lpetrich
Contributor
Also transcendental: the Euler 
Beta function B(a,b) for a, b, and a+b non-integer rational numbers:
\( \displaystyle{ \Beta(z_1,z_2) = \int_0^1 t^{z_1} (1-t)^{z_2} \, dt = \frac{\Gamma(z_1)\Gamma(z_2)}{\Gamma(z_1+z_2)} } \)
Some more partial results are for the Riemann zeta function and its relatives, the Dirichlet eta, lambda, and beta functions:
\( \displaystyle{ \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} } \)
\( \displaystyle{ \eta(s) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^s} } \)
\( \displaystyle{ \lambda(s) = \sum_{n=0}^{\infty} \frac{1}{(2n+1)^s} } \)
\( \displaystyle{ \beta(s) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)^s} } \)
Interrelationships:
\( \lambda(s) = (1 - 2^{-s}) \zeta(s) \text{ ... } \eta(s) = (1 - 2^{1-s}) \zeta(s) \)
None between the Dirichlet beta function and the others, however.
The Riemann zeta function has special values
\( \displaystyle{ \zeta(2n) = \frac{(-1)^{n+1} B_{2n} (2\pi)^{2n} }{ 2(2n)! } } \)
for n a positive integer and B the Bernoulli numbers, all rational numbers. This is obviously transcendental.
But for zeta(2n+1), it is known that zeta(3) is irrational, and not known whether any of the others is irreational. However, an infinite number of the zeta(2n+1) are irrational, and at least one of zeta(5), zeta(7), ..., zeta(21).
The Dirichlet beta function has special values
\( \displaystyle{ \beta(2n+1) = \frac{(-1)^n E_{2n} (\pi/2)^{2n+1} }{ 2(2n)! } } \)
for n a nonnegative integer and E the Euler numbers, all integers. This is obviously transcendental.
But for beta(2n), it is not known whether any of them are irrational.
Beta function B(a,b) for a, b, and a+b non-integer rational numbers:\( \displaystyle{ \Beta(z_1,z_2) = \int_0^1 t^{z_1} (1-t)^{z_2} \, dt = \frac{\Gamma(z_1)\Gamma(z_2)}{\Gamma(z_1+z_2)} } \)
Some more partial results are for the Riemann zeta function and its relatives, the Dirichlet eta, lambda, and beta functions:
\( \displaystyle{ \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} } \)
\( \displaystyle{ \eta(s) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^s} } \)
\( \displaystyle{ \lambda(s) = \sum_{n=0}^{\infty} \frac{1}{(2n+1)^s} } \)
\( \displaystyle{ \beta(s) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)^s} } \)
Interrelationships:
\( \lambda(s) = (1 - 2^{-s}) \zeta(s) \text{ ... } \eta(s) = (1 - 2^{1-s}) \zeta(s) \)
None between the Dirichlet beta function and the others, however.
The Riemann zeta function has special values
\( \displaystyle{ \zeta(2n) = \frac{(-1)^{n+1} B_{2n} (2\pi)^{2n} }{ 2(2n)! } } \)
for n a positive integer and B the Bernoulli numbers, all rational numbers. This is obviously transcendental.
But for zeta(2n+1), it is known that zeta(3) is irrational, and not known whether any of the others is irreational. However, an infinite number of the zeta(2n+1) are irrational, and at least one of zeta(5), zeta(7), ..., zeta(21).
The Dirichlet beta function has special values
\( \displaystyle{ \beta(2n+1) = \frac{(-1)^n E_{2n} (\pi/2)^{2n+1} }{ 2(2n)! } } \)
for n a nonnegative integer and E the Euler numbers, all integers. This is obviously transcendental.
But for beta(2n), it is not known whether any of them are irrational.