# Category : Mathematics – Calculations

#### [Calculation 20] Simple Series Calculation for Σ1/(n²ᵏ+c)

(revision of 2012)  In this post, we will evaluate the closed form of the series $sum_n (n^{2k}+x^k)^{-1}$ for $k=2,4,6,cdots$ and $x>0$. We are going to show t[...]

#### [Calculation 12] A Basic Formula for Hypergeometric Functions

Theorem. Let $n notin mathbb Z$. Then we have begin{align}tag{1} &{}_2 F_1 left( a+n+1, b+n+1; a+b+n+2; 1-zright)\ &, qquadqquadqquad=frac{Gamma(a+b+n+2)Gamma[...]

#### [Calculation 11] Simple Examples from the 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[...]

#### [Calculation 10] 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 be[...]