Post History
#7: Post edited
by
DNB
·
2022-01-25T17:54:07Z (almost 3 years ago)
How can a 16 year old construe the LHS of Generalized Vandermonde's Identity, when it lacks summation limits and a summation index?
- 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](https://math.stackexchange.com/q/3572304) (henceforth RHI)
$\sum_{k=0}^n\frac{x}{x+kz}{x+kz \choose k}\frac{y}{y+(n-k)z}{y+(n-k)z \choose n-k}=\frac{x+y}{x+y+nz}{x+y+nz \choose n}$- generalizes [Generalized Vandermonde's Identity](https://math.stackexchange.com/q/1131450) (henceforth GVI),
$\sum_{k_1+\cdots +k_p = m} {n_1\choose k_1} {n_2\choose k_2} \cdots {n_p\choose k_p} = { n_1+\dots +n_p \choose m }$RHI is more intelligible than GVI for my 16 year old. A 16 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 16 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!
- Paradoxically, though [Rothe-Hagen Identity](https://math.stackexchange.com/q/3572304) (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](https://math.stackexchange.com/q/1131450) (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!
#6: Post edited
by
DNB
·
2022-01-25T17:51:57Z (almost 3 years ago)
Paradoxically, though [Rothe-Hagen Identity](https://en.wikipedia.org/wiki/Rothe%E2%80%93Hagen_identity)- $\sum_{k=0}^n\frac{x}{x+kz}{x+kz \choose k}\frac{y}{y+(n-k)z}{y+(n-k)z \choose n-k}=\frac{x+y}{x+y+nz}{x+y+nz \choose n}$
generalizes [Generalized Vandermonde's Identity](https://math.stackexchange.com/q/1131450),- $\sum_{k_1+\cdots +k_p = m} {n_1\choose k_1} {n_2\choose k_2} \cdots {n_p\choose k_p} = { n_1+\dots +n_p \choose m }$
the former is more intelligible than the latter for a 16 year old. A 16 y.o. can effortlessly write any particular term of Rothe-Hagen Identity, 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 16 y.o. interpret the LHS of Generalized Vandermonde's Identity? How can a 16 y.o. even write the first few terms of the LHS of Generalized Vandermonde's Identity? It contains no lower limit of summation, no upper limit of summation, and no summation index (no k, unlike Rothe-Hagen Identity)!
- Paradoxically, though [Rothe-Hagen Identity](https://math.stackexchange.com/q/3572304) (henceforth RHI)
- $\sum_{k=0}^n\frac{x}{x+kz}{x+kz \choose k}\frac{y}{y+(n-k)z}{y+(n-k)z \choose n-k}=\frac{x+y}{x+y+nz}{x+y+nz \choose n}$
- generalizes [Generalized Vandermonde's Identity](https://math.stackexchange.com/q/1131450) (henceforth GVI),
- $\sum_{k_1+\cdots +k_p = m} {n_1\choose k_1} {n_2\choose k_2} \cdots {n_p\choose k_p} = { n_1+\dots +n_p \choose m }$
- RHI is more intelligible than GVI for my 16 year old. A 16 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 16 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!
#5: Post edited
by
DNB
·
2022-01-25T17:32:58Z (almost 3 years ago)
- Paradoxically, though [Rothe-Hagen Identity](https://en.wikipedia.org/wiki/Rothe%E2%80%93Hagen_identity)
- $\sum_{k=0}^n\frac{x}{x+kz}{x+kz \choose k}\frac{y}{y+(n-k)z}{y+(n-k)z \choose n-k}=\frac{x+y}{x+y+nz}{x+y+nz \choose n}$
generalizes [Generalized Vandermonde's Identity](https://en.wikipedia.org/wiki/Vandermonde%27s_identity#Generalized_Vandermonde's_identity),- $\sum_{k_1+\cdots +k_p = m} {n_1\choose k_1} {n_2\choose k_2} \cdots {n_p\choose k_p} = { n_1+\dots +n_p \choose m }$
- the former is more intelligible than the latter for a 16 year old. A 16 y.o. can effortlessly write any particular term of Rothe-Hagen Identity, 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 16 y.o. interpret the LHS of Generalized Vandermonde's Identity? How can a 16 y.o. even write the first few terms of the LHS of Generalized Vandermonde's Identity? It contains no lower limit of summation, no upper limit of summation, and no summation index (no k, unlike Rothe-Hagen Identity)!
