0
votes
0answers
75 views

Simplifying a finite difference sum that represents a model posterior to avoid numerical issues.

Problem Is it possible to simplify/rewrite the following expression, preferably without explicit sums, such that it can be computed without numerical issues when the $n_*$ are in the range of ...
3
votes
0answers
142 views

Iterated Kumaraswamy distributions

The Kumaraswamy distribution has cdf $F(x;a,b) = 1-(1-x^a)^b$. Does anyone know any formulas or properties relating to iterations of this on itself, meaning $$ F_i(x;a,b) = 1-(1-F_{i-1}^a)^b$$ If ...
0
votes
1answer
745 views

Generalizations of a product formula for the gamma function

Hello and Happy holidays. I am interested in generalizations of the following product formula for the gamma function $\Gamma(z)= \int_{0}^{\infty} t^{z-1}e^{-t}dt$ when $n \geq 2$: \begin{align} ...
3
votes
3answers
469 views

Reference request for a “well-known identity” in a paper of Shepp and Lloyd

I ran into a "well-known identity" on page 345 of Shepp and Lloyd's On ordered cycle lengths in a random permutation: $$\int_x^{\infty} \frac{\exp(-y)}y dy = \int_0^x \frac{1-\exp(-y)}y dy - \log x - ...
4
votes
2answers
532 views

Generalized binomial coefficients and Gaussian density

I ran into an expression calculating the expected value of $\exp(i t \sigma)$ where $\sigma$ is the total number of cycles in a uniformly chosen $S_n$ element. The expression is $$E_n (\exp(i t ...