Christopher, try these. The trick is to escape every asterisk and underscore.

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\\[P\_n= \sum\_{k=0}^n \hat{P}\_{n-k}\*X\_k\\]

\\[P\_n= P\_n\*{X\_0} + \sum\_{k=1}^{n} \hat{P}\_{n-k}\*{X\_k} \\]

\\[\hat{P}\_n = P\_n - \sum\_{k=1}^{n} \hat{P}\_{n-k}\*X\_k\\]

\\[\hat{P}\_n = P\_n - \sum\_{k=0}^{n-1} \hat{P}\_{n-1-k}\*X\_{k+1}\\]

\\[\hat{P}\_n = P\_n - ((\hat{P}\circ(n \mapsto n-1)) \ast (X \circ (k \mapsto k+1))\_n\\]