Bound Product of Sine

aleclarsen12 · 2010-03-05 12:25:45

How do I go about expressing the following product without any

's or 's? Can it even be done?

bobbym · 2010-03-05 12:39:42

Hi aleclarsen12;

No closed form that I am aware of. Do you know something about a and n? I could provide a numerical answer then.

aleclarsen12 · 2010-03-05 14:23:22

No, unfortunaly this is a small part of a much larger product where both n and a are dependant varbles.

Is it possible to express this series in terms of

only?

bobbym · 2010-03-06 08:39:16

Hi;

Any product can be immediately turned into a sum. That doesn't mean the the sum is doable either.

aleclarsen12 · 2010-03-06 09:09:39

I don't know if I am helping my cause but there using the laws of Sine. I can show:

It is clear the denominator of the fraction is increasing by powers of 2 and the outside signs of the trig functions are powers of -1. The inside of the trig functions apear to also be some kind of modulation so that every possible "combination" of addition is expressed. The trig function used is alternating between sine and cosine. Knowing this I am not sure how to express this in terms of a single

. Is it possible to do so?

Last edited by aleclarsen12 (2010-03-06 09:10:41)

bobbym · 2010-03-06 09:41:16

Hi;

The inside signs are just every possible permutation of + and - for instance.

a (+ -) b (+ -) c (+ -) d that's why there are 8. 2^3 = 8. The outside sign I am not getting offhand.

It looks like you are going to get a sigma inside each cos or sin. I don't think that is going to make it easy,
unless you can get a closed form for each.

aleclarsen12 · 2010-03-06 10:22:46

Again, I'm not sure how helpful this is but I am trying to post my progress in the problem so that if anyone has any corrections or something to add they can.

Substituting

for I get.

Where

is the q-Pochhammer.

Although this appears to be a closed form, the q-Pochhammer is defined in terms of

(see http://mathworld.wolfram.com/q-PochhammerSymbol.html).

The q-Pochhammer can be expressed differently depending upon the range of

in . Since the product above starts at 2 and moves through n respectivally. It stands to reason that the product would be undefined if . That being said, much of the definiton can be canceled.

So for the purposes above let q-Pochhammer be defined as

.

Ok! Wooh. I feel like I just gave birth to an accountant. Please check over everything I did. I probably made a mistake somewhere. If you have any suggestions or ideas of your own please let me know. Thanks for putting up with my rediculious problem. It seems like the more I try to simplify this the more complex it gets.

Last edited by aleclarsen12 (2010-03-06 12:41:03)

bobbym · 2010-03-06 10:40:10

Hi;

I can't verify your identity:

Try substituting x = 0:

aleclarsen12 · 2010-03-06 12:35:17

Sorry, there was a typo in my last message. The identity is corrected now.

aleclarsen12 · 2010-03-06 13:24:32

How do I evaluate a product in the form of

?

bobbym · 2010-03-06 13:58:10

Hi;

One question at a time, I am still wrestling with your original problem.

A q-pochhammer isn't exactly an elementary form.

Try this:

I have done a lot of simplifying here so check this.

aleclarsen12 · 2010-03-06 14:47:13

The sigma is geometric and can be expressed without bounds as

.

Subsituting this in and simplifying yeilds

I can't verify that statement. I tried n=3 and received 2 different values. I could, however, be wrong.

Last edited by aleclarsen12 (2010-03-06 14:47:44)

bobbym · 2010-03-06 15:05:27

Hi;

I simplified the RHS of the last expression:

A mistake has been made somewhere!?

bobbym · 2010-03-06 15:36:22

Hi;

Found the error:

aleclarsen12 · 2010-03-06 16:37:15

Hmm... Again I am receiving different values for n=3.

bobbym · 2010-03-07 13:40:37

Hi;

Where?

aleclarsen12 · 2010-03-07 13:51:38

(substituting n=3 into your equasion)

I do not belive this is the correct answer.

Last edited by aleclarsen12 (2010-03-07 13:53:32)

bobbym · 2010-03-07 14:09:09

Hi;

Checked the results for n = 3 to 50 digits.

Looks okay, just a disagreement inthe last place, which is to be expected.

aleclarsen12 · 2010-03-07 15:17:06

Oh wow. I'm stupid I mis-evaluated it.

Thats an awsome defenition! How did you derive that?

Also, is there any way to use this identiy to solve my original problem (see my first post in this thread)?

Last edited by aleclarsen12 (2010-03-07 15:22:18)

bobbym · 2010-03-07 15:53:06

H alec;

Also, is there any way to use this identiy to solve my original problem (see my first post in this thread)?

start with your substitution in post 7.

Thats an awsome defenition! How did you derive that?

Alec your not serious are you? It is total CAS work. I was nothing but a package jockey here.
The only intelligence I showed was in getting the syntax right and coaxing 2 packages into
giving me an answer. Yes, they were awesome. Thanks too, Jerry Keipfer, Doron Zeilberger, William Gosper,
Herbert S. Wilf. Daniel LichtBlau, Ilan Vardi, Marko Petskovek ... the geniuses who helped write them.

Math Is Fun Forum

#1 2010-03-05 12:25:45

Bound Product of Sine

#2 2010-03-05 12:39:42

Re: Bound Product of Sine

#3 2010-03-05 14:23:22

Re: Bound Product of Sine

#4 2010-03-06 08:39:16

Re: Bound Product of Sine

#5 2010-03-06 09:09:39

Re: Bound Product of Sine

#6 2010-03-06 09:41:16

Re: Bound Product of Sine

#7 2010-03-06 10:22:46

Re: Bound Product of Sine

#8 2010-03-06 10:40:10

Re: Bound Product of Sine

#9 2010-03-06 12:35:17

Re: Bound Product of Sine

#10 2010-03-06 13:24:32

Re: Bound Product of Sine

#11 2010-03-06 13:58:10

Re: Bound Product of Sine

#12 2010-03-06 14:47:13

Re: Bound Product of Sine

#13 2010-03-06 15:05:27

Re: Bound Product of Sine

#14 2010-03-06 15:36:22

Re: Bound Product of Sine

#15 2010-03-06 16:37:15

Re: Bound Product of Sine

#16 2010-03-07 13:40:37

Re: Bound Product of Sine

#17 2010-03-07 13:51:38

Re: Bound Product of Sine

#18 2010-03-07 14:09:09

Re: Bound Product of Sine

#19 2010-03-07 15:17:06

Re: Bound Product of Sine

#20 2010-03-07 15:53:06

Re: Bound Product of Sine

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