# Monster Limit 3

Suppose,

$\displaystyle f(x) = \sum_{n=0}^{\infty} f_n x^n$, $\displaystyle F(n) = \sum_{r=0}^{n} f_{r}$ and $\displaystyle \lim_{n \to \infty} \dfrac{F(n)}{n^{r_f}} = \text{d}_f$

Similarly,

$\displaystyle g(x) = \sum_{n=0}^{\infty} g_n x^n$, $\displaystyle G(n) = \sum_{r=0}^n g_r$ and $\displaystyle \lim_{n \to \infty} \dfrac{G(n)}{n^{r_g}} = \text{d}_g$

From Monster Limit 2, we have,

$\begin{cases} \lim_{x \to 1^{-}} (1-x)^{r_f} \ f(x) = \text{d}_f \ \Gamma(1+r_f)\\ \lim_{x \to 1^{-}} (1-x)^{r_g} \ g(x) = \text{d}_g \ \Gamma(1+r_g) \end{cases}$

Now, consider the same quantities for $fg(x) = f(x)g(x)$, i.e,

$\displaystyle fg(x) = \sum_{n=0}^{\infty} {fg}_n x^n$, $\displaystyle FG(n) = \sum_{r=0}^{n} {fg}_{r}$ and $\displaystyle \lim_{n \to \infty} \dfrac{FG(n)}{n^{r_{fg}}} = \text{d}_{fg}$

Again,

$\displaystyle \lim_{x \to 1^{-}} (1-x)^{r_{fg}} \ fg(x) = \text{d}_{fg} \ \Gamma(1+r_{fg})$

Prove That
$\displaystyle \text{d}_{fg} = \text{d}_{f} \text{d}_{g} \dfrac{\Gamma(1+r_f) \Gamma(1+r_g)}{\Gamma(1+r_f + r_g)}$

Note by Ishan Singh
1 year, 5 months ago

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SMO is a molecular target of the teratogen cyclopamine and has also been shown to bind to actin Costal-2 and play a role in the localization of the Ci complex. Studies have shown that SMO can act as a proto-oncogene, and activation of SMO mutations can lead to activation of the unregulated hedgehog pathway. SMO abnormalities are closely related to diseases such as -Jones syndrome and basal cell carcinoma. As its function, SMO is an attractive targeted-drug in cancer treatment.

https://www.creative-biogene.com/crispr-cas9/solution/smo-gene-editing.html

- 4 months, 2 weeks ago