2
votes
1answer
35 views

Approximation of a factorial

In books I can find a lot of approximations of factorial, which doesn't require the a lot of multiplications, like for example Stirling's approximation. Very popular is also the Euler's solution which ...
3
votes
1answer
127 views

More identities of the Ramanujan Double factorial type.

Ramanujan discovered the following identity $$x=\sum_{n=0}^\infty (-1)^n ...
8
votes
3answers
122 views

Factorial limit from gamma function calculation

I want to show that $$\lim_{n\rightarrow\infty}\dfrac{\Gamma\left(n+\frac12\right)}{\sqrt{n}\Gamma(n)}=1$$ Using the formula for $\Gamma\left(n+\frac12\right)$ here, it reduces to ...
1
vote
1answer
60 views

How to derive $\frac{d}{dn}n! = \Gamma(1 + n)\psi^{(0)}(n+ 1)$?

How can you derive $\frac{d}{dn}n! = \Gamma(1 + n)\psi^{(0)}(n + 1)$? I have tried checking Wolfram Alpha for a step-by-step solution, but none is given. Moreover, of what is the second function, ...
4
votes
2answers
418 views

Proof that the gamma function is an extension of the factorial function

I've already proved that $$\Gamma (n)= (n-1)!$$ but I donĀ“t really know what else to do to verify that $\Gamma$ is an extension of the factorial function for real numbers (positive) Thank you! And ...
1
vote
2answers
64 views

How $\alpha(\alpha+1)\ldots(\alpha+k-1)=\frac{\Gamma(\alpha+k)}{\Gamma(\alpha)}$?

Probability function of Negative Binomial Distribution, $NB(\alpha,p)$, is $$P(X=k)=\binom{\alpha+k-1}{k}(1-p)^{\alpha}p^k,\quad \alpha>0$$ Probability generating function of Negative Binomial ...
5
votes
3answers
182 views

How Many Ways to Build a 6-Pack

There is a beverage company here that claims to have a selection of 200 different beers. They have a special deal where you can build your own six pack at a discount. They advertise that there are ...
1
vote
1answer
1k views

How do we calculate factorials for numbers with decimal places? [duplicate]

I was playing with my calculator when I tried $1.5!$. It came out to be $1.32934038817$. Now my question is that isn't factorial for natural numbers only? Like $2!$ is $2\times1$, but how do we ...
0
votes
1answer
39 views

Non-equality with Gamma functions

Let $n \in N$, $k \in Z_+$. Show that $$ \frac{\Gamma\left(k+\frac 12\right)}{\Gamma\left(k+\frac ...
4
votes
4answers
227 views

$\left(-\frac{1}{2}\right)! = \sqrt{\pi}?$ [duplicate]

I recently learned that $\left(-\frac{1}{2}\right)! = \sqrt{\pi}$ but I don't understand how that makes sense. Can someone please explain how this is possible? Thanks!
1
vote
1answer
274 views

Summation with factorial terms (involving Laguerre polynomials)

As part of an exercise including gamma functions and Laguerre polynomials, I need to show that for a Laguerre polynomial $L_n(x)$, $$\int\limits_0^\infty L_n(x)x^ke^{-x}dx = 0 \textrm{ with } n \in ...
11
votes
2answers
6k views

How to find the factorial of a fraction?

From what I know, the factorial function is defined as follows: $$n! = n(n-1)(n-2) \cdots(3)(2)(1)$$ And $0! = 1$. However, this page seems to be saying that you can take the factorial of a ...
1
vote
1answer
143 views

Gamma function question

Gamma function is also known as generalized factorial function . why does the term "generalized" have been used ? Again, why is the Gamma function called Euler's second integral?
4
votes
0answers
91 views

Generalizing Jitsuro Nagura's result: my resulting upper bound for the second chebyshev function is too low. What am I doing wrong?

I've been reading through Jitsuro Nagura's classic proof that there is a prime between $x$ and $\frac{6x}{5}$ and it seems to me that it should be possible to improve on his upper bound for the second ...
0
votes
0answers
78 views

Using Stirling's Formula to approximate a difference of logarithms of factorials in the same way as Jitsuro Nagura.

In Jitsuro Nagura's classic proof of a prime existing between $x$ and $\frac{6x}{5}$, he uses Stirling's formula to show that: $$T\left(x\right) - T\left(\frac{x}{2}\right) - ...
0
votes
0answers
49 views

Do these inequalities regarding the gamma function and factorials work?

I am seeing if it is possible to generalize lemma 1 in Nagura's proof that there is always a prime between $x$ and $\frac{6x}{5}$. In a previous question, I asked whether the following inequality is ...
2
votes
0answers
60 views

Trying to generalize an inequality from Jitsuro Nagura: Does this work?

