Transcendence Criterion Research Articles

Transcendence criterion with $$(\beta ,{\mathcal {A}})$$-representations in some quadratic integer bases

Transcendence criterion with $$(\beta ,{\mathcal {A}})$$-representations in some quadratic integer bases

Principal Solutions of Recurrence Relations and Irrationality Questions in Number Theory

We apply the theory of disconjugate linear recurrence relations to the study of irrational quantities in number theory. In particular, for an irrational number associated with solutions of linear three-term recurrence relations, we show that there exists a four-term linear recurrence relation whose solutions show that the number has an irrational square if and only if the four-term recurrence relation has a principal solution of a certain type. The result is extended to higher-order recurrence relations, and a transcendence criterion can also be formulated in terms of these principal solutions. The method generates new series expansions of positive integer powers of ζ(3) and ζ(2) in terms of Apéry’s now classic sequences.

On simultaneous approximation of algebraic numbers

Let $\Gamma\subset \bar{\mathbb Q}^{\times}$ be a finitely generated multiplicative group of algebraic numbers. Let $\alpha_1,\ldots,\alpha_r\in\bar{\mathbb Q}^\times$ be algebraic numbers which are $\mathbb{Q}$-linearly independent and let $\epsilon>0$ be a given real number. One of the main results that we prove in this article is as follows; There exist only finitely many tuples $(u, q, p_1,\ldots,p_r)\in\Gamma\times\mathbb{Z}^{r+1}$ with $d = [\mathbb{Q}(u):\mathbb{Q}]$ for some integer $d\geq 1$ satisfying $|\alpha_i q u|>1$, $\alpha_i q u$ is not a pseudo-Pisot number for some integer $i\in\{1, \ldots, r\}$ and $$ 0<|\alpha_j qu-p_j|<\frac{1}{H^\epsilon(u)|q|^{\frac{d}{r}+\varepsilon}} $$ for all integers $j = 1, 2,\ldots, r$, where $H(u)$ is the absolute Weil height. In particular, when $r =1$, this result was proved by Corvaja and Zannier in [3]. As an application of our result, we also prove a transcendence criterion which generalizes a result of Han\v{c}l, Kolouch, Pulcerov\'a and \v{S}t\v{e}pni\v{c}ka in [4]. The proofs rely on the clever use of the subspace theorem and the underlying ideas from the work of Corvaja and Zannier.

Remark on the transcendence of real numbers generated by Thue–Morse along squares

In 1929, Mahler proved that the real number generated by Thue–Morse sequence is transcendental. Later, Adamczewski and Bugeaud gave a different proof of the transcendence of this number using a combinatorial transcendence criterion. Moreover, Kumar and Meher gave the generalization of the combinatorial transcendence criterion under the subspace Lang conjecture. In this paper, we prove under the subspace Lang conjecture that the real number generated by Thue–Morse along squares is transcendental by using the combinatorial transcendence criterion of Kumar and Meher.

Palindromic p-adic continued fractions

The aim of this paper is to establish new transcendence criteria of p-adic continued fractions. We prove that a p-adic number whose sequence of partial quotients is bounded in Qp and begins with arbitrarily long palindromes is either quadratic or transcendental.

Irrationality and Transcendence Criteria for Infinite Series in Isabelle/HOL

ABSTRACT We give an overview of our formalizations in the proof assistant Isabelle/HOL of certain irrationality and transcendence criteria for infinite series from three different research papers: by Erdős and Straus, Hančl, and Hančl and Rucki. Our formalizations in Isabelle/HOL can be found on the Archive of Formal Proofs. Here we describe selected aspects of the formalization and discuss what this reveals about the use and potential of Isabelle/HOL in formalizing modern mathematical research, particularly in these parts of number theory and analysis.

Differential transcendence criteria for second-order linear difference equations and elliptic hypergeometric functions

We develop general criteria that ensure that any non-zero solution of a given second-order difference equation is differentially transcendental, which apply uniformly in particular cases of interest, such as shift difference equations, q-dilation difference equations, Mahler difference equations, and elliptic difference equations. These criteria are obtained as an application of differential Galois theory for difference equations. We apply our criteria to prove a new result to the effect that most elliptic hypergeometric functions are differentially transcendental.

Generalized Lambert series and arithmetic nature of odd zeta values

AbstractIt is pointed out that the generalized Lambert series $\sum\nolimits_{n = 1}^\infty {[(n^{N-2h})/(e^{n^Nx}-1)]} $ studied by Kanemitsu, Tanigawa and Yoshimoto can be found on page 332 of Ramanujan's Lost Notebook in a slightly more general form. We extend an important transformation of this series obtained by Kanemitsu, Tanigawa and Yoshimoto by removing restrictions on the parameters N and h that they impose. From our extension we deduce a beautiful new generalization of Ramanujan's famous formula for odd zeta values which, for N odd and m &gt; 0, gives a relation between ζ(2m + 1) and ζ(2Nm + 1). A result complementary to the aforementioned generalization is obtained for any even N and m ∈ ℤ. It generalizes a transformation of Wigert and can be regarded as a formula for ζ(2m + 1 − 1/N). Applications of these transformations include a generalization of the transformation for the logarithm of Dedekind eta-function η(z), Zudilin- and Rivoal-type results on transcendence of certain values, and a transcendence criterion for Euler's constant γ.

