Kummer's Hypergeometric Theorem/Lemma 2

Lemma for Kummer's Hypergeometric Theorem

 * $\ds \lim_{y \mathop \to \infty} \dfrac {\paren {y + \dfrac n 2 + 1}^{\overline x} } {\paren {y + n + 1}^{\overline x} } = 1$

where $y^{\overline x}$ denotes the $x$th rising factorial of $y$.

Proof
From Properties of Limit at Infinity of Real Function: Product Rule, we have:

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 kth limit, we have $\ds \map {f_k} y = \paren {y + \dfrac n 2 + x}$ and $\ds \map {g_k} y = \paren {y + n + x}$

Therefore taking the derivative of the numerator $\map {f_k} y$ and denominator $\map {g_k} y$ $y$, we obtain:
 * $\ds \lim_{y \mathop \to \infty}\paren {\dfrac {\paren {y + \dfrac n 2 + x} } {\paren {y + n + x} } } = \lim_{y \mathop \to \infty} \dfrac 1 1 = 1$

Therefore: