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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 ${\sum }_{r=1}^{\infty }{H}_r(x₁,...,x_s)\in C^{\infty }\left({ℝ}^s\right)$ of analytic functions $H_r$, each solving the same system of universal partial differential equations, namely $P(x_{\sigma };H_r,\partial H_r/\partial x_{\sigma },...,\partial ⁵H_r/\partial x{⁵}_{\sigma }⁵)=0$ (σ = 1,..., s).

We compute upper and lower bounds for the approximation of hyperbolic functions at points $1/s$ $(s=1,2,\cdots )$ by rationals $x/y$, such that $x,y$ satisfy a quadratic equation. For instance, all positive integers $x,y$ with $y\equiv 0(mod2)$ solving the Pythagorean equation $x^2+y^2=z^2$ satisfy $$|ysinh(1/s)-x|\gg \frac{loglogy}{logy}.$$ Conversely, for every $s=1,2,\cdots$ there are infinitely many coprime integers $x,y$, such that $$|ysinh(1/s)-x|\ll \frac{loglogy}{logy}$$ 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 $h\left(t\right)$...

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.