Need mathematical explanation for different musical notes sound different on different instruments I am not expert in music.
There are number of musical instruments. One (especially a person who knows about music) can blindly recognize which instrument is being played just by listening to it. 
I need mathematical explanation about following

Every instrument (say Guitar, Harmonium , Violin like that) seems to be unique while listening. My basic question is why is it so?
Why different musical notes sound different on different instruments? 

 A: 1) A vibrating string produces not only the basic (fundamental) frequency, i.e. vibrates up-down, but also lots of other (multiple) frequencies, i.e. double frequency when two halves of the string vibrates in the counter-phase etc. See more on the video.
2) Each acoustic instrument has a resonating body where the sound waves with all those frequencies travels around and undergoes multiple reflections, producing a complex resulting wave in which all such reflections are added up with some frequencies having been amplified and other attenuated, depending on how the wave and the reflected wave tops meet in a point.
A human ear distinguishes this particular spectral character of the signal with different amplitudes on different frequencies. Different sounds of different instruments comes mainly from the geometry of their resonators.
P.S. With modern digital sound processing it is easy to synthesize any spectrum of the wave and to mimic whatever instrument.
A: Pitch, in music, is equivalent to frequency.  How often the wavefore cycles.  This is usually defined by length, i.e. how long the string is, how long the pipe is, etc.  It can also be affected by the tension (how tight the string is.)
Timbre, the sound of a specific instrument, is defined by the "shape" of the wavefore, whether spikes, round, square, or whatever other shape that instrument makes.  This wavefore shape is defined by the construction of the instrument.  A trumpet, for instance, makes a very "spiky" shape that just sounds "brassy" to an experienced ear.
A: When you pluck a string or hit a drum or sound a not on a flute, the instrument and the air in and around it vibrate and this vibration propagates as sound waves in the air to your hear drum.
When you hear an instrument being played, what you recognise as the note is the base frequency. 'C' corresponds to $261.6$ Hz and is the same for a piano or a guitar. But a 'C' played from a guitar, played from a piano or simply a $261.6$ Hz sounwave played from a computer speaker sound totally different. This is because of the overtones.
Let's look at the case of a string for a concrete example.
If you pluck the 'C' string on the guitar, you will hear the characteristic sound of it makes. This is because the string is vibrating at $261.6$ Hz, but also at a bunch of higher frequencies. These higher frequencies are called "overtones", and they are determined by the shape and build of the body of the guitar as well as the way you set the string in motion.
This is guitars with different shapes sound different. You can also try plucking or strumming the guitar string in various different places, and you will hear different tones of 'C'.
Overtones of the vibrating instrument are what makes each instrument (or voice, for that matter) sound different. The material, shape, and way the instrument is played all contribute to determine which overtones will be present.
The reason instruments sound more similar while holding a long note is that the overtones dissipate energy faster. Higher frequency vibrations generally lose energy quicker. So once the string is plucked or hit, the overtones start losing energy (thus lowering volume) faster than the base note, and after a while you only hear the base note.
TL;DR:
Instrument sound different because of the overtones they produce. These are higher frequencies than the note being played, and are determined by factors such as shape, material and way of playing the instrument and give the characteristic flavour to each instrument.
