How can i get or caculate the parameter "S" from the given 4 parameters "Q, gain, boost, f0/fs" for shelving iir filter in the sigmasudio based on adau1701?

The image above shows that the filter 6 have 4 given parameters, "fc/fs, Q_factor, gain, boost". To get the coefficients-- "a1,a2, b0,b1,b2", there is a templet in the literature "1st-2nd-Order_IIR_Coefficient_Calculator" about highShelf and lowShelf filter, like this:

Now, the original 4 given parameters become 5 control values. "S" is one and a new member, the original "Q" becoming a intermediate value derived from S and A.

My question focus on two:

1. Is the intermediate value “Q” the same as the original "Q" which is one of the 4 given parameter.

2. where is the value "S" coming from? Because there is no such parameter among the original 4 given parameters. HOW CAN I GET IT?

The value "S" is most important I think, it decides all the 5 coefficients in the following algorithm. In the literature "Cookbook formulae for audio EQ biquad filter coefficients", there is a

equation about "S" and" Q":

1/Q = sqrt((A + 1/A)*(1/S - 1) + 2); (1)
And in the literature "1st-2nd-Order_IIR_Coefficient_Calculator" , the equation is somewhat different:

1/Q = sqrt((A + 1/A)^(1/S - 1) + 2); (2)
I don't know which one is correct! pls tell me which one I should choose.
The key point is how can i get "S" first. I have tried many ways to get this "S" , but all failed. For example, I tried to get "S" from equation (1) or (2) ,
assuming the intermediate value "Q"is the original given parameter "Q". However the result made me frustrated. Maybe the reason is because the "Q" here is totally
different from the original given one.
In one word, how can I get "S" from the given 4 parameters "Q", "fc/fs", "boost", and "gain". Please give me the detailed algorithm.

Thanks a lot!

Shihong Sun.

Audio Signal Processors

SigmaDSP Audio Processors

adau1701