How to describe the game of blackjack in mathematical terms?

For some time, I have been trying to solve several aspects of the game of blackjack using computer simulation. However, one can directly describe blackjack in terms of discrete, direct analysis on a computer. Not only is direct computation faster, it is more accurate than that of monte carlo simulation!

However, to describe what is going on “under the hood”, so to speak, defining the mechanics of blackjack using mathematical symbolism and logic has so far escaped me. I can come up with simplified ideas like describing enumerating a player hand:

E_opt[X|i] = Σ[n = 1; 11] P[n] * E_opt[X + n| i + n];

Where we are recursively finding the optimal expectation for hand X, give some hand total i. Something like the above.

The questions are:

1.) How can one describe the structure of the game of blackjack with maths? That is, how does one mathematically describe a hand and its properties?

2.) How would one describe and define algorithms that compute certain properties of the game itself? (Things like dealer probabilities, stand, hit, double, and split expectation plus how dealer probabilities and each action previously listed are connected.)

3.) Certain actions, like splitting, require a more rigorous treatment. How would I go about making sure any definitions and/or functions are correct?

4.) What, besides probability theory and statistics, would be a requirement to mathematically define blackjack?

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