- Paradoxically, though [Rothe-Hagen Identity](https://en.wikipedia.org/wiki/Rothe%E2%80%93Hagen_identity)
- $\sum_{k=0}^n\frac{x}{x+kz}{x+kz \choose k}\frac{y}{y+(n-k)z}{y+(n-k)z \choose n-k}=\frac{x+y}{x+y+nz}{x+y+nz \choose n}$
- generalizes [Generalized Vandermonde's Identity](https://math.stackexchange.com/q/1131450),
- $\sum_{k_1+\cdots +k_p = m} {n_1\choose k_1} {n_2\choose k_2} \cdots {n_p\choose k_p} = { n_1+\dots +n_p \choose m }$
- the former is more intelligible than the latter for a 16 year old. A 16 y.o. can effortlessly write any particular term of Rothe-Hagen Identity, 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 16 y.o. interpret the LHS of Generalized Vandermonde's Identity? How can a 16 y.o. even write the first few terms of the LHS of Generalized Vandermonde's Identity? It contains no lower limit of summation, no upper limit of summation, and no summation index (no k, unlike Rothe-Hagen Identity)!
#4: Post edited
by
DNB
·
2022-01-25T17:13:08Z (almost 3 years ago)
- Paradoxically, though [Rothe-Hagen Identity](https://en.wikipedia.org/wiki/Rothe%E2%80%93Hagen_identity)
- $\sum_{k=0}^n\frac{x}{x+kz}{x+kz \choose k}\frac{y}{y+(n-k)z}{y+(n-k)z \choose n-k}=\frac{x+y}{x+y+nz}{x+y+nz \choose n}$
- generalizes [Generalized Vandermonde's Identity](https://en.wikipedia.org/wiki/Vandermonde%27s_identity#Generalized_Vandermonde's_identity),
- $\sum_{k_1+\cdots +k_p = m} {n_1\choose k_1} {n_2\choose k_2} \cdots {n_p\choose k_p} = { n_1+\dots +n_p \choose m }$
- the former is more intelligible than the latter for a 16 year old. A 16 y.o. can effortlessly write any particular term of Rothe-Hagen Identity, 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 16 y.o. interpret the LHS of Generalized Vandermonde's Identity? How can a 16 y.o. even write t the first few terms of the LHS of Generalized Vandermonde's Identity? It contains no lower limit of summation, no upper limit of summation, and no summation index!
- Paradoxically, though [Rothe-Hagen Identity](https://en.wikipedia.org/wiki/Rothe%E2%80%93Hagen_identity)
- $\sum_{k=0}^n\frac{x}{x+kz}{x+kz \choose k}\frac{y}{y+(n-k)z}{y+(n-k)z \choose n-k}=\frac{x+y}{x+y+nz}{x+y+nz \choose n}$
- generalizes [Generalized Vandermonde's Identity](https://en.wikipedia.org/wiki/Vandermonde%27s_identity#Generalized_Vandermonde's_identity),
- $\sum_{k_1+\cdots +k_p = m} {n_1\choose k_1} {n_2\choose k_2} \cdots {n_p\choose k_p} = { n_1+\dots +n_p \choose m }$
- the former is more intelligible than the latter for a 16 year old. A 16 y.o. can effortlessly write any particular term of Rothe-Hagen Identity, 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 16 y.o. interpret the LHS of Generalized Vandermonde's Identity? How can a 16 y.o. even write the first few terms of the LHS of Generalized Vandermonde's Identity? It contains no lower limit of summation, no upper limit of summation, and no summation index (no k, unlike Rothe-Hagen Identity)!
