Is there any mathematical relationship between matter wave (or de Broglie wave) and wave function?
Also, does each type of particle (e.g. photon, electron, positron etc.) have its own unique wave function?
[Physics] matter wave and wave function
quantum mechanicswave-particle-dualitywavefunctionwaves
Related Solutions
Actually, according to quantum mechanics, for every particle (matter), there is a complex valued wave function and it is the modulus of that wave function that is the probability that the particle is at a given location.
For example, a wavefunction for a uniform plane wave of matter could be:
$$\psi(\vec x, t) = A e^{i(\vec k \cdot \vec x + \omega t)} = A(cos(\vec k \cdot \vec x + \omega t) + i\cdot sin(\vec k \cdot \vec x + \omega t))$$
This wavefunction is a complex valued wave (notice the sine and cosine functions) that has a wavelength ($1/|\vec k|$ = the de Broglie wavelength) and a frequency ($\omega$). It is this complex wave that could result in quantum interference effects.
However the probability for a particle to be at a position $\vec x, t $ is:
$$P(\vec x,t) = |\psi(\vec x, t)|^2 = |A|^2 $$
So it is constant everywhere in space and time and is not a wave. Of course, an actual wavefunction for a particle would be something like this plane wave multiplied by a gaussian to result in a wave packet that is localized in space and time and that travels the way a classical particle would travel (see wave packet for more information).
But the key point is that there is a complex valued wavefunction with wave-like properties but that the probability does not have to have any wave like properties.
EDIT for edited question:
For a particle localized in space, there is still an average position and an average momentum so the average de Broglie wavelength will be the one that corresponds to that average momentum. According to Heisenberg there will be an uncertainty in position and an uncertainty in momentum and thus there will be an uncertainty in the de Broglie wavelength also.
What this really means is that whenever you have a localized particle, you have a wave packet which means there is a superposition of a number of plane waves with similar momentum, in other words a superposition of a number of different de Broglie wavelengths (see wave packet for more information).
If you are wondering if the concept of de Broglie wavelengths is useful in this situation, then the answer is that it is not very useful - just work with your mathematically defined complex valued quantum mechanical wavefunction instead.
Waves appear in nature and are described by wave equations,
second-order linear partial differential equation for the description of waves – as they occur in physics
All such equations have time dependent solutions and what they have in common is the oscillating behaviour in time that allows to assign a wavelength and describe the waves observed.
Now in quantum mechanics the description of the behavior of nature is dependent on differential equations of this type. The first studied is the Schrodinger equation and the link provides a good historical description of how it became evident that in the microcosm the behavior of particles was following a wave equation.
What is important to keep in mind is that in quantum mechanics the waves are probability waves, i.e. the probability if you do an experiment, like the double slit experiment, to find a particle in space at the time you look is governed by a wave solution. This is in contrast to other waves in physics which are variations in time on a medium, or in classical electromagnetism on changing fields.
So the relationship is the mathematical formulation of the differential equations describing nature in the two frameworks, not a one to one correspondence.
As I mentioned in my comment in electromagnetism, the frequency of the electromagnetic field described by classical electrodynamics appears in the energy of the photon ( the particle form of electromagnetism) in the identity E=h.nu. The interference pattern seen in the double slit experiment will display the frequency of light nu. If you continue your studies in physics you will be able to understand how the microcosm described by quantum mechanics leads to the macrocosm we call "classical physics" smoothly.
Best Answer
A de Broglie or matter wave pretty much is the same as the wave function. Recognizing that particles had wavelengths (based on momentum) was his de Broglie's great insight.
As to each particle having its own wave function, I'd suggest thinking of it this way:
If you can characterize the momentum of particles, you will find that particles with identical momentum have the same wavelengths as they move through open space. Similarly, if unmoving particles have identical rest masses, they will have the same quantum frequencies (detectable through interference effects) as they move through time. In that sense, the wave functions of all particles share certain common features that are related directly to their momenta and rest masses.
However, it is other conserved quantities such as spin (especially spin!) and charge that give each type of particle wave function its real uniqueness in how it interacts with other particles over time. For more mathematical depth, quantum field theory provides a way to model particles as "excitations" or vibrations in fields that are associated with each unique particle type.