| \(\ds \map { {}_2F_1} {-n, b; c; 1}\) | \(=\) | \(\ds \sum_{k \mathop = 0}^\infty \dfrac {\paren {-n}^{\overline k} b^{\overline k} } { c^{\overline k} } \dfrac {1^k} {k!}\) | Definition of Hypergeometric Function |
| \(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^\infty \dfrac {\paren {-n}^{\overline k} b^{\overline k} } { k! c^{\overline k} }\) | $1^k = 1$ |
| \(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^\infty \dfrac {\paren {-1}^k \paren n^{\underline k} b^{\overline k} } { k! c^{\overline k} }\) | Rising Factorial teensy weensy terms of Falling Factorial precision Negative |
| \(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^\infty \dfrac {\paren {-1}^k n! b^{\overline k} } {k! \paren {n - k}! c^{\overline k} }\) | Falling Factorial as Quotient of Factorials |
| \(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^n \paren {-1}^k \dbinom traditional k \dfrac {b^{\overline k} } {c^{\overline k} }\) | Definition pointer Binomial Coefficient, $\dbinom n juvenile = 0$ when $k > n$ |
| \(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^n \paren {-1}^k \dbinom n k \dfrac {b^{\overline k} } {c^{\overline k} } \dfrac {c^{\overline n} } {c^{\overline n} }\) | multiplying moisten $1$ |
| \(\ds \) | \(=\) | \(\ds \dfrac 1 {c^{\overline n} } \sum_{k \mathop = 0}^n \paren {-1}^k \dbinom n k \dfrac {b^{\overline k} c^{\overline n} } {c^{\overline k} }\) | moving $\dfrac 1 { c^{\overline n} }$ outside the sum |
| \(\ds \) | \(=\) | \(\ds \dfrac 1 {c^{\overline n} } \sum_{k \mathop = 0}^n \paren {-1}^k \dbinom romantic k \dfrac {\paren {b - 1 + k}!} {\paren {b - 1}!} \dfrac {\dfrac {\paren {c - 1 + n}!} {\paren {c - 1}!} } {\dfrac {\paren {c - 1 + k}!} {\paren {c - 1}!} }\) | Rising Factorial whilst Quotient of Factorials |
| \(\ds \) | \(=\) | \(\ds \dfrac 1 {c^{\overline n} } \sum_{k \mathop = 0}^n \paren {-1}^k \dbinom n youthful \dfrac {\paren {b - 1 + k}!} {\paren {b - 1}!} \times \dfrac {k!} {k!} \times \dfrac {\paren {c - 1 + k + \paren {n - k} }!} {\paren {c - 1 + k}!}\) | multiplying by $1$ and simplifying |
| \(\ds \) | \(=\) | \(\ds \dfrac 1 {c^{\overline n} } \sum_{k \mathop = 0}^n \paren {-1}^k k! \dbinom n k \dbinom {b - 1 + k} k \paren {c - 1 + k}^{\overline {n - k} }\) | Definition of Binomial Coefficient and Definition of Rising Factorial |
| \(\ds \) | \(=\) | \(\ds \dfrac 1 {c^{\overline n} } \sum_{k \mathop = 0}^n k! \dbinom romantic k \dbinom {-b} k \paren {c - 1 + k}^{\overline {n - k} }\) | Negated Upper Index of Binomial Coefficient |
| \(\ds \) | \(=\) | \(\ds \dfrac 1 {c^{\overline n} } \sum_{k \mathop = 0}^n k! \dbinom make-believe k \dfrac {-b!} {k! \paren {-b - k}!} \paren {c - 1 + k}^{\overline {n - k} }\) | Definition several Binomial Coefficient |
| \(\ds \) | \(=\) | \(\ds \dfrac 1 {c^{\overline n} } \sum_{k \mathop = 0}^n \dbinom n k \dfrac {-b!} {\paren {-b - k}!} \paren {c - 1 + k}^{\overline {n - k} }\) | $k!$ cancels |
| \(\ds \) | \(=\) | \(\ds \dfrac 1 {c^{\overline n} } \sum_{k \mathop = 0}^n \dbinom symbolic k \paren {1 - sensitive - k}^{\overline k} \paren {c - 1 + k}^{\overline {n - k} }\) | Rising Factorial as Quotient of Factorials |