# What is the meaning of the delay in a FIR filter

I'm using a C# library that has the FIR filter implemented and I'd like to know what's the meaning of the ${z^{-1}}$ operation (delay) mathematically. The triangle is multiplication, the circle is a sum, but what does the delay means in terms of mathematical operations?

In mathematical terms, time discrete systems are are most often expressed in terms of difference equations. A delay is simply $y[n]=x[n-1]$ so the difference equation for your FIR filter would be $$y[n]=\sum_{i=0}^{M}b_{i}\cdot x[n-i]$$.
The $z^{-1}$ notation comes from the Z-transform which is the discrete-time equivalent of the Laplace transform. On it's own, the only mathematical meaning that multiplying by $z^{-1}$ holds is a delay of one sample. However, as part of a transfer function it is a very powerful notation which allows you to study the frequency response of the system based on its zeros and poles. Transfer functions expressed in this way can be useful for both system analysis and system design.
If you are just trying to implement a specific FIR filter with given coefficients, the only thing you need to know about $z^{-1}$ is that it represents a delay. You could just as easily represent the filter in your question by representing the whole row of $z^{-1}$ elements as a serial-in, parallel-out shift register. At each cycle of the "clock," the delay elements accept a new value on their inputs and hold that value until the next clock cycle.