A transcendence criterion for Cantor series

A transcendence criterion for Cantor series

Transcendental continued β-fraction with quadratic pisot basis over Fq((x-1))

Let Fq be a finite field and Fq((x-1)) is the field of formal power series with coefficients in Fq. Let ??Fq((x-1)) be a quadratic Pisot series with deg(?) = 2. We establish a transcendence criterion depending on the continued ?-fraction of one element of Fq((x-1)).

Transcendence Criteria with Negative Base

Transcendence Criteria with Negative Base

Transcendental p-adic continued fractions

We establish a new transcendence criterion of $p$-adic continued fractions which are called Ruban continued fractions. By this result, we give explicit transcendental Ruban continued fractions with bounded $p$-adic absolute value of partial quotients. This is $p$-adic analogy of Baker's result. We also prove that $p$-adic analogy of Lagrange Theorem for Ruban continued fractions is not true.

Number of digit changes in $${\beta}$$ β -expansion of unity in $${\mathbb{F}_{q}((x^{-1}))}$$ F q ( ( x - 1 ) )

Let \({\beta}\) be an algebraic power series over a finite field \({\mathbb{F}_{q}}\) such that \({\beta}\) is not Pisot or Salem series. We study the asymptotic behaviour of the function that counts the number of digit changes in the \({\beta}\)-expansion of 1. As a consequence, we get a transcendence criterion in \({\mathbb{F}_{q}(x)}\) .

A transcendence criterion for continued fraction expansions in positive characteristic

We exhibit a family of transcendental continued fractions of formal power series over a finite field through some specific irregularities of its partial quotients.

Continued Fractions and Transcendence of Formal Power Series Over a Finite Field

The aim of this paper was to establish a new transcendence criterion for a class of quasi-periodic continued fractions of formal power series over a finite field depending only on the length of specific blocks appearing in the sequence of partial quotients.

Automatic continued fractions are transcendental or quadratic

We establish new combinatorial transcendence criteria for continued fraction expansions. Let $\alpha = [0; a_1, a_2,...]$ be an algebraic number of degree at least three. One of our criteria implies that the sequence of partial quotients $(a_{\ell})_{\ell \ge 1}$ of $\alpha$ cannot be generated by a finite automaton, and that the complexity function of $(a_{\ell})_{\ell \ge 1}$ cannot increase too slowly.

Applications of a transcendence criterion for reciprocal sums of binary linear recurrences

Abstract Let {R n } n≥0 be a binary recurrence defined by R n+2 = A R n+1 + B Rn (n ≥ 0), where A, B, R 0, R 1 are integers and Δ = A 2 + 4B &gt; 0 is not a perfect square. We show a necessary and sufficient condition for the transcendency of the sum ∑ k ≥ 0 , R r k ≠ - b k a k R r k + b k , $ {\sum _{k\ge 0, {R_{r^k}\ne -b_k}}\frac{a_k}{R_{r^k}+b_k}}, $ where r ≥ 2 is an integer and {a k } k≥0 and {b k } k≥0 are sequences of algebraic numbers satisfying certain growth conditions. For the proof, we apply the transcendence criterion for Mahler type functions of several variables which gives equivalence between the rationality of the function and the algebraicity of its values at algebraic points.

Transcendence in positive characteristic and special values of hypergeometric functions

We prove a simple transcendence criterion suitable for function field arithmetic. We apply it to show the transcendence of special values at non-zero rational arguments (or more generally, at algebraic arguments which generate extension of the rational function field with less than q places at infinity) of the entire hypergeometric functions in the function field (over 𝔽q) context, and to obtain a new proof of the transcendence of special values at non-natural p-adic integers of the Carlitz–Goss gamma function. We also characterize in the balanced case the algebraicity of hypergeometric functions, giving an analog of the result of F. R. Villegas, based on Beukers–Heckman results and E. Landau's method in the classical hypergeometric case.

Transcendence Criterion for Values of Certain Functions of Several Variables

Let \({\Phi_0(\boldmath{z})}\) be the function defined by $$\Phi_0({\boldmath z}) = \Phi _{0}(z_1,\ldots, z_m)=\sum_{k\geq 0}\frac{E_k(z_1^{r^k},\ldots,z_m^{r^k})}{F_k(z_1^{r^k},\ldots,z_m^{r^k})},$$ where \({E_k(\boldmath{z})}\) and \({F_k(\boldmath{z})}\) are polynomials in m variables \({\boldmath{z} = (z_1,\ldots, z_m)}\) with coefficients satisfying a weak growth condition and r ≥ 2 a fixed integer. For an algebraic point \({\boldmath{\alpha}}\) satisfying some conditions, we prove that \({\Phi_{0}(\boldmath{\alpha})}\) is algebraic if and only if \({\Phi_{0}(\boldmath{z})}\) is a rational function. This is a generalization of the transcendence criterion of Duverney and Nishioka in one variable case. As applications, we give some examples of transcendental numbers.

Small value estimates for the multiplicative group

We generalize Gel'fond's transcendence criterion to the context of a sequence of polynomials whose first derivatives take small values on large subsets of a fixed subgroup of the multiplicative group C* of the field of complex numbers.