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5
votes
5answers
735 views

Summation of an expression

Hi, Does anyone have an idea about an exact or approximate formulae for the following summation? $$ \sum_{j=1}^n \frac{j^k}{(j-1)!} $$ where k is a positive integer (the denominator of the j^th term ...
0
votes
1answer
855 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} ...
2
votes
1answer
627 views

What are conditions to make f(x) defined by f(x)=f(x-1)*x + 1/e unique(for instance convex)?

[Background:] Looking at the powerseries for the gamma-function $ \Gamma(1+x) = 1 + a_1 x + a_2 x^2 - a_3 * x^3 + ... $ then we can arrive at a decomposition $ \Gamma(1+x) = r(x) + g(x) $ ...
3
votes
1answer
618 views

Connection between Bernoulli polynomials and polygamma function

There is an intricate connection between Hurwitz Zeta and the (traditional) polygamma function: $$\psi_n(z)=(-1)^{n+1}n!\zeta(n+1,z)$$ If to use a generalization for Bernoulli numbers, this can be ...
2
votes
0answers
367 views

Can any antidifference (indefinite sum) of a function be expressed in elementary functions and generalized polygamma function if its integral can be expressed in elementary functions?

If the integral or multiplicative integral of a function can be expressed with elementary functions, does it mean its indefinite sum (antidifference) or indefinite product respectively can be ...
6
votes
2answers
1k views

Multiplicative integral of $\Gamma(x)$

A recent question on the notion and notation of multiplicative integrals ( What is the standard notation for a multiplicative integral? ) induced me to play with the Riemann products of the Gamma ...
14
votes
2answers
2k views

Importance of Log Convexity of the Gamma Function

The Bohr-Mollerup theorem states that the Gamma function is the unique function that satisfies: 1) f(x+1) = x*f(x) 2) f(1) = 1 3) ln(f(x)) is convex The Gamma function is meant to interpolate the ...
83
votes
10answers
11k views

Why is the Gamma function shifted from the factorial by 1?

I've asked this question in every math class where the teacher has introduced the Gamma function, and never gotten a satisfactory answer. Not only does it seem more natural to extend the factorial ...
8
votes
1answer
1k views

Errata for Emil Artin's 'The Gamma Function'?

In the English translation of The Gamma Function by Emil Artin (1964 - Holt, Rinehart and Winston) there appears to be a mistake in the formula given for the gamma function on page 24: $$\Gamma(x) = ...
3
votes
3answers
493 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
544 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 ...
2
votes
2answers
1k views

Hadamard's Gamma function

Hi, I'm looking for a link to a derivation of some of the basic properties of Hadamard's Gamma function. For instance that it satisfies $H(x+1)=xH(x)+\frac{1}{\Gamma(1-x)}$ I've been looking on the ...
14
votes
2answers
2k views

Why does the Gamma function satisfy a functional equation?

In question #7656, Peter Arndt asked why the Gamma function completes the Riemann zeta function in the sense that it makes the functional equation easy to write down. Several of the answers were from ...
5
votes
1answer
722 views

Gamma function versions of combinatorial identites?

We can extend the binomial coefficient $\binom{n}{k}$ to $\mathbb{R}$ or $\mathbb{C}$ by defining $\binom{x}{y}=\frac{\Gamma(x+1)}{\Gamma(y+1)\Gamma(x-y+1)}$. Do any the standard binomial coefficient ...
8
votes
3answers
868 views

No simple duplication formula for factorials?

Many special functions including the gamma function have a duplication formula of some sorts. In the case of the gamma function it reads: Gamma(2z) = Gamma(z) Gamma(z+1/2) 22z-1/Gamma(1/2) On ...