Several articles in this website describe the Summing Amplifier. In one of these articles, Solving the Summing Amplifier, I showed a numeric method to design a non-inverting summing amplifier based on its input and output voltage range requirements.
This article shows how to design a summing amplifier calculator and the mathematical relations it uses. You can find the calculator here:
Type the input voltage range, output range, a reference voltage and a choice of two resistors. The calculator gives you the answer for the remaining two resistors. The default values are for a bipolar to unipolar converter, which is explained in Design a Bipolar to Unipolar Converter to Drive an ADC.
What are the underlying equations?