Comments on How can a 15 year old construe the LHS of Generalized Vandermonde's Identity, when it lacks summation limits and a summation index?
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How can a 15 year old construe the LHS of Generalized Vandermonde's Identity, when it lacks summation limits and a summation index?
Paradoxically, though Rothe-Hagen Identity (henceforth RHI)
$\sum\limits_{k=0}^n\frac{x}{x+kz}{x+kz \choose k}\frac{y}{y+(n-k)z}{y+(n-k)z \choose n-k}=\frac{x+y}{x+y+nz}{x+y+nz \choose n}$
generalizes Generalized Vandermonde's Identity (henceforth GVI),
$\sum\limits_{k_1+\cdots +k_p = m} {n_1\choose k_1} {n_2\choose k_2} \cdots {n_p\choose k_p} = { n_1+\dots +n_p \choose m }$
RHI is more intelligible than GVI for my 15 year old. A 15 y.o. can effortlessly write any term of RHI, by substituting the lower limits for all $k$ in the addend. When $k = 0$, just input $k = 0$ in the addend. When $k = n$, just swap all $k$'s in the addend with $n$'s!
But how can a 15 y.o. interpret the LHS of GVI? Or even write the first few terms of the LHS of GVI? It contains no lower and upper limits of summation, and no summation index. GVI contains no $k$, unlike RHI!
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