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


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)*...*(f2)*(f1)*f;
((1))_k = (k)_k = k! = 1*2*3*4*5*...*(k2)*(k1)*k;
or, (k+r1)! 1*2*3*...*(r1)*r*(r+1)*...*(k+r1) ((r))_k = ======== = ===================================, (r1)! 1*2*3*...*(r1)
= r*(r+1)*(r+2)*...*(r+(k3))*(r+(k2))*(r+(k1)), = (k'+1)*(k'+2)*(k'+3)*...*(k'+k); ("rising" factorial)
f! 1*2*3*...*(fk)*(fk+1)*(fk+2)*...*f (f)_k = ====== = =====================================, (fk)! 1*2*3*...*(fk)
= (f(k1))*(f(k2))*...*(f2)*(f1)*f, = (k'+1)*...*(k'+k2)*(k'+k1)*(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, cofactorial 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 "nonnatural" 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 multisetbinomial coefficents would be "sequential product functions".
~Kaimbridge~
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