I'm looking for combinatorial proofs of the convolutional identity COP below and its specializations I) and II).
(Edit 6/2/2023: A combinatorial proof is sketched in a blog post by Mike Spivey of a Rothe-Hagen generalized Chu-Vandermonde convolution identity that encompasses specialization I--"Combinatorial Proofs of Two Hagen-Rothe Identities in Concrete Mathematics".)
For $m$ any integer, the sets $[A^{(m)}]$ of (m)-associahedra partition polynomials (ParPs) $A_n^{(m)}(u_1,...,u_n)$ and the sets $[N^{(m)}]$ of (m)-noncrossing partitions / (m)-refined Narayana / (m)-parking function ParPs $N_n^{(m)}(u_1,...,u_2)$ are characterized in my pdf "As Above, So Below: (m)-associahedra and (m)-noncrossing partitions polynomials, Section 1" at my mini-arXiv with explicit multinomial expressions for the coefficients of the monomials of the ParPs.
Given $[A^{(1)}]=[A]$, the refined Euler characteristic partition polynomials, or signed, refined face polynomials, of the convex associahedra polytopes and $[N^{(1)}]= [N]$, the associated signed, refined h-polynomials (or given one of these and the reciprocal ParPs $[A^{(0)}] =[R]$, refined binomial, or Pascal, ParPs for multiplicative inversion of power series / o.g.f.s), the other sets are determined by the core identity
$$[A^{(m)}] = [N^{(m)}][R] = [N]^m[R],$$
denoting substitution of $R_k(u_1,...,u_k)$ for the indeterminate $u_k$ of $N^{(m)}_n(u_1,...,u_n)$ for $1 \leq k \leq n.$ The $[N]^m$ are iterated self-substitutions of $[N]$ into itself with $[N]^{-1}= [N^{(-1)}]$, the inverse to $[N]$ under substitution.
It follows from their relations to compositional inversion of power and Laurent series that $[A^{(m)}]$ are involutive, so
$$ [A^{(m)}] = [A^{(m)}]^{-1} = ([N^{(m)}][R])^{-1} =[R] [N]^{-m}.$$
An equivalent rep of this identity with, e.g., $A^{(m)}(x) = \sum_{n \geq 0} A_n^{(m)}(u_1,...,u_n) x^n$
is the reciprocal identity
$$A^{(m)}(x) = \frac{1}{N^{(-m)}(x)},$$
or
$$1 = A^{(m)}(x) N^{(-m)}(x).$$
The Cauchy convolution identity applies to give the
convolution identity of the partition polynomials (COP):
$$\delta_n = \sum_{k=0}^n A_k^{(m)}(u_1,...,u_n)N_{n-k}^{(-m)}(u_1,...,u_n).$$
Specializations give identities among the (m)-Fuss-Catalan and (m)-Fuss-Narayana number sequences. Be aware that in the literature the names are often interchanged, and, in fact, the two sets are shifted versions of each other with (m)-Fuss-Catalan equal to (m+1)-Fuss-Narayana numbers (absolute values). I define them in terms of the diagonal numbers, or coefficients of the n'th order monomials, of the ParPs with $(-1)^n DA_n^{(m)} = DN_n^{(m+1)}$. (See explicit formulas for these sequences below.)
(To add to potential conflation, the (-1)-Fuss-Catalan numbers are often called the positive Catalan numbers by cluster theorists for geometric reasons.)
Specializations / reductions:
I) For diagonal coefficients, the (m)-Fuss-Catalan and (m)-Fuss-Narayana numbers:
$\delta_n = \sum_{k=0}^n A_k^{(m)}(t,0,...,0)N_{n-k}^{(-m)}(t,0,...,0) = t^n \sum_{k=0}^n DA_k^{(m)}DN_{n-k}^{(-m)},$
or
$\delta_n = \sum_{k=0}^n DA_k^{(m)}DN_{n-k}^{(-m)} = \sum_{k=0}^n(-1)^k DA_k^{(m)}DA_{n-k}^{(-m-1)} = \sum_{k=0}^n (-1)^k DN_k^{(m+1)} DN_{n-k}^{(-m)} .$
II) For the associated single-variable polynomials:
$\delta_n = \sum_{k=0}^n A_k^{(m)}(t,t,...,t)N_{n-k}^{(-m)}(t,t,...,t)= \sum_{k=0}^n TA_k^{(m)}(t)TN_{n-k}^{(-m)}(t) .$
III) Sums of coefficients of the polynomials give I) again, but provide a different avenue of interpretation:
$\delta_n = \sum_{k=0}^n A_k^{(m)}(1,1,...,1)N_{n-k}^{(-m)}(1,1,...,1) =\sum_{k=0}^n DN_k^{(m)}DA^{(-m)}_{n-k} . $
Explicit expressions for the Fuss-Catalan / Fuss-Narayana numbers
Initially,
$DA_0^{(m)} = DN_0^{(m)}= DN_1^{(m)} = -DA_1^{(m)} = 1$ for all integer $m$.
