Closed form of the following Recurrence Relation

Let $L\colon\mathbb{N}^3 \to \mathbb{N}$ satisfy the following recurrence relationship,

$$ L(a,b,c) = 1 + \sum_{i=0}^{a-1} \sum_{j=0}^{b-1} \sum_{k=0}^{c-1} L(i,j,k), $$

With "initial conditions" $L(0,a,b) = L(c,0,d) = L(0,e,f) = 0$. I am interested in knowing a closed form of $L$.

Work I have done:

I have investigated a simpler case $ G\colon \mathbb{N}^2\to \mathbb{N} $, satisfying $$G(a,b) = 1 + \sum_{i=0}^{a-1} \sum_{j=0}^{b-1} G(i,j)$$ with similar "initial conditions" and can obtain $$G(a,b) = \binom{a+b-2}{a-1}= \binom{a+b-2}{b-1}= \frac{(a-b-2)!}{(a-1)!(b-1)!},$$ so I would have guessed that $$L(a,b,c) = \frac{(a+b+c-3)!}{(a-1)!(b-1)!(c-1)!}, $$ but this isn't correct. I would appreciate any hints on how to find a closed form solution for this.

Continuing according to Fred's very clever deduction of the multiple z-tranform ( the credit should go to him, so I invite him to post it, and votes to be casted on it as well) $$ F(x,y,z) = \frac{{\frac{{x\,y\,z}} {{\left( {1 - x} \right)\left( {1 - y} \right)\left( {1 - z} \right)}}}} {{1 - \frac{{x\,y\,z}} {{\left( {1 - x} \right)\left( {1 - y} \right)\left( {1 - z} \right)}}}} = \sum\limits_{1\, \leqslant \,n} {\left( {\frac{{x\,y\,z}} {{\left( {1 - x} \right)\left( {1 - y} \right)\left( {1 - z} \right)}}} \right)^{\,n} } $$ and considering that $$ \frac{{z^n }} {{\left( {1 - z} \right)^n }} = \sum\limits_{0\, \leqslant \,k} {\left( \begin{gathered} n - 1 + k \\ k \\ \end{gathered} \right)\,\,z^{\,n + k} } = \sum\limits_{0\, \leqslant \,j} {\left( \begin{gathered} j - 1 \\ j - n \\ \end{gathered} \right)\,\,z^{\,j} } $$ then we have $$ \begin{gathered} \sum\limits_{1\, \leqslant \,n} {\left( {\frac{{x\,y\,z}} {{\left( {1 - x} \right)\left( {1 - y} \right)\left( {1 - z} \right)}}} \right)^n } = \hfill \\ = \sum\limits_{0\, \leqslant \,a,\;b,\;c\;} {\left( {\sum\limits_{1\, \leqslant \,n\,\left( { \leqslant \,\min \left( {a,b,c} \right)} \right)} {\left( \begin{gathered} a - 1 \\ a - n \\ \end{gathered} \right)\left( \begin{gathered} b - 1 \\ b - n \\ \end{gathered} \right)\left( \begin{gathered} c - 1 \\ c - n \\ \end{gathered} \right)} } \right)x^{\,a} \,y^{\,b} \,z^{\,c} } \hfill \\ \end{gathered} $$ i.e. $$ \begin{gathered} L(a,b,c) = \sum\limits_{1\, \leqslant \,n\,\left( { \leqslant \,\min \left( {a,b,c} \right)} \right)} {\left( \begin{gathered} a - 1 \\ a - n \\ \end{gathered} \right)\left( \begin{gathered} b - 1 \\ b - n \\ \end{gathered} \right)\left( \begin{gathered} c - 1 \\ c - n \\ \end{gathered} \right)} = \hfill \\ = \sum\limits_{1\, \leqslant \,n\,\left( { \leqslant \,\min \left( {a,b,c} \right)} \right)} {\left( \begin{gathered} a - 1 \\ n - 1 \\ \end{gathered} \right)\left( \begin{gathered} b - 1 \\ n - 1 \\ \end{gathered} \right)\left( \begin{gathered} c - 1 \\ n - 1 \\ \end{gathered} \right)} \hfill \\ \end{gathered} $$ Note that:

• in 1D it becomes $L(a,b,c) = 2^{\,a - 1} $
• in 2D $$ \begin{gathered} L(a,b) = \sum\limits_{1\, \leqslant \,n\,\left( { \leqslant \,\min \left( {a,b} \right)} \right)} {\left( \begin{gathered} a - 1 \\ a - n \\ \end{gathered} \right)\left( \begin{gathered} b - 1 \\ b - n \\ \end{gathered} \right)} = \hfill \\ = \sum\limits_{1\, \leqslant \,n\,\left( { \leqslant \,\min \left( {a,b} \right)} \right)} {\left( \begin{gathered} a - 1 \\ n - 1 \\ \end{gathered} \right)\left( \begin{gathered} b - 1 \\ b - n \\ \end{gathered} \right)} = \left( \begin{gathered} a + b - 2 \\ b - 1 \\ \end{gathered} \right) \hfill \\ \end{gathered} $$
• but in 3D I do not know if there is a closed form.

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Addendum Again thanks to Fred's hint, actually $L(a,b,c)$ can also be expressed in terms of Hypergeometric Function, as $$ \begin{gathered} L(a,b,c) = \sum\limits_{1\, \leqslant \,n\,\left( { \leqslant \,\min \left( {a,b,c} \right)} \right)} {\left( \begin{gathered} a - 1 \\ a - n \\ \end{gathered} \right)\left( \begin{gathered} b - 1 \\ b - n \\ \end{gathered} \right)\left( \begin{gathered} c - 1 \\ c - n \\ \end{gathered} \right)} = \hfill \\ = \sum\limits_{0\, \leqslant \,n\,\left( { \leqslant \,\min \left( {a,b,c} \right) - 1} \right)} {\left( \begin{gathered} a - 1 \\ n \\ \end{gathered} \right)\left( \begin{gathered} b - 1 \\ n \\ \end{gathered} \right)\left( \begin{gathered} c - 1 \\ n \\ \end{gathered} \right)} = \hfill \\ = \sum\limits_{0\, \leqslant \,n\,\left( { \leqslant \,\min \left( {a,b,c} \right) - 1} \right)} {\left( { - 1} \right)^{\,n} \left( \begin{gathered} n - a \\ n \\ \end{gathered} \right)\left( { - 1} \right)^{\,n} \left( \begin{gathered} n - b \\ n \\ \end{gathered} \right)\left( { - 1} \right)^{\,n} \left( \begin{gathered} n - c \\ n \\ \end{gathered} \right)} = \hfill \\ = \sum\limits_{0\, \leqslant \,n\,\left( { \leqslant \,\min \left( {a,b,c} \right) - 1} \right)} {\frac{{\left( {1 - a} \right)^{\,\overline {\,n\,} } \left( {1 - b} \right)^{\,\overline {\,n\,} } \left( {1 - c} \right)^{\,\overline {\,n\,} } }} {{1^{\,\overline {\,n\,} } \;1^{\,\overline {\,n\,} } }}\frac{{\left( { - 1} \right)^{\,n} }} {{n!}}} = \hfill \\ = {}_3F_{\,2} \left( {\left( {1 - a} \right),\left( {1 - b} \right),\left( {1 - c} \right);\;\;1,1;\;\; - 1} \right) \hfill \\ \end{gathered} $$ although, having the variable $z$ fixed at $-1$, we are not much exploiting that function.