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### On the approximation of real continuous functions by series of solutions of a single system of partial differential equations

Colloquium Mathematicae

We prove the existence of an effectively computable integer polynomial P(x,t₀,...,t₅) having the following property. Every continuous function $f:{ℝ}^{s}\to ℝ$ can be approximated with arbitrary accuracy by an infinite sum ${\sum }_{r=1}^{\infty }{H}_{r}\left(x₁,...,{x}_{s}\right)\in {C}^{\infty }\left({ℝ}^{s}\right)$ of analytic functions ${H}_{r}$, each solving the same system of universal partial differential equations, namely $P\left({x}_{\sigma };{H}_{r},\partial {H}_{r}/\partial {x}_{\sigma },...,\partial ⁵{H}_{r}/\partial x{⁵}_{\sigma }⁵\right)=0$ (σ = 1,..., s).

### Approximation of values of hypergeometric functions by restricted rationals

Journal de Théorie des Nombres de Bordeaux

We compute upper and lower bounds for the approximation of hyperbolic functions at points $1/s$ $\left(s=1,2,\cdots \right)$ by rationals $x/y$, such that $x,y$ satisfy a quadratic equation. For instance, all positive integers $x,y$ with $y\equiv 0\phantom{\rule{4.44443pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}2\right)$ solving the Pythagorean equation ${x}^{2}+{y}^{2}={z}^{2}$ satisfy $|ysinh\left(1/s\right)-x|\phantom{\rule{0.166667em}{0ex}}\gg \frac{loglogy}{logy}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}.$ Conversely, for every $s=1,2,\cdots$ there are infinitely many coprime integers $x,y$, such that $|ysinh\left(1/s\right)-x|\phantom{\rule{0.166667em}{0ex}}\ll \frac{loglogy}{logy}$ and ${x}^{2}+{y}^{2}={z}^{2}$ hold simultaneously for some integer $z$. A generalization to the approximation of $h\left({e}^{1/s}\right)$ for rational functions $h\left(t\right)$...

### Asymptotic representations for Fibonacci reciprocal sums and Euler’s formulas for zeta values

Journal de Théorie des Nombres de Bordeaux

We present asymptotic representations for certain reciprocal sums of Fibonacci numbers and of Lucas numbers as a parameter tends to a critical value. As limiting cases of our results, we obtain Euler’s formulas for values of zeta functions.

### On rational approximations to Euler's constant $\gamma$ and to $\gamma +log\left(a/b\right)$.

International Journal of Mathematics and Mathematical Sciences

### On prime-detecting sequences from Apéry's recurrence formulae for $\zeta \left(3\right)$ and $\zeta \left(2\right)$.

Journal of Integer Sequences [electronic only]

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