For $n > 1$ evaluate the following expressions with positively signed $m$ for $m \geq 0$ and other expressions otherwise;
$DN_n^{(m)} = \frac{1}{mn+1} \binom{mn+1}{n} = (-1)^n \frac{1}{mn+1} \binom{(-m+1)n-2}{n}$
$ = \frac{1}{n} \binom{mn}{n-1} = (-1)^{n+1} \frac{1}{n} \binom{(-m+1)n-2}{n-1}$
and
$DA_n^{(m)} =(-1)^n \frac{1}{mn+1} \binom{(m+1)n}{n} = \frac{1}{mn+1} \binom{-mn-1}{n}$
$ = (-1)^n \frac{1}{n} \binom{(m+1)n}{n-1} = -\frac{1}{n} \binom{-mn-2}{n-1} .$
Spot checks / illustrations:
For $m=1$,
$\delta_n = \sum_{k=0}^n A_k^{(1)}(u_1,0,...,0)N_{n-k}^{(-1)}(u_1,0,...,0) = \sum_{k=0}^n DA_k^{(1)}DN_{n-k}^{(-1)}$
Diagonal monomials of $[A^{(1)}]$ are $(1,-1u_1,2u_1^2,-5u_1^3, 14u_1^4,...)$, signed Catalans.
Diagonal monomials of $[N^{(-1)}]$ are $(1,1u_1,-1u_1^2,2u_1^3, -5u_1^4, 14u_1^5,...)$, shifted, signed Catalans.
$DA(t) = \frac{-1 + \sqrt{1+4t}}{2t} = 1 - t + 2 t^2 - 5 t^3 + 14 t^4 - 42 t^5 + \cdots$,
and
$DN^{(-1)}(t) = \frac{1 + \sqrt{1+4t}}{2} = 1 + t - t^2 + 2 t^3 - 5 t^4 + 14 t^5 +\cdots,$
so
$DA(t)DN^{(-1)}(t) = 1.$
For $m=-1$,
$\delta_n = \sum_{k=0}^n A_k^{(-1)}(u_1,0,...,0)N_{n-k}^{(1)}(u_1,0,...,0) = \sum_{k=0}^n DA_k^{(-1)}DN_{n-k}^{(1)}$
Diagonal monomials of $A^{(-1)}]$ are $(1,-u_1,0,0,0,...)$ with associated o.g.f. $1-t$.
Diagonal monomials of $[N^{(1)}]$ are $(1,u_1,u_1^2,...)$ with associated o.g.f. $1/(1-t)$.
For $n >1$
$\sum_{k=0}^n A_k^{(-1)}(1,0,...,0) \cdot 1 = \sum_{k=0}^n DA_k^{(-1)} \cdot 1 = 1-1 = 0.$
For $m=0$,
Diagonal monomials of $[A^{(0)}] = [R]$ are $(1,-1u_1,1u_1^2,-1u_1^3, 1u_1^4,...)$ with associated o.g.f. $\frac{1}{1+t}.$
Diagonal monomials of $[N^{(0)}] = [I]$, the identity under substitution, are $(1,u_1,0,...)$ with associated o.g.f. $1+t$.
For $n \geq 0$,
$\sum_{k=0}^n DA^{(0)}_{n-k} \cdot DN_k^{(0)} = \delta_n .$
For $m= 2$,
convolution for $n=4$:
$[DA^{(2)}] = (1,-1,3,-12,55,...) = $ signed OEIS A001764,
$[DN^{(-2)}] = (1,1,-2,7,-30,...) = $ signed (1,A006013).
$\sum_{0}^4 DA_k^{(2)}DN_{n-k}^{(-2)} = (1)(-30)+(-1)(7)+(3)(-2)+ (-12)(1) +(55) (1) = 0 $.
Edit 6/3/2023:
I think the gist of a proof can be found in weaving together Theorem I of “Some convolution series identities” by Raina and Srivastava for a very general convolution identity for pairs of series with discussion of the Lagrange inversion formula 2.1.2 (pg. 2) and 4.2.1 (pg. 23) in "Lagrange inversion" by Gessel. The function $R$ in Gessel corresponds to $x/f^{(-1)}(x)$ in his notation in Theorem 2.1.1, so, to $N^{(-m)}(x)$ here. See also identity (2) on p. 2 of "Lagrange inversion for species" by Gessel and Labelle and Lemma 5.6.4 on pg. 68 of the thesis “Lattice path enumeration and factorization” by Vidales. Mohanty in his book Lattice Path Counting and Applications discusses more specialized convolution identities surrounding eqns. 6.1 and 6.2 on pg. 166 and 6.6 on pg. 168.
