# Binomial identities,combinatorial, equivalence

1. Dec 1, 2011

### zmth

NOte this is not a homework nor related to any course nor any test problem etc. - entirely my own interest and study.
Re\: text by Biedenharn and Louck "Angular momentum in Q.Physics" . I derive an expression for the norm squared wrt a certain expression in Boson calculus. You don't really need to get into this in order to answer my question. Anyway below in the following is my two summation index form and other equivalent expressions later. Now the authors don't derive how they arrived at their expression but just give the answer as

(k+j+1)!*(k-j)!/(2*j+1)

and in fact this is equivalent(equal) to my more complicated expressions and yes my expressions also have two extra variables or call them parameters if you like which are m and mp in addtion to j and k . Now my question is why is it that the m and mp drop out of my expressions and what binomial/ combinatorial identities etc. are used to get from my multiple summation expressions with the two extra m and mp terms to his ofcourse much simpler reduced and preferred expression again being (k+j+1)!*(k-j)!/(2*j+1) with no summations and without m and mp. Now we begin with my equivalent expressions of which anyone can verify reduce exactly to the author's answer for any and all choices of integer variables under the constraints .Here for simplicity just consider k and j positive integers where k >= j ; m and mp can be positive or negative with maximum abs. value j so
-j <= m,mp <= j and in these latter parameters case you may notice like pertaining or reference to quantum angular momentum to which these pertain and actually k,j,m,mp could all be half an odd integer but to keep it simpler just assume they are all integer. Here are my expressions.

(j-m)*(j+m)*(j-mp)*(j+mp)*sum_{s=max(0,mp-m)}^{min(mp,-m)+k} *(sum_{sp=max(mp-m,0,s-k+j)}^{min(j-m,j+mp,s)}(-1)^(s-sp)* binom(k-j ,s-sp)/((j+mp-sp)!*sp!*(m-mp+sp)!*(j-m-sp)!))^2*(k-m-s)!*(k+mp-s)! *s!*(m-mp+s)!

Or equivalently using 3 summation indexes expanding the square written as:

(j+mp)!*(j-mp)!*(k-j)!^2*sum_{s=max(0,mp-m)}^{min(mp,-m)+k} *sum_{sp=max(mp-m,0,s-k+j)}^{min(j-m,j+mp,s)}(-1)^(s+sp)*binom(s,sp)*binom(k-m-s,j-m-sp)*binom(j+m,j+mp-sp) *sum_{s1=max(mp-m,0,s-k+j)}^{min(j-m,j+mp,s)}(-1)^(s+s1)*binom(s,s1)*binom(k-m-s,j-m-s1)*binom(j+m,j+mp-s1)

OR as :

(j+m)!*(j-m)!*(k-j)!^2*sum_{s=max(0,mp-m)}^{min(mp,-m)+k} *sum_{sp=max(mp-m,0,s-k+j)}^{min(j-m,j+mp,s)}(-1)^(s+sp)*binom(s,sp)*binom(k+mp-s,j+mp-sp)*binom(j-mp,j-m-sp) *sum_{s1=max(mp-m,0,s-k+j)}^{min(j-m,j+mp,s)}(-1)^(s+s1)*binom(s+m-mp ,s-s1)*binom(k-m-s,j-m-s1)*binom(j+mp,s1)

Here binom(a,b) means the usual a!/b!/(a-b)! with a slightly different interpretation when some or one of the variables are negative integers.That is all 3 of these are all equivalent as you may verify by trials esp. if you have Macsyma(unfortunately they have been out of business for quite a few years now) but I assume Maple or some other symbolic math software could also do. In fact I could go on and on writing numerous other different looking but equivalent 3 sum index expression forms.

The issue is the mathematical(quantum) physics of the problem gives me directly these two and/or 3 index summations forms and the authors give no hint as how they arrived at their reduced expression. ANd yes again there are more variables, here m and mp, in my summations versions than the reduced simple form from which somehow m and mp drop out or cancel out in the reduced form. This is my question - why and how ? I can't for the life of me show how to get from the double or triple summation forms to the simple reduced (no summations) form which again is :

(k+j+1)!*(k-j)!/(2*j+1)

and if one does it correctly substituting and evaluating my multiple sum versions one will get exactly this value for any and all choices of integer variables within the prescribed constraints. If someone thinks they may have a solution or can see thru the "forest" but does not get values that agree then i could send a .tex or .dvi or even .pdf files which may show your errors in interpreting my ascii only attempt at writing my mathematical expressions.

2. May 20, 2012

### zmth

Note(I am the poster zmth) there are errors here and I will repost. In particular left off the factorial sign in 4 places in the first expression:

(j-m)*(j+m)*(j-mp)*(j+mp)*sum_{s=max(0,mp-m)}^{min(mp,-m)+k} *(sum_{sp=max(mp-m,0,s-k+j)}^{min(j-m,j+mp,s)}(-1)^(s-sp)* binom(k-j ,s-sp)/((j+mp-sp)!*sp!*(m-mp+sp)!*(j-m-sp)!))^2*(k-m-s)!*(k+mp-s)! *s!*(m-mp+s)!

(j-m)!*(j+m)!*(j-mp)!*(j+mp)!*sum_{s=max(0,mp-m)}^{min(mp,-m)+k} *(sum_{sp=max(mp-m,0,s-k+j)}^{min(j-m,j+mp,s)}(-1)^(s-sp)* binom(k-j ,s-sp)
/((j+mp-sp)!*sp!*(m-mp+sp)!*(j-m-sp)!))^2*(k-m-s)!*(k+mp-s)! *s!*(m-mp+s)!

Also seems there is some error in the expression:

(j+mp)!*(j-mp)!*(k-j)!^2*sum_{s=max(0,mp-m)}^{min(mp,-m)+k} *sum_{sp=max(mp-m,0,s-k+j)}^{min(j-m,j+mp,s)}(-1)^(s+sp)*binom(s,sp)*binom(k-m-s,j-m-sp)*binom(j+m,j+mp-sp) *sum_{s1=max(mp-m,0,s-k+j)}^{min(j-m,j+mp,s)}(-1)^(s+s1)*binom(s,s1)*binom(k-m-s,j-m-s1)*binom(j+m,j+mp-s1)

so just leave that out. NOw the expression after that which reads:

OR as :

(j+m)!*(j-m)!*(k-j)!^2*sum_{s=max(0,mp-m)}^{min(mp,-m)+k} *sum_{sp=max(mp-m,0,s-k+j)}^{min(j-m,j+mp,s)}(-1)^(s+sp)*binom(s,sp)*binom(k+mp-s,j+mp-sp)*binom(j-mp,j-m-sp) *sum_{s1=max(mp-m,0,s-k+j)}^{min(j-m,j+mp,s)}(-1)^(s+s1)*binom(s+m-mp ,s-s1)*binom(k-m-s,j-m-s1)*binom(j+mp,s1)

IS correct as stands.