I am investigating the generality of Lemma 1 in Nagura's proof that there is always a prime between $x$ and $\frac{6x}{5}$. In Lemma 1, Nagura establishes when $n > 1$, $x \ge 1$: ...
0
votes
0answers
33 views

Does it follow that if $\{\frac{x}{2}\} \ge \frac{1}{2} + \frac{\{x\}}{2}$, $\log(\lfloor\frac{x}{2}\rfloor!) \le \log\Gamma(\frac{x+1}{2})$?

The answer seems to be yes. Here's my reasoning. Let $\{x\} = x - \lfloor{x}\rfloor$ Assume $\{\frac{x}{2}\} \ge \frac{1}{2} + \frac{\{x\}}{2}$ $$\log(\lfloor\frac{x}{2}\rfloor!) = ...
0
votes
1answer
110 views

Proving that a specific gamma function is a guaranteed lower bound for a factorial function

In reviewing Ramanujan's proof of Bertrand's postulate, Ramanujan observes that: $$\ln\Gamma(x) - 2\ln\Gamma(\frac{x+1}{2}) \le \ln(\lfloor{x}\rfloor!) - 2\ln(\lfloor\frac{x}{2}\rfloor!)$$ I have ...
0
votes
1answer
39 views

Is it possible to generalize Ramanujan's lower bound for factorials when $\{\frac{x}{b_2}\} + \{\frac{x}{b_3}\} < 1$?

This is the second attempt at a proof. My first attempt had a flaw in its logic. After reviewing the mistake in logic, I believe that with a revised logic, the argument can be saved. The revision ...
1
vote
0answers
33 views

Is it possible to generalize Ramanujan's lower bound for factorials when $\{\frac{x}{b_2}\} + \{\frac{x}{b_3}\} \ge 1$?

This is the second attempt at a proof. My first attempt had a flaw in its logic. After reviewing the mistake in logic, I believe that with a revised logic, the argument can be saved. The revision ...
8
votes
1answer
222 views

Is it possible to generalize Ramanujan's lower bound for factorials that he used in his proof of Bertrand's Postulate?

I am starting to feel more confident in my understanding of Ramanujan's proof of Bertrand's postulate. I hope that I am not getting overconfident. In particular, Ramanujan's does the following ...
1
vote
0answers
137 views

Using the gamma function as an upper and lower bound to the logarithm of a factorial function.

I am trying to find an upper and lower bound for the following function: $$f(x) = \ln(\lfloor\frac{x}{b_1}\rfloor!) - \ln(\lfloor\frac{x}{b_2}\rfloor!) - \ln(\lfloor\frac{x}{b_3}\rfloor!)$$ where ...
5
votes
1answer
356 views

Question about Ramanujan's proof of Bertrand's Postulate

I am reviewing Ramanujan's proof of Bertrand's Postulate which can be found here. At step #7, he writes: "But it is easy to see that..." $\log\Gamma(x) - 2\log\Gamma\left(\frac{1}{2}x + ...
10
votes
1answer
333 views

How to solve $x!=5^x$?

Or, more generally, $$\Gamma (x+1)=\int_0^{\infty}t^{x}e^{-t}dt=p^x$$ with $p \in \mathbb{Z}^+$ and $x \in \mathbb{C}$. Perhaps begin with $\large p^x=p^x \lim_{n \rightarrow ...
5
votes
3answers
201 views

Limits defined for negative factorials (i.e. $(-n)!,\space n\in\mathbb{N}$)

I apoligize if this is a stupid/obvious question, but last night I was wondering how we can compute limits for factorials of negative integers, for instance, how do we evaluate: ...
1
vote
2answers
197 views

Relating Gamma and factorial function for non-integer values.

We have $$\Gamma(n+1)=n!,\ \ \ \ \ \Gamma(n+2)=(n+1)!$$ for integers, so if $\Delta$ is some real value with $$0<\Delta<1,$$ then $$n!\ <\ \Gamma(n+1+\Delta)\ <\ (n+1)!,$$ because ...
4
votes
2answers
116 views

Summation of a binomial style sum, equal to gamma function?

I am trying to prove the following for a very, very long time: $$\sum_{k=0}^j \frac{ (-1)^k}{k! (j-k)!} \frac{1}{2k+1} = \frac{1}{2} \frac{\sqrt{\pi}}{\Gamma(3/2 + j)}$$ or, equivalently ...
13
votes
6answers
2k views

Stirling's formula: proof?

Suppose we want to show that $$ n! \sim \sqrt{2 \pi} n^{n+(1/2)}e^{-n}$$ Instead we could show that $$\lim_{n \to \infty} \frac{n!}{n^{n+(1/2)}e^{-n}} = C$$ where $C$ is a constant. Maybe $C = ...
5
votes
3answers
584 views

Asymptotics of terms and errors in Stirling's Approximation

I have two related questions. Both are related to the asymptotics of Stirling's approximation, which is why I have included them in the same question. I will separate the questions if it is deemed ...
27
votes
1answer
610 views

A series problem by Knuth

I came across the following problem, known as Knuth's Series which originally was an American Mathematical Monthly problem. Prove that $$\sum_{n=1}^\infty ...