What we call "Hybridization" is really just a mathematical transformation of an approximated wavefunction. Your basic summation formula describes how to build some sort of customized wavefunction by forming a linear combination of some set of "basis" orbitals; you will find this equation over and over if you look up "basis set expansion". Now, of course you can't just use any random coefficients $c_{ij}$ of your liking for this custom wavefunction; you have to adhere to some physical rules. Most importantly, the overall energy that the wavefunction represents must not change. I won't go into detail about the constraints, but if you are interested, google "unitary transformation".
For orbital hybridization, you first have to find a set of basis orbitals from a physically sound starting wavefunction. If you only look at an isolated atom, then atomic orbitals are obviously a useful choice. From there, you can form combinations of those orbitals (by some criteria that really aren't uniquely defined) while adhering to the rules of the unitary transformation. In the end, you will arrive at your preferred hybrid orbitals that fulfill your chosen criterion as best as they can while keeping the wavefunction energetically equivalent to the original one. Some of the coefficients $c_{ij}$ in your equations may come out to be 0, typically for reasons of symmetry. For example, an $sp^2$ hybrid that is oriented along the $y$ axis of your coordinate system would obviously only have contributions from the $p_y$ orbital, not from $p_x$ and $p_z$. But the exact values of the cofficients will generally depend on the specifics of your problem, most importantly the definition of your coordinate system. So you really can't say per se which coefficients will drop out to be 0 and which won't -- it's one of those "it depends" situations.
General Thoughts
Unfortunately, hybridization is a concept that is so misleading that I would consider its general use in chemistry to be outright physically incorrect. People invoke it to suggest and visualize bonding situations in molecules and to explain the resulting molecular shapes, but unfortunately always seem to forget some very basic undergraduate quantum chemistry in this process.
Hybrid orbitals are linear combinations of eigenfunctions to different energy eigenvalues of the underlying one-electron Hamiltonian; think $s$ and $p$ orbitals. As such, they are not solutions to the time-independent Schrödinger Equation anymore, which in turn means that they are time-dependent functions. This implies two things:
1.) Hybrid orbitals do not have a fixed shape; what you see visualized is just a single snapshot in time. The orbital shapes periodically "morph" back and forth and can at other points in time look nothing like those suggestive localized "bonding" orbitals anymore. In fact, the hybrid orbitals can even change places among each other over time if the symmetry of the system permits.
To better drive this point home, here are some (semi-quantitative) visualizations of the time development of the electron densities in the bonding and non-bonding orbitals in water; the C-C σ and π "banana" bonds in ethylene; and an sp³ hybrid in methane. In each case, the orbitals transform over time into something that is actually quite the opposite to their original "purpose": Bonding electron densities shift outwards to resemble something of anti-bonding character, and symmetry-equivalent orbitals migrate from one bonded pair of nuclei to another. At each point throughout these videos, one could stop the animation and take a freeze-frame of the orbital at that time, and it would still describe the exact same "localized electron pair" that was used in the beginning to explain how electron density is confined with a certain purpose to some specific region of the molecule.
2.) Hybrid orbitals do not have defined energies, since they are not solutions to the time-independent Schrödinger Equation. If you were to do any sort of "energy measurement" on a hybrid orbital, you would end up with a Schrödinger's Cat situation where you observe the energies of the constituent atomic orbitals with certain probabilities that are connected to the expansion coefficients $c_{ij}$.
3.) Hybridization is not stringently mathematically defined if one regards it as an orbital localization procedure. Different for concepts exist for this localization process, which can yield qualitatively different results (see, for example, the discussion about "rabbit ears" lone pairs in water). This is also what ultimately gives rise to the weird fractional mixtures of s-p hybridization ratios that one sometimes finds discussed in the literature.
"MO and VB Theory are ultimately equivalent"
Quantum chemistry deals with two major theories to describe the electronic structure of molecules: Molecular orbital (MO) theory, which gives rise to canonical MOs that can generally be delocalized over an entire molecule and are thus very unappealing from a classical chemical perspective; and Valence-Bond (VB) theory, which attempts to reconcile the idea of isolated chemical bonds and electron pairs with quantum mechanics by postulating the existence of the localized hybrid orbitals that are in question here.
