I may be asking a stupid question. In this website, the proof of this theorem is posted. But in the last step, it uses that \begin{equation*} _3F_2 \left[ \begin{array}{cc} -x,-y,-z \\ n+1,-x-y-z-n \end{array} ; 1\right] =\sum_{k \mathop = 0}^n \dfrac {a^{\overline k} b^{\overline k} (-n)^{\overline k} } {k! c^{\overline k} (1 + a + b - c - n)^{\overline k} }, \end{equation*} however in the definition of generalized hypergeometric function, the upper limit should be infinity. The answer in previous question has the same question. In the second answer, formula(3)'s upper limit should be n, not infinity. Am I wrong?
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If $n>k$, $(-n)_k$ is vanished. – Username_qs Jul 19 '24 at 05:45
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@Username_qs oh thanks – Tttttt Jul 20 '24 at 14:14