Kummer's Hypergeometric Theorem/Lemma 1
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Lemma for Kummer's Hypergeometric Theorem
- $\ds \lim_{y \mathop \to \infty} \dfrac {y^{\underline k} } {\paren {y + n + 1}^{\overline k} } = 1$
where $y^{\underline k}$ denotes the $k$th falling factorial of $y$.
Proof
From Properties of Limit at Infinity of Real Function: Product Rule, we have:
| \(\ds \lim_{y \mathop \to \infty} \dfrac {y^{\underline k} } {\paren {y + n + 1}^{\overline k} }\) | \(=\) | \(\ds \lim_{y \mathop \to \infty} \paren {\paren {\dfrac y {\paren {y + n + 1} } } \paren {\dfrac {\paren {y - 1} } {\paren {y + n + 2} } } \cdots \paren {\dfrac {\paren {y + 1 - k} } {\paren {y + n + k} } } }\) | Definition of Rising Factorial and Definition of Falling Factorial | |||||||||||
| \(\ds \) | \(=\) | \(\ds \lim_{y \mathop \to \infty} \paren {\dfrac y {\paren {y + n + 1} } } \lim_{y \mathop \to \infty} \paren {\dfrac {\paren {y - 1} } {\paren {y + n + 2} } } \cdots \lim_{y \mathop \to \infty} \paren {\dfrac {\paren {y + 1 - k} } {\paren {y + n + k} } }\) | Properties of Limit at Infinity of Real Function: Product Rule |
From L'Hôpital's Rule:Corollary 2, we have:
- $\ds \lim_{y \mathop \to a^+} \frac {\map f y} {\map g y} = \lim_{y \mathop \to a^+} \frac {\map {f'} y} {\map {g'} y}$
In the present example, for the $k$th limit, we have:
- $\map {f_k} y = \paren {y + 1 - k}$
- $\map {g_k} y = \paren {y + n + k}$
Therefore taking the derivative of the numerator $\map {f_k} y$ and denominator $\map {g_k} y$ with respect to $y$, we proceed:
| \(\ds \lim_{y \mathop \to \infty} \dfrac {y + 1 - k} {y + n + k}\) | \(=\) | \(\ds \lim_{y \mathop \to \infty} \dfrac {\map {\frac \d {\d y} } {y + 1 - k} } {\map {\frac \d {\d y} } {y + n + k} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \lim_{y \mathop \to \infty} \dfrac {1 + 0 + 0} {1 + 0 + 0}\) | Derivative of Identity Function, Derivative of Constant | |||||||||||
| \(\ds \) | \(=\) | \(\ds 1\) | trivially |
Therefore:
| \(\ds \lim_{y \mathop \to \infty} \dfrac {y^{\underline k} } {\paren {y + n + 1}^{\overline k} }\) | \(=\) | \(\ds \lim_{y \mathop \to \infty} \paren {\paren {\dfrac y {\paren {y + n + 1} } } \paren {\dfrac {\paren {y - 1} } {\paren {y + n + 2} } } \cdots \paren {\dfrac {\paren {y + 1 - k} } {\paren {y + n + k} } } }\) | Definition of Rising Factorial and Definition of Falling Factorial | |||||||||||
| \(\ds \) | \(=\) | \(\ds \lim_{y \mathop \to \infty} \paren {\dfrac y {\paren {y + n + 1} } } \lim_{y \mathop \to \infty} \paren {\dfrac {\paren {y - 1} } {\paren {y + n + 2} } } \cdots \lim_{y \mathop \to \infty} \paren {\dfrac {\paren {y + 1 - k} } {\paren {y + n + k} } }\) | Properties of Limit at Infinity of Real Function: Product Rule | |||||||||||
| \(\ds \) | \(=\) | \(\ds \lim_{y \mathop \to \infty} \dfrac 1 1 \lim_{y \mathop \to \infty} \dfrac 1 1 \cdots \lim_{y \mathop \to \infty} \dfrac 1 1\) | L'Hôpital's Rule:Corollary 2 | |||||||||||
| \(\ds \) | \(=\) | \(\ds 1^k\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 1\) |
$\blacksquare$