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Topic: Binomial where a=1?
Replies: 8   Last Post: Apr 5, 2013 2:07 PM

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Kaimbridge M. GoldChild

Posts: 79
From: 42.57°N/70.89°W; FN42nn (North Shore, Massachusetts, USA)
Registered: 3/28/05
Re: Binomial where a=1?
Posted: Apr 5, 2013 2:07 PM
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On Apr 3, 11:40 pm, David Bernier <david250@videotron.ca> wrote:

> On 04/03/2013 06:50 PM, Bart Goddard wrote:
>

>> Anonymous <nobody@remailer.paranoici.org> wrote in
>> news:741da947a07ad28bd260dc8dcfe67e8b@remailer.paranoici.org:

>
>>> Is there a special name for a binomial where a=1:
>
>>> (a+b)^n = (1+b)^n ?
>
>>> Something like "uninomial", "unomial" or "anomial"?
>
>> "Geometric series."
>
> (1+b)^n = sum_{k=0 ... n} C(n, k) b^k ;
>
> so, I think we still have the binomial coefficients in the
> expansion, and don't see that it's a genuine geometric series
> on the right hand side ...


Which brings out the usually misrepresented denotation and nature of
the *bi*nomial coefficient.

Where f and r are natural numbers, let

f = k + k'= k + (r - 1) = k'+ (r'-1) = r + r'- 2;

r = f - (r'- 2) = f - (k - 1) = k'+ 1;
r'= f - (r - 2) = f - (k'- 1) = k + 1;

k = f - k'= f - (r - 1) = r'- 1;

k'= f - k = f - (r'- 1) = r - 1;

[ k"= f - (k + k') ];


============================
Sequential Product Functions
============================

Permutations (or, ((r))ising, (f)alling factorials)
---------------------------------------------------
((r))_k = (f)_k = P(f,k) = r*(r+1)*(r+2)*...*(f-2)*(f-1)*f;

((1))_k = (k)_k = k! = 1*2*3*4*5*...*(k-2)*(k-1)*k;

or,
(k+r-1)! 1*2*3*...*(r-1)*r*(r+1)*...*(k+r-1)
((r))_k = ======== = ===================================,
(r-1)! 1*2*3*...*(r-1)

= r*(r+1)*(r+2)*...*(r+(k-3))*(r+(k-2))*(r+(k-1)),
= (k'+1)*(k'+2)*(k'+3)*...*(k'+k);
("rising" factorial)

f! 1*2*3*...*(f-k)*(f-k+1)*(f-k+2)*...*f
(f)_k = ====== = =====================================,
(f-k)! 1*2*3*...*(f-k)

= (f-(k-1))*(f-(k-2))*...*(f-2)*(f-1)*f,
= (k'+1)*...*(k'+k-2)*(k'+k-1)*(k'+k);
("falling" factorial)

[where "(f)_k" is the Pochhammer symbol, with the
reverse denotation traditionally being "r^(k)", but
"((r))_k" is introduced here, both for consistency with
the corresponding coefficient notation, and to eliminate
any potential conflict when exponents become involved.]

Trinomial Coefficient
---------------------
f!
(f__k,k',k") = C(f;k,k',k") = ========;
k!k'!k"!

((Multiset)), (Binomial) Coefficients
-------------------------------------
f!
(f__k,k') = C(f;k,k') = =====,
k!k'!

(f)_f ((1))_f
= ============= = ================,
(k)_k*(k')_k' ((1))_k*((1))_k'

(f)_k (f)_k'
= (f__k) = ===== = ======= = (f__k'),
(k)_k (k')_k'

((r))_k ((r'))_k'
= ((r__k)) = ======= = ========= = ((r'__k')),
(k)_k (k')_k'

(r+r'-2)!
= ========= = ((r,r'__k,k')) = C((r,r';k,k'));
k!k'!

thus (e.g.),
r' r f
1*2*3 * 4*5*6*7*8*9 * 10*11*12
((10,4__3,9)) = ============================== = (12__3,9),
1*2*3 * 1*2*3 * 4*5*6*7*8*9
k k'

((1)_12 12! (12)_12
= =============== = ==== = ===========,
((1))_3*((1))_9 3!9! (3)_3*(9)_9

4*5*6*7*8*9* 10*11*12
= ((4__9)) = ===================== = (12__9),
1*2*3 *4*5*6*7*8*9

((4))_9 (12)_9 (12)_3 ((10))_3
= ======= = ====== = ====== = ========,
((1))_9 (9)_9 (3)_3 ((1))_3

10*11*12
= ((10__3)) = ======== = (12__3);
1*2*3

But these double, co-factorial cases only exist when the sequential
factors are natural numbers??if f is negative or a
decimal, then it is only a reduced (simplified?) binomial coefficient
(unless you want to bring the Gamma function
into it P=).
Is there a specific name for a "non-natural" binomial coefficient,
i.e., in the same way that a "natural" bc is a "combination"?

(Well, hey, before you say "why *would* there be?", there
is a name (and modified notation) for the rising endpoint
argument bc....The "multiset coefficient"!)

Likewise, does a permutation inherently mean a natural numbered factor
sequence? If so, is there a name for the general
*concept* of "(f)_k", not the notation's specific name
("Pochhammer symbol")?
I would say the conceptual name of permutations, combinations and
multiset-binomial coefficents would be "sequential product functions".

~Kaimbridge~

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