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## #1 2017-04-25 01:18:34

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,856
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### Binomial Coefficent Sums

How do I prove this identity?

Last edited by Agnishom (2017-04-25 01:18:48)

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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## #2 2017-04-25 01:22:46

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Binomial Coefficent Sums

Possibly by snake oil...

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #3 2017-04-25 01:42:18

zetafunc
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Registered: 2014-05-21
Posts: 2,196
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### Re: Binomial Coefficent Sums

Induction will probably work, though the right hand side will likely be messy.

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## #4 2017-04-25 03:49:16

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,856
Website

### Re: Binomial Coefficent Sums

Hm, but how do I start?

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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## #5 2017-04-25 09:53:03

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Binomial Coefficent Sums

Just to interject this though:

Wilf Petskovek Zeilberger Nemes wrote:

The problem of finding simple evaluations of major classes of sums that involve factorials,
binomial coefficients,  and their q-analogues,  has  been  completely solved.   Sums  that  have
the rather general form specified in Section 3 can all be done algorithmically, that is to say,
you can do them on your own PC. Your computer evaluates the sum as a simple formula,
if that's possible, and gives you a proof that you can check, or gives you a proof that your
sum cannot be done" in simple closed form, if that is the case.

If you are to start doing lots of these then I suggest A=B and Generatingfunctionology and picking up the appropriate software from those books.

As a beginning course, let us examine the work of the first person to ever attack these sums in a systematic manner, Sister Mary Celine:

https://www.andrew.cmu.edu/course/15-35 … ture02.pdf

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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