I'm looking into expanding my knowledge in ways to show that some constant is irrational. I'm gonna give some examples of irrationality proofs, and I'm interested in learning what strategies you guys know and if such strategies can be expanded to other constants.

I'll ignore algebraic numbers such as $n^{1/m}$ where $n,m$ are integers or rationals because I believe such proofs abuse properties of integers and these proofs cannot be expanded to other constants. The strategies below concern constants that are transcendental or conjectured to be transcendentals.

(Fast converging series) One way to show that a constant is irrational is to use a series that converges really fast to it. Example: We can use the truncated power series for $e$ to show that $$n!(e-\sum_{k=0}^n 1/k!) <1/n$$ for every positive integer $n$. If we assume $e=p/q$ we find an integer in $(0,1)$. This shows that the power series is so fast that as $n$ goes to infinity, $e-\sum_{k=0}^n 1/k!$ reaches $0$ faster than $n!$ goes to infinity. This can be expanded to $e^r$, where $r$ is rational. The other proof that uses a fast converging series is Apery proof of the irrationality of $\zeta(2),\zeta(3)$. Although, I don't remember seeing this approach to other constants.

(Niven's polynomial) This approach uses the fact that the derivative of $e^x$ is itself to show that $e$ is irrational. Since the second derivative of $\sin x$ is minus itself this approach also works to show that $\pi$ is irrational, and $\sin, \cos, \exp$ are irrational in rational arguments, and so are the hyperbolics functions. I believe this approach only works for these constants because it uses the fact that the derivatives of said functions are themselves. For example see Niven's paper 'A simple proof that $\pi$ is irrational'.

(Beukers integrals) I'll call a Beukers integral, the integral $$I_n=\int_0^1 x^nf(x)dx = \frac{a_n\xi+b_n}{d_n}$$ where $\xi$ is the number we want to show that is irrational and $a,b,d$ are sequence of integers. The challenge in using this approach is to find a suitable function $f(x)$. The strategy is to show that $I_n$ is not null, and as $n\to\infty$, $d_nI_n$ goes to $0$. Note that, we can change $x^n$ for a polynomial with degree $n$ and integer coefficients that the integral will still have the same form. Thus we can choose the Legendre polynomial. More details here Legendre polynomials in irrationality proofs. by Beukers We can use this approach to show that $\ln 2, e^r,\pi^2, \zeta(2),\zeta(3)$ are irrational. I don't think this approach have been used to other constants, but it seems it is the most promising strategy so far.

In the book Making Transcendence Transparent, it contains a proof that $e^\pi$ is irrational (also transcendental) which I did not know existed without using the Gelfond theorem. However I'm still learning about it, but wanted to mention it.

Please, share other methods and strategies that you're aware of, or share a link to a paper that demonstrates the strategy.

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