[Calculation 4] Saalschütz’s Theorem

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Theorem. (Saalschütz’s Theorem) $$ {}_3 F_2 (a,b,-n;c,1+a+b-c-n;1) = \frac{(c-a)_n (c-b)_n}{(c)_n (c-a-b)_n} $$ holds for $n\in \mathbb N_0$.   Proof. We recall the Euler Transformation Formula: $$(1-z)^{a+b-c} {}_2 F_1 (a,b;c;z) = {}_2 F_1 (c-a,c-b;c;z).$$ Equating the coefficients of $z^n$ from both … Continue reading