multiple harmonic sums. As contrast to the study of multiple harmonic sums where Bernoulli numbers and Bernoulli polynomials…
This addendum devotes to a detailed proof for the inequality (9.14) in our joint work: Arithmetic exponent pairs for algebraic trace functions and applications, with an appendix by Will Sawin, arXiv:1603.07060 [math.NT], which will appear…
In this technical report, certain interesting classification of arithmetical functions is proposed. The notion of additively decomposable and multiplicatively decomposable arithmetical functions is proposed. The concepts of arithmetical…
Some notes and observations on analytic functions defined on an annulus
We give estimates for the convolution product of an arbitrary number of endlessly continuable functions. This allows us to deal with nonlinear operations for the corresponding resurgent series, e.g. substitution into a convergent power…
In this course of lectures we give an account of the growth theory of subharmonic functions, which is directed towards its applications to entire functions of one and several complex variables.
We generalize and unify the proofs of several results on algebraic in- dependence of arithmetic functions and Dirichlet series by a theorem of Ax on differential Schanuel conjecture. Along the way, we find counter-examples to some results…
In this supplementary appendix we provide additional results, omitted proofs and extensive simulations that complement the analysis of the main text (arXiv:1201.0224).
The Additive Transform of an arithmetic function represents a novel approach to examining the interplay between multiplicative arithmetic function and additive functions. This transform concept introduces a method to systematically generate…
The connection between q-analogs of special functions and representations of quantum algebras has been developed recently. It has led to advances in the theory of q-special functions that we here review.
We prove some extensions of Andrews inequality.
We modify the proof of the basic lemma of a paper of Saks and Zygmund on additive functions of rectangles.
In this paper, we show the existence of a transcendental function $f\in\mathbb{Z}\{z\}$ with coefficients that are almost all bounded such that $f$ and all its derivatives assume algebraic values at algebraic points. Furthermore, we…
Arithmetic differential equations are analogues of algebraic differential equations in which derivative operators acting on functions are replaced by Fermat quotient operators acting on numbers. Now, various remarkable transcendental…