Excess Kurtosis of Beta Distribution/Lemma 2

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Lemma for Excess Kurtosis of Beta Distribution

\(\ds 4 \alpha \paren {\alpha + 1} \paren {\alpha + 2} \paren {\alpha + \beta}^2 \paren {\alpha + \beta + 3} \paren {\alpha + \beta + 1}\) \(=\) \(\ds 4 \alpha^7 + \paren {16 \beta + 28} \alpha^6 + \paren {24 \beta^2 + 96 \beta + 68 } \alpha^5 + \paren {16 \beta^3 + 120 \beta^2 + 200 \beta + 68} \alpha^4\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \paren {4 \beta^4 + 64 \beta^3 + 204 \beta^2 + 168 \beta + 24} \alpha^3 + \paren {12 \beta^4 + 80 \beta^3 + 132 \beta^2 + 48 \beta} \alpha^2 + \paren {8 \beta^4 + 32 \beta^3 + 24 \beta^2 } \alpha\)


Proof

\(\ds 4 \alpha \paren {\alpha + 1} \paren {\alpha + 2} \paren {\alpha + \beta}^2 \paren {\alpha + \beta + 3} \paren {\alpha + \beta + 1}\) \(=\) \(\ds 4 \alpha \paren {\alpha + 1} \paren {\alpha + 2} \paren {\alpha + \beta}^2 \paren {\paren {\alpha + \beta }^2 + 4\paren {\alpha + \beta } + 3}\) Group $\paren {\alpha + \beta}$
\(\ds \) \(=\) \(\ds 4 \alpha \paren {\alpha + 1} \paren {\alpha + 2} \paren {\paren {\alpha + \beta }^4 + 4\paren {\alpha + \beta }^3 + 3 \paren {\alpha + \beta}^2}\) Distribute $\paren {\alpha + \beta }^2$
\(\ds \) \(=\) \(\ds \paren {4\alpha^3 + 12 \alpha^2 + 8 \alpha} \paren {\paren {\alpha + \beta }^4 + 4\paren {\alpha + \beta }^3 + 3 \paren {\alpha + \beta}^2}\) Grouping non $\paren {\alpha + \beta }$ terms
\(\ds \) \(=\) \(\ds 4 \alpha^3 \paren {\paren {\alpha + \beta }^4 + 4\paren {\alpha + \beta }^3 + 3 \paren {\alpha + \beta}^2}\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds 12 \alpha^2 \paren {\paren {\alpha + \beta }^4 + 4\paren {\alpha + \beta }^3 + 3 \paren {\alpha + \beta}^2}\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds 8 \alpha \paren {\paren {\alpha + \beta }^4 + 4\paren {\alpha + \beta }^3 + 3 \paren {\alpha + \beta}^2}\)
\(\ds \) \(=\) \(\ds 4 \alpha^3 \paren {\paren {\alpha^4 + 4 \alpha^3 \beta + 6 \alpha^2 \beta^2 + 4 \alpha \beta^3 + \beta^4 } + 4\paren {\alpha^3 + 3\alpha^2 \beta + 3 \alpha \beta^2 + \beta^3 } + 3 \paren {\alpha^2 + 2 \alpha \beta + \beta^2} }\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds 12 \alpha^2 \paren {\paren {\alpha^4 + 4 \alpha^3 \beta + 6 \alpha^2 \beta^2 + 4 \alpha \beta^3 + \beta^4 } + 4\paren {\alpha^3 + 3\alpha^2 \beta + 3 \alpha \beta^2 + \beta^3 } + 3 \paren {\alpha^2 + 2 \alpha \beta + \beta^2} }\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds 8 \alpha \paren {\paren {\alpha^4 + 4 \alpha^3 \beta + 6 \alpha^2 \beta^2 + 4 \alpha \beta^3 + \beta^4 } + 4\paren {\alpha^3 + 3\alpha^2 \beta + 3 \alpha \beta^2 + \beta^3 } + 3 \paren {\alpha^2 + 2 \alpha \beta + \beta^2} }\)
\(\ds \) \(=\) \(\ds 4 \alpha^7 + \paren {16 \beta + 16 + 12} \alpha^6 + \paren {24 \beta^2 + 48 \beta + 12 + 48 \beta + 48 + 8} \alpha^5 + \paren {16 \beta^3 + 48 \beta^2 + 24 \beta + 72 \beta^2 + 144 \beta + 36 + 32 \beta + 32} \alpha^4\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \paren {4 \beta^4 + 16 \beta^3 + 12 \beta^2 + 48 \beta^3 + 144 \beta^2 + 72 \beta + 48 \beta^2 + 96 \beta + 24} \alpha^3 + \paren {12 \beta^4 + 48 \beta^3 + 36 \beta^2 + 32 \beta^3 + 96 \beta^2 + 48 \beta} \alpha^2 + \paren {8 \beta^4 + 32 \beta^3 + 24 \beta^2 } \alpha\)
\(\ds \) \(=\) \(\ds 4 \alpha^7 + \paren {16 \beta + 28} \alpha^6 + \paren {24 \beta^2 + 96 \beta + 68 } \alpha^5 + \paren {16 \beta^3 + 120 \beta^2 + 200 \beta + 68} \alpha^4\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \paren {4 \beta^4 + 64 \beta^3 + 204 \beta^2 + 168 \beta + 24} \alpha^3 + \paren {12 \beta^4 + 80 \beta^3 + 132 \beta^2 + 48 \beta} \alpha^2 + \paren {8 \beta^4 + 32 \beta^3 + 24 \beta^2 } \alpha\)

$\Box$

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