[Calculation 11] Simple Examples from the Dixon Theorem

Dixon Theorem

Here we introduce some examples from the Dixon Theorem. By setting suitable coefficients, we can obtain simple formulas of infinite series, which are related to the Gamma functions.   Example. From the Dixon Theorem, we have \begin{align*} &\text{(i) } 1 … Continue reading

[Calculation 10] Dixon Theorem

Dixon Theorem

Here we note Dixon’s theorem, which gives some special values of ${}_3 F_2$, since the proof is almost automatic by using Gauss and Kummer’s formulas which we’ve shown before. Theorem. (Dixon’s Theorem) $$ {}_3 F_2 (a,b,c;1+a-b,1+a-c;1) = \frac{\Gamma(1+\frac{a}{2})\Gamma(1+a-b)\Gamma(1+a-c)\Gamma(1+\frac{a}{2}-b-c)}{\Gamma(1+a)\Gamma(1+\frac{a}{2}-b)\Gamma(1+\frac{a}{2}-c)\Gamma(1+a-b-c)} $$   … Continue reading