Now, in almost any discussion on the topic, it is at some point claimed that the two theories are ultimately equivalent and related to one another only through mathematical transformations. This is technically correct, and one can arrive at the total overall wavefunction and energy of the molecule either way. However, it is again also very misleading, because it suggests that the two theories sort of "meet in the middle" if you apply them in some strict, physically rigorous way. But what this statement actually means is that appropriately constructed hybrid orbitals can be used as a basis set for developing static wavefunctions with valid energy solutions to a one-electron molecular Hamiltonian; and that, when taking the totality of all hybrid orbitals together, you arrive at the same wavefunction as MO theory. In other words, "accurate" VB theory is simply MO theory with extra intermediate steps! The two theories are equivalent in the sense that at the invoked "limit of correctness", VB theory becomes MO theory; the latter never has to budge in the process, as it is a physically and mathematically sound treatment of the molecule to begin with.
Experimental (Counter-)Evidence
Up to this point, all arguments have been made from a purely theoretical perspective. The one concession that can be readily made to VB theory is that a combination of all hybrid orbitals again yields the total MO wavefunction for the molecule. As chemists, however, we are very often not just interested in a "holistic" description of molecules as a whole, but rather its individual electrons (or electron pairs). In that case, this equivalence is of no help. What we would really like to have is a direct look into whether the orbital hybridization concept holds up as a model in physical reality for the constituent electron pairs inside a molecule. In other words, can we interrogate electrons as to "what kind of orbitals they are in"?
Usually, this question immediately prompts responses that hinge on Photoelectron Spectroscopy (PES), where one measures the photon energies required to eject electrons from a molecule. An exemplary case of this is again the methane molecule: MO theory predicts a 3:1 ratio of two energetically distinct orbital types in the valence shell, whereas VB theory postulates its equivalent hybrid orbitals. The actual photoelectron spectrum reveals two distinct peaks at roughly a 3:1 intensity ratio, which is often taken as direct evidence for the veracity of the MO picture. VB proponents typically counter these claims by invoking involvement of the ionic product state of methane in the spectrum, and by requiring that it is not just one hybrid orbital that contributes to the spectrum, but a combination of all four of them (which, again, is simply the reconstruction of MOs).
Since the interpretation of these findings is contentious, they don't serve us well to find an answer as to whether hybrid or molecular orbitals are actually observable in any way. Luckily, there is another method we can use to investigate the nature of bound electrons, called Electron Momentum Spectroscopy (EMS). In essence, this technique allows the measurement of the momentum distribution of valence electrons in the initial state of the molecule without significant involvement of the ionic product, thus addressing one of the criticisms of the PES experiments. When compared to MOs or the extremely similar Kohn-Sham orbitals (KSOs) from Density Functional Theory (which are essentially MOs with inclusion of some electron correlation), it can be seen that the measurement data is almost perfectly explained by MOs or KSOs, but qualitatively incompatible with hybrid orbitals. (1,2)
Obviously, these observations lend strong support to the idea that orbitals can indeed be interpreted as a physically meaningful constructs for describing the behaviour of specific electrons in a molecule. In order to be observable in this way however, they must be eigenfunctions to the molecular Hamiltonian, which -- with the arguments above -- directly explains why MOs (or KSOs), but not hybrid orbitals, are able to describe these orbitals.
From my own perspective, the concept of orbital hybridization thus has a very convincing body of both theoretical and experimental evidence speaking against it, or at least against the way that it is usually invoked to explain molecular shapes and chemical bonding. As mentioned above, you can use VB theory and its hybrid orbitals in a sense that makes them physically rigorous; but it just means that you're actually back to doing MO theory again.
Unfortunately, at least some of the points I have addressed above -- especially the time dependence and lack of energy eigenvalues -- seem to rarely ever be mentioned when it comes to discussions of hybrid orbitals. This is undoubtedly because these topics are simply too advanced for a lot of the teaching contexts that hybrid orbitals appear in (although even high-level disputes in quantum chemistry journals generally fail to address these points too). But especially with regards to chemical education, we must ask ourselves whether teaching a concept "not because it is correct, but because it is simple" is really an appropriate thing to do. Obviously, the answer should be no; and I think that chemistry as an entire discipline needs to abandon the idea of hybrid orbitals in the same sense, and for the same reasons, that we abandoned Bohr's atom model.
r&l&r&l&r. $\endgroup$