# Better way to compute triple (or higher order) convolution

I have an identity that I want to verify:

$$\sum_{i+j+k\ge 2}\binom{n}{i,j,k}{B_{N,i}B_{N,j}B_{N,k}}=\left(\frac{2N^2+n^2-3nN}{2N^2}\right)B_{N,n}+\left(\frac{3n^2-2n-4nN}{2N^2}\right)B_{N,n-1}+\left(\frac{n^2-n}{N^2}\right)B_{N,n-2}$$

First, this is my identity and this is my notation. However, because Mathematica has the N function, when I write the expression in Mathematica I've changed N to m. The $B_{N,p}$ term is the p-th coefficient in the power series representation of the following function.

$$\sum_{k=0}^\infty{\frac{B_{N,k}}{k!}x^k}=\frac{x^N/N!}{e^x-1-x-...-x^{N-1}/(N-1)!}$$

In order to obtain the the $B_{N,k}$, I used the following code:

 T[m_, x_] := Sum[x^j/j!, {j, 0, m}]
g[m_, x_, z_] := (x^m E^(x z))/(m! (E^x - T[m - 1, x]))
b[m_, z_, n_, M_] := Coefficient[ n! Normal[Series[g[m, x, z], {x, 0, M}]], x, n]
B[m_, n_, M_] := b[m, 0, n, M]


Now, I've verified the identity using the following code:

 t3[m_] := Table[ ((2 m^2 + n^2 - 3 n m )/(2 m^2) ) B[m, n, 100] + ((
3 n^2 - 2 n - 4 n m)/(2 m^2) ) B[m, n - 1, 100] + ((n^2 - n)/
m^2) B[m, n - 2, 100] - n! Sum[Sum[B[m, j, 100]/j! B[m, k - j, 100]/(k - j)!, {j, 0, k}] B[m, n - k, 100]/(n - k)!, {k, 0, n} ], {n, 2, 10}]


Notice that my objective in the code is to see if all inputs give me zero. This will verify the identity for small numbers, and while it doesn't constitute a proof I get a sense of its viability and most likely its correctness by calculating the first few values of m.

My question is this: is there a simpler way to verify such identities? My ultimate goal is higher convolutions and I feel that this technique becomes extremely difficult for higher convolution since you are just simply tacking on more and more nested summations.

• There is an entire zoo of combinatorial identities involving sums of products of binomial coefficients and various other animal species, and honestly it's all over my head. However, I have heard that Petkovsek, Wilf, and Zeilberger's book "A = B" is a fantastic resource for the computer-assisted automated proving of a variety of such identites. I don't know whether the problem you're trying to solve falls under the family of identities which are vulnerable to the methods provided in the book because I don't understand it, but if you haven't already read it, it might be a valuable resource. Commented Nov 15, 2014 at 0:13
• Thank you. I just downloaded it from UPenn and will start thumbing through. Commented Nov 15, 2014 at 0:29
• Quite compact: b[n_Integer?Positive, k_Integer?NonNegative] := k! SeriesCoefficient[x^n Exp[-x]/(n Gamma[n, 0, x]), {x, 0, k}]; in actual code, replace x with something like \[FormalX]. Commented Aug 27, 2015 at 11:24