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?
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.
A "matter wave" is usually what people call the wavefunction of a single particle, but it can have other meanings.
A "matter wave" can also be a lot of bosonic particles which together act as a classical wave. Such a colletion of atoms is called a Bose-Einstein condensate, if of photons, it's an electromagnetic wave. These are a lot of particles which are coherent with each other, and travel together described by the same wave.
There is no separate wavefunction for each particle, the wavefunction is a function of all possible worlds, of all possible configurations, so there's only one wavefunction for the entire system under consideration, not one for each particle in the system. This is very important, it is a recurring confusion.
But this doesn't mean you can't have separate waves for different collections of particles which are coherent. You can have electromagnetic wave here and a Bose Einstein condensate there and a gravitational wave somewhere else. So this type of classical wave comes in many kinds and can be waving in different places.
There is another sense in which every particle has a wave-- every particle is associated with a classical field whose quantum motion describes all particles of that type.