An application of Jacobi type polynomials to irrationality measures

  title={An application of Jacobi type polynomials to irrationality measures},
  author={Ari Heimonen and Tapani Matala-aho and K. V{\"a}{\"a}n{\"a}nen},
  journal={Bulletin of The Australian Mathematical Society},
The paper provides irrationality measures for certain values of binomial functions and definite integrals of some rational functions. The results are obtained using Jacobi type polynomials and divisibility considerations of their coefficients. 
A note on pseudo Jacobi polynomials
Abstract The present paper is a study of pseudo-Jacobi polynomials which have been defined on the pattern of Shively’s pseudo-Laguerre polynomials. The paper contains generating functions, RodriguesExpand
Irrationality Measures of log 2 and π/√3
Using a class of polynomials that generalizes Legendre polynmials, this work unify previous works of E. V. Chudnovsky about irrationality measures of log 2 and π/√3. Expand
Approximation of values of the Gauss hypergeometric function by rational fractions
AbstractWe consider a new approach to estimating the irrationality measure of numbers that are values of the Gauss hypergeometric function. Some of the previous results are improved, in particular,Expand
Effective measures of irrationality for certain algebraic numbers
In this paper, we derive a number of explicit lower bounds for rational approximation to certain cubic irrationalities, proving, for example, that 1 _,< for any non-zero integers p and q. A number ofExpand
Diophantine approximations of the number π by numbers from the field ℚ(√3)
We prove an estimate of the irrationality measure of any nonzero number of the form r1π + r2π/√3, r1, r2 ∊ ℚ(√3)
Estimates for approximations to logarithms of rational numbers by rational numbers and quadratic irrationalities are established.
On the irrationality measure of certain numbers
The paper presents upper estimates for the irrationality measure and the non-quadraticity measure for the numbers $\alpha_k=\sqrt{2k+1}\ln\frac{\sqrt{2k+1}-1}{\sqrt{2k+1}+1}, \ k\in\mathbb N.$
Simultaneous Approximation to Pairs of Algebraic Numbers
The author uses an elementary lemma on primes dividing binomial coefficients and estimates for primes in arithmetic progressions to sharpen a theorem of J. Rickert on simultaneous approximation toExpand
Solving certain Thue equations with the aid
The paper gives a computational method for solving Diophantine equations aa;^ - 6^^ = K for certain a, 6 and A". The method is based on an effective irrationality measure result for \/a/b and on theExpand
Diophantine approximations for a constant related to elliptic functions
This paper is devoted to the study of rational approximations of the ratio h…l†=o…l†, where o…l† and h…l† are the real period and real quasi-period, re- spectively, of the elliptic curve y 2 ˆ x…x yExpand


On irrationality measures of the values of Gauss hypergeometric function
The paper gives irrationality measures for the values of some Gauss hypergeometric functions both in the archimedean andp-adic case. Further, an improvement of general results is obtained in the caseExpand
Number Theoretic Applications of Polynomials with Rational Coefficients Defined by Extremality Conditions
It is well known that classes of polynomials in one variable defined by various extremality conditions play an extremely important role in complex analysis. Among these classes we find orthogonalExpand
On the arithmetic nature of definite integrals of rational functions.
A. Baker's theorems on linear forms in the logarithms of algebraic numbers imply information on the arithmetic nature of definite integrals of rational functions. This paper pro vides a convenientExpand
Effective irrationality measure for certain algebraic numbers
A result of Chudnovsky concerning rational approximation to certain algebraic numbers is reworked to provide a quantitative result in which all constants are explicitly given. More particularly, PadeExpand
This paper compares 21 methods to distinguish prime numbers from composite numbers. It answers the following questions for each method: Does the method certify primality? Conjecturally certifyExpand
Irrationalité de certaines integrales hypergéométriques
Resume Dans cet article nous appliquons la theorie des approximants de Pade a l'etude des approximations diophantiennes des integrales hypergeometriques 2 F 1 1, 1 k 1+ 1 k ϵx k , pour k entier ≥ 2Expand
Zur Approximation gewisser p-adischer algebraischer Zahlen durch rationale Zahlen.
Allerdings läßt sich das Problem, für j?-adische algebraische oc eines Grades s ^>3 und für Werte mit 2 < < s die Konstante c(<x, ) effektiv anzugeben, mit der RothRidoutschen Methode prinzipiellExpand