Let $d=4$, $T\ge d$ and $p\ge 0$ be integers.

By solving Spectral densities of finite dimensional sample covariance matrices we stumbled on a following identity.

\begin{eqnarray} &&\sum\limits_{j=1}^d \left| \begin{array}{l} \left( (\frac{T-d-1}{2}+\delta_{1,j} p+2)^{(\eta-2)} \right)_{\eta=1}^d\\ \left(2^{-\eta} (T-d+1)^{((\delta_{2,j}+\delta_{1,j})p+\eta-1)} \cdot F_{2,1}[\begin{array}{r} 1 & T-d+(\delta_{2,j}+\delta_{1,j})p+\eta \\ & \frac{T-d-1}{2} +\delta_{1,j} p+2\end{array};\frac{1}{2}] \right)_{\eta=1}^d \\ \left( (\frac{T-d-1}{2}+\delta_{3,j} p+2)^{(\eta-2)} \right)_{\eta=1}^d\\ \left(2^{-\eta} (T-d+1)^{((\delta_{4,j}+\delta_{3,j})p+\eta-1)} \cdot F_{2,1}[\begin{array}{r} 1 & T-d+(\delta_{4,j}+\delta_{3,j})p+\eta \\ & \frac{T-d-1}{2} +\delta_{3,j} p+2\end{array};\frac{1}{2}] \right)_{\eta=1}^d \end{array} \right|=\\ &&-\frac{p^2}{16} (T-3)^{(p)} \left( T+2 p^2+2 p-3\right) \end{eqnarray} Here $a^{(n)} := \prod\limits_{j=0}^{n-1} (a+j)$ is the upper Pochhammer symbol.

We discovered this identity by fixing $p$ and then interpolating a polynomial in $T$ to the left hand side, factoring this polynomial and then recognizing a pattern as a function of $p$.

Now, how would you go about proving this identity in a rigorous manner?

Hint: From the generalized Gauss' second summation theorem we have: \begin{eqnarray} F_{2,1}\left[ \begin{array}{r} 1 & T+n \\ & \frac{T+m}{2} \\ \end{array} ;\frac{1}{2} \right] = 2^{T+n-1} \cdot \left\{ \begin{array}{r} \Gamma(\frac{T-m}{2} + n+1)\left(\frac{T+m-2}{2}\right) \sum\limits_{l=0}^{m-n-2} \binom{m-n-2}{l} (-1)^{m-n-2-l} \frac{\Gamma(\frac{T+n+l}{2})}{\Gamma(T+n)\Gamma(\frac{4-m+n+l}{2})} & \mbox{if $m \ge n+2$}\\ \Gamma(\frac{T+m}{2}) \sum\limits_{l=0}^{n+2-m} \binom{n+2-m}{l} (+1)^{m-n-2-l} \frac{\Gamma(\frac{T+n+l}{2})}{\Gamma(T+n) \Gamma(\frac{m-n+l}{2})} & \mbox{otherwise} \end{array} \right. \end{eqnarray} where $n,m \in {\mathbb Z}$.


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