Thread: Product of arithmetic sequence in term of x.

1. Re: Product of arithmetic sequence in term of x.

Also can I ask you another question:can a factorial of x can be said that is a product of arithmetic sequence for the first x terms starting from x with diffrence of -1?Thanks again!

2. Re: Product of arithmetic sequence in term of x.

Originally Posted by lai001
Also can I ask you another question:can a factorial of x can be said that is a product of arithmetic sequence for the first x terms starting from x with diffrence of -1?Thanks again!
Depends... how many terms? If you take the product of the first x terms, then yes.

3. Re: Product of arithmetic sequence in term of x.

I already stated that is the first x terms....Is this just a special case for the products of arithmetic sequence?Also, it is possible to calculate gamma function of a fraction using a scientific calculator(like casio 570)?Thanks.

4. Re: Product of arithmetic sequence in term of x.

Given the sequence: $(a_n)_{n\ge 1}$ defined by $a_n = x+1-n$, the product

$\displaystyle \prod_{n=1}^k a_n = \begin{cases}\dfrac{x!}{(x-k)!} & k\ge x \\ 0 & k>x\end{cases}$

To use a scientific calculator for the Gamma function, I'd recommend the Lanczos approximation:
https://en.wikipedia.org/wiki/Lanczos_approximation

5. Re: Product of arithmetic sequence in term of x.

Actually, is a factorial of a non interger positive number and gamma function has any usage in daily life?Thanks.

6. Re: Product of arithmetic sequence in term of x.

Originally Posted by lai001
Actually, is a factorial of a non interger positive number and gamma function has any usage in daily life?Thanks.
That really depends on what you do for a living. Some probability distributions are defined directly in terms of the Gamma function. It shows up in many unrelated areas of mathematics. Notice that the simplification I provided in post #14 produced a strange $\sqrt{\pi}$? So, there is a connection between the Gamma function and circles. It is a very useful function in many areas of mathematics.

$\sqrt{\pi} = \dfrac{4^{x-1}\Gamma\left(x-\dfrac{1}{2}\right)\Gamma(x)}{\Gamma(2x-1)}$

The right hand side of the equation does not look like a constant function, but it is! Somehow!

7. Re: Product of arithmetic sequence in term of x.

I had viewed the wikipedia page about Lanczos approximation,but I still cannot understand it.(maybe I am a fool...)Can you explain for me about Lanczos aporoximation as elementary as possible( in functions of a scientific calculator) and also provide an example for me?Sorry for asking so much question and bothering you. Thanks a lot.

8. Re: Product of arithmetic sequence in term of x.

Originally Posted by lai001
I had viewed the wikipedia page about Lanczos approximation,but I still cannot understand it.(maybe I am a fool...)Can you explain for me about Lanczos aporoximation as elementary as possible( in functions of a scientific calculator) and also provide an example for me?Sorry for asking so much question and bothering you. Thanks a lot.
You are not a fool. Estimating the Gamma function is not for the faint of heart. It typically requires a computer algebra system. There are other approximations. For example, Stirling's approximation. For small values of $x$, you will get relatively close to the actual factorial. Stirling's approximation is probably easier to use, but far less accurate of an approximation.

9. Re: Product of arithmetic sequence in term of x.

So it is not possible to express Lanczos aporoximation using functions of a scientific calculator?or must use a math software to calculate it?

YES!

11. Re: Product of arithmetic sequence in term of x.

Is this means that finding product of an arithmetic sequence(if very large that not able to multiply it one by one) needs a math software, since the product is in terms of gamma function and uses Lanczos approximation to get a numerical value?

12. Re: Product of arithmetic sequence in term of x.

All of these functions become difficult to approximate because of how large the numbers get. It depends on how accurate you need for your approximation. The Lanczos approximation CAN be used with a scientific calculator. You would need to study the math behind it, fully understand it, and then you would be able to use it. There are probably papers written that you could read that could even give you the correct algorithm for how to plug it into a scientific calculator, but that is not my area of mathematics, nor do I have any interest in doing something like that, so I have never taken the time to look.

13. Re: Product of arithmetic sequence in term of x.

If I am in the level of able to understand mathematics behind the Lanczos approximation and read mathematics papers maybe I will not ask question about this..... Anyway, maybe I can find out what means in the formula in the wikipedia by myself....... Thanks.

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