Spectral series formulas for hydrogen-like atoms

[Hawkingo]

I was studying about atomic spectra of mono-electron species and in the pic it describes the $4$ series (principal, sharp, diffuse and fundamental).

However I'm a little confused by the formula.Here my doubts are:

1) In the Wikipedia about the sharp series. There is $+ve$ sign between $p/q$ (energy levels) and $\alpha$ and $\beta$ (characteristics constants) but in the book it is $-ve$ sign. So which is correct?

2) In the sharp series it says $p=2$ and $q\ge2$,but it writes in fomula ($1- \alpha$) where $1$ is supposed to be the value of $p$. So why is it like that?

3) In the sharp series it says $p=2$ and $q\ge2$,but in the diffuse series it says $p=2$ and $q\ge3$.So wouldn't the diffuse series come under sharp series because what is $\ge2$ is obviously $\ge3$

Thanks for reading. [1]:
[2]: https://en.m.wikipedia.org/wiki/Sharp_series

 

[Dr_Nate]

1) You could try calculating the wavenumbers for both and comparing them to experiment.
2) Eq. 2.15 seems to show you that Eq. 2.12 is a typo.
3) Each of those equations has a $\beta$ with a different subscript, so they are going to give different results.

 

[Hawkingo]

1) You could try calculating the wavenumbers for both and comparing them to experiment.
2) Eq. 2.15 seems to show you that Eq. 2.12 is a typo.
3) Each of those equations has a $\beta$ with a different subscript, so they are going to give different results.

Can you give any source material on this concept containing the formulas? In Wikipedia there are only 1st 2 series

 

[Dr_Nate]

Can you give any source material on this concept containing the formulas? In Wikipedia there are only 1st 2 series

You might want to search for "sharp principal diffuse fundamental series equations" or "alkali spectra".

Here, here and https://www.tcd.ie/Physics/people/Peter.Gallagher/lectures/js_atomic/JS_atomic_lecture8_9.pdf might be useful links.