Communities

Writing
Writing
Codidact Meta
Codidact Meta
The Great Outdoors
The Great Outdoors
Photography & Video
Photography & Video
Scientific Speculation
Scientific Speculation
Cooking
Cooking
Electrical Engineering
Electrical Engineering
Judaism
Judaism
Languages & Linguistics
Languages & Linguistics
Software Development
Software Development
Mathematics
Mathematics
Christianity
Christianity
Code Golf
Code Golf
Music
Music
Physics
Physics
Linux Systems
Linux Systems
Power Users
Power Users
Tabletop RPGs
Tabletop RPGs
Community Proposals
Community Proposals
tag:snake search within a tag
answers:0 unanswered questions
user:xxxx search by author id
score:0.5 posts with 0.5+ score
"snake oil" exact phrase
votes:4 posts with 4+ votes
created:<1w created < 1 week ago
post_type:xxxx type of post
Search help
Notifications
Mark all as read See all your notifications »
Q&A

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?

Post

How can a 15 year old construe the LHS of Generalized Vandermonde's Identity, when it lacks summation limits and a summation index?

+0
−1

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!

History
Why does this post require attention from curators or moderators?
You might want to add some details to your flag.
Why should this post be closed?

1 comment thread

Rothe-Hagen is a distraction (1 comment)
Rothe-Hagen is a distraction
r~~‭ wrote almost 3 years ago

This is just a question about Vandermonde, not Rothe-Hagen; I suggest removing the Rothe-Hagen content.

(It's not the case that either of the two identities quoted are generalizations of each other; both are different ways to generalize the original Vandermonde identity. But this is a total aside from your actual question about how to interpret the sum notation in generalized Vandermonde.)