Kummer's Hypergeometric Theorem/Lemma 2

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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:

\(\ds \lim_{y \mathop \to \infty} \dfrac {\paren {y + \dfrac n 2 + 1}^{\overline x} } {\paren {y + n + 1}^{\overline x} }\) \(=\) \(\ds \lim_{y \mathop \to \infty} \paren {\paren {\dfrac {\paren {y + \dfrac n 2 + 1} } {\paren {y + n + 1} } } \paren {\dfrac {\paren {y + \dfrac n 2 + 2} } {\paren {y + n + 2} } } \cdots \paren {\dfrac {\paren {y + \dfrac n 2 + x} } {\paren {y + n + x} } } }\) Definition of Rising Factorial
\(\ds \) \(=\) \(\ds \lim_{y \mathop \to \infty} \paren {\dfrac {\paren {y + \dfrac n 2 + 1} } {\paren {y + n + 1} } } \lim_{y \mathop \to \infty} \paren {\dfrac {\paren {y + \dfrac n 2 + 2} } {\paren {y + n + 2} } } \cdots \lim_{y \mathop \to \infty} \paren {\dfrac {\paren {y + \dfrac n 2 + x} } {\paren {y + n + x} } }\) 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 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 $\map {g_k} y$ with respect to $y$, we proceed:

\(\ds \lim_{y \mathop \to \infty} \dfrac {y + \dfrac n 2 + x} {y + n + x}\) \(=\) \(\ds \lim_{y \mathop \to \infty} \dfrac {\map {\frac \d {\d y} } {y + \dfrac n 2 + x} } {\map {\frac \d {\d y} } {y + n + x} }\)
\(\ds \) \(=\) \(\ds \lim_{y \mathop \to \infty} \dfrac {1 + 0 + 0} {1 + 0 + 0}\) Derivative of Identity Function, Derivative of Constant
\(\ds \) \(=\) \(\ds 1\) trivially


Therefore:

\(\ds \lim_{y \mathop \to \infty} \dfrac {\paren {y + \dfrac n 2 + 1}^{\overline x} } {\paren {y + n + 1}^{\overline x} }\) \(=\) \(\ds \lim_{y \mathop \to \infty} \paren {\paren {\dfrac {\paren {y + \dfrac n 2 + 1} } {\paren {y + n + 1} } } \paren {\dfrac {\paren {y + \dfrac n 2 + 2} } {\paren {y + n + 2} } } \cdots \paren {\dfrac {\paren {y + \dfrac n 2 + x} } {\paren {y + n + x} } } }\) Definition of Rising Factorial
\(\ds \) \(=\) \(\ds \lim_{y \mathop \to \infty} \paren {\dfrac {\paren {y + \dfrac n 2 + 1} } {\paren {y + n + 1} } } \lim_{y \mathop \to \infty} \paren {\dfrac {\paren {y + \dfrac n 2 + 2} } {\paren {y + n + 2} } } \cdots \lim_{y \mathop \to \infty} \paren {\dfrac {\paren {y + \dfrac n 2 + x} } {\paren {y + n + x} } }\) 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
\(\ds \) \(=\) \(\ds 1^x\)
\(\ds \) \(=\) \(\ds 1\)

$\blacksquare$