On spectrum of irrationality exponents of Mahler numbers

We consider Mahler functions $f(z)$ which solve the functional equation $f(z) = \frac{A(z)}{B(z)} f(z^d)$ where $\frac{A(z)}{B(z)}\in \mathbb{Q}(z)$ and $d\ge 2$ is integer. We prove that for any integer $b$ with $|b|\ge 2$ either $f(b)$ is rational or its irrationality exponent is rational. We also compute the exact value of the irrationality exponent for $f(b)$ as soon as the continued fraction for the corresponding Mahler function is known. This improves the result of Bugeaud, Han, Wei and Yao where only an upper bound for the irrationality exponent was provided

Irrationality proofs \`a la Hermite

As rewards of reading two great papers of Hermite from 1873, we trace the historical origin of the integral Niven used in his well-known proof of the irrationality of $\pi$, uncover a rarely acknowledged simple proof by Hermite of the irrationality of $\pi^2$, give a new proof of the irrationality of $r\tan r$ for nonzero rational $r^2$, and generalize it to a proof of the irrationality of certain ratios of Bessel functions.Comment: 8 page

On simultaneous diophantine approximations to ζ(2)\zeta(2) and ζ(3)\zeta(3)

We present a hypergeometric construction of rational approximations to $\zeta(2)$ and $\zeta(3)$ which allows one to demonstrate simultaneously the irrationality of each of the zeta values, as well as to estimate from below certain linear forms in 1, $\zeta(2)$ and $\zeta(3)$ with rational coefficients. A new notion of (simultaneous) diophantine exponent is introduced to formalise the arithmetic structure of these specific linear forms. Finally, the properties of this newer concept are studied and linked to the classical irrationality exponent and its generalisations given recently by S. Fischler.Comment: 23 pages; v2: new subsection 4.5 adde

Recurrent proofs of the irrationality of certain trigonometric values

We use recurrences of integrals to give new and elementary proofs of the irrationality of pi, tan(r) for all nonzero rational r, and cos(r) for all nonzero rational r^2. Immediate consequences to other values of the elementary transcendental functions are also discussed

But next time, I will win: On the relation between irrationality and probability estimates in a game of chance

Based on Rational Emotive Behavior Therapy (REBT) we tested the hitherto unexplored assumption that irrationality as conceptualized by REBT (demandingness, self evaluation, low frustration tolerance), is associated with erroneous statistical reasoning. We assessed trait irrationality of 216 respondents and individual estimates of future winning probabilities in the context of the Wortman (1975) perceived control design. Results indicate that an increased (i.e., unrealistically optimistic) as well as a decreased (i.e., unrealistically pessimistic) estimation of future winnings is associated with irrationality. Findings substantiate an association between erroneous probability estimates and therapeutically relevant cognitions which do not imply any mathematical or statistical contents

A neuroeconomic theory of rational addiction and\ud nonlinear time-perception.

Neuroeconomic conditions for “rational addiction” (Becker and Murphy, 1988) have\ud been unknown. This paper derived the conditions for “rational addiction” by utilizing a\ud nonlinear time-perception theory of “hyperbolic” discounting, which is mathematically\ud equivalent to the q-exponential intertemporal choice model based on Tsallis' statistics. It\ud is shown that (i) Arrow-Pratt measure for temporal cognition corresponds to the degree\ud of irrationality (i.e., Prelec’s “decreasing impatience” parameter of temporal\ud discounting) and (ii) rationality in addicts is controlled by a nondimensionalization\ud parameter of the logarithmic time-perception function. Furthermore, the present theory\ud illustrates the possibility that addictive drugs increase impulsivity via dopaminergic\ud neuroadaptation without increasing irrationality. Future directions in the application of\ud the model to studies in neuroeconomics are discussed

Irrationality proof of a qq-extension of ζ(2)\zeta(2) using little qq-Jacobi polynomials

We show how one can use Hermite-Pad\'{e} approximation and little $q$-Jacobi polynomials to construct rational approximants for $\zeta_q(2)$. These numbers are $q$-analogues of the well known $\zeta(2)$. Here $q=\frac{1}{p}$, with $p$ an integer greater than one. These approximants are good enough to show the irrationality of $\zeta_q(2)$ and they allow us to calculate an upper bound for its measure of irrationality: $\mu(\zeta_q(2))\leq 10\pi^2/(5\pi^2-24) \approx 3.8936$. This is sharper than the upper bound given by Zudilin (\textit{On the irrationality measure for a $q$-analogue of $\zeta(2)$}, Mat. Sb. \textbf{193} (2002), no. 8, 49--70).Comment: 13 pages, one reference was corrected, two were adde