#3: Post edited
by
DNB
·
2022-01-25T06:10:31Z (almost 3 years ago)
How can a 16 year old construe $\sum_{k_1+\cdots +k_p = m} {n_1\choose k_1} {n_2\choose k_2} \cdots {n_p\choose k_p} = { n_1+\dots +n_p \choose m }$?
- How can a 16 year old construe the LHS of Generalized Vandermonde's Identity, when it lacks summation limits and a summation index?
#2: Post edited
by
DNB
·
2022-01-25T06:09:55Z (almost 3 years ago)
- Paradoxically, though [Rothe-Hagen Identity](https://en.wikipedia.org/wiki/Rothe%E2%80%93Hagen_identity)
- $\sum_{k=0}^n\frac{x}{x+kz}{x+kz \choose k}\frac{y}{y+(n-k)z}{y+(n-k)z \choose n-k}=\frac{x+y}{x+y+nz}{x+y+nz \choose n}$
- generalizes [Generalized Vandermonde's Identity](https://en.wikipedia.org/wiki/Vandermonde%27s_identity#Generalized_Vandermonde's_identity),
- $\sum_{k_1+\cdots +k_p = m} {n_1\choose k_1} {n_2\choose k_2} \cdots {n_p\choose k_p} = { n_1+\dots +n_p \choose m }$
the former is more intelligible than the latter for a 16 year old. A 16 y.o. can effortlessly write the terms of Rothe-Hagen Identity, by substituting the lower limits into the addend.But how would you write the first few terms of the LHS of Generalized Vandermonde's Identity? It contains no lower limit of summation, no upper limit of summation, and no summation index!
- Paradoxically, though [Rothe-Hagen Identity](https://en.wikipedia.org/wiki/Rothe%E2%80%93Hagen_identity)
- $\sum_{k=0}^n\frac{x}{x+kz}{x+kz \choose k}\frac{y}{y+(n-k)z}{y+(n-k)z \choose n-k}=\frac{x+y}{x+y+nz}{x+y+nz \choose n}$
- generalizes [Generalized Vandermonde's Identity](https://en.wikipedia.org/wiki/Vandermonde%27s_identity#Generalized_Vandermonde's_identity),
- $\sum_{k_1+\cdots +k_p = m} {n_1\choose k_1} {n_2\choose k_2} \cdots {n_p\choose k_p} = { n_1+\dots +n_p \choose m }$
- the former is more intelligible than the latter for a 16 year old. A 16 y.o. can effortlessly write any particular term of Rothe-Hagen Identity, 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 16 y.o. interpret the LHS of Generalized Vandermonde's Identity? How can a 16 y.o. even write t the first few terms of the LHS of Generalized Vandermonde's Identity? It contains no lower limit of summation, no upper limit of summation, and no summation index!
#1: Initial revision
by
DNB
·
2022-01-25T06:07:44Z (almost 3 years ago)
How can a 16 year old construe $\sum_{k_1+\cdots +k_p = m} {n_1\choose k_1} {n_2\choose k_2} \cdots {n_p\choose k_p} = { n_1+\dots +n_p \choose m }$?
Paradoxically, though [Rothe-Hagen Identity](https://en.wikipedia.org/wiki/Rothe%E2%80%93Hagen_identity)
$\sum_{k=0}^n\frac{x}{x+kz}{x+kz \choose k}\frac{y}{y+(n-k)z}{y+(n-k)z \choose n-k}=\frac{x+y}{x+y+nz}{x+y+nz \choose n}$
generalizes [Generalized Vandermonde's Identity](https://en.wikipedia.org/wiki/Vandermonde%27s_identity#Generalized_Vandermonde's_identity),
$\sum_{k_1+\cdots +k_p = m} {n_1\choose k_1} {n_2\choose k_2} \cdots {n_p\choose k_p} = { n_1+\dots +n_p \choose m }$
the former is more intelligible than the latter for a 16 year old. A 16 y.o. can effortlessly write the terms of Rothe-Hagen Identity, by substituting the lower limits into the addend.
But how would you write the first few terms of the LHS of Generalized Vandermonde's Identity? It contains no lower limit of summation, no upper limit of summation, and no summation index!