How to expand product of $n$ factors.
up vote
2
down vote
favorite
I have a product say
begin{equation}
F(a,n,x) = prod _{j=0}^{n}(1-{a}^{n-2,j}x)
end{equation}
I want to expand and hope to have general terms of the coefficients. I did for $n= 2,3,4,5,6,7,8...$ I
see it will be different for $n$ even or $n$ odd. We have
begin{equation*}
F(a,2,x)= 1-{x}^{3}+ left( {a}^{2}+1+{a}^{-2} right) {x}^{2}+ left( -{a}^{2}-
1-{a}^{-2} right) x
end{equation*}
begin{equation}
F(a,3,x)= 1+{x}^{4}+ left( -{a}^{3}-a-{a}^{-3}-{a}^{-1} right) {x}^{3}+
left( {a}^{4}+2+{a}^{-2}+{a}^{2}+{a}^{-4} right) {x}^{2}+ left( -{
a}^{3}-a-{a}^{-3}-{a}^{-1} right) x
end{equation}
The coefficients of $a$ gets more interesting as $n$ grows. I am interested in the coefficient of $a.$ Does anyone know how to expand product of $n$ factors.
binomial-coefficients products multinomial-coefficients
asked Nov 11 at 23:31
Learner
404310
add a comment |
up vote
2
down vote
favorite
I have a product say
begin{equation}
F(a,n,x) = prod _{j=0}^{n}(1-{a}^{n-2,j}x)
end{equation}
I want to expand and hope to have general terms of the coefficients. I did for $n= 2,3,4,5,6,7,8...$ I
see it will be different for $n$ even or $n$ odd. We have
begin{equation*}
F(a,2,x)= 1-{x}^{3}+ left( {a}^{2}+1+{a}^{-2} right) {x}^{2}+ left( -{a}^{2}-
1-{a}^{-2} right) x
end{equation*}
begin{equation}
F(a,3,x)= 1+{x}^{4}+ left( -{a}^{3}-a-{a}^{-3}-{a}^{-1} right) {x}^{3}+
left( {a}^{4}+2+{a}^{-2}+{a}^{2}+{a}^{-4} right) {x}^{2}+ left( -{
a}^{3}-a-{a}^{-3}-{a}^{-1} right) x
end{equation}
The coefficients of $a$ gets more interesting as $n$ grows. I am interested in the coefficient of $a.$ Does anyone know how to expand product of $n$ factors.
binomial-coefficients products multinomial-coefficients
asked Nov 11 at 23:31
Learner
404310
By "coefficients of $a$" do you mean "coefficients of $x^k$ in terms of $a$"?
– YiFan
Nov 12 at 1:06
@ YiFan yes you can say that.
– Learner
Nov 12 at 1:23
There should be a development of this product involving the $a^2$-binomial coefficients. See here.
– René Gy
Nov 12 at 18:38
@Thanks René Gy
– Learner
Nov 13 at 2:52
Apply the $q$-binomial formula $prodlimits_{k=0}^{n-1} left(1+q^k tright) = sumlimits_{k=0}^n q^{kleft(k-1right)/2} dbinom{n}{k}_q t^k$ to $n+1$, $a^n x$ and $a^{-2}$ instead of $n$, $t$ and $q$.
– darij grinberg
Nov 14 at 0:44
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
I have a product say
begin{equation}
F(a,n,x) = prod _{j=0}^{n}(1-{a}^{n-2,j}x)
end{equation}
I want to expand and hope to have general terms of the coefficients. I did for $n= 2,3,4,5,6,7,8...$ I
see it will be different for $n$ even or $n$ odd. We have
begin{equation*}
F(a,2,x)= 1-{x}^{3}+ left( {a}^{2}+1+{a}^{-2} right) {x}^{2}+ left( -{a}^{2}-
1-{a}^{-2} right) x
end{equation*}
begin{equation}
F(a,3,x)= 1+{x}^{4}+ left( -{a}^{3}-a-{a}^{-3}-{a}^{-1} right) {x}^{3}+
left( {a}^{4}+2+{a}^{-2}+{a}^{2}+{a}^{-4} right) {x}^{2}+ left( -{
a}^{3}-a-{a}^{-3}-{a}^{-1} right) x
end{equation}
The coefficients of $a$ gets more interesting as $n$ grows. I am interested in the coefficient of $a.$ Does anyone know how to expand product of $n$ factors.
binomial-coefficients products multinomial-coefficients
asked Nov 11 at 23:31
Learner
404310
I have a product say
begin{equation}
F(a,n,x) = prod _{j=0}^{n}(1-{a}^{n-2,j}x)
end{equation}
I want to expand and hope to have general terms of the coefficients. I did for $n= 2,3,4,5,6,7,8...$ I
see it will be different for $n$ even or $n$ odd. We have
begin{equation*}
F(a,2,x)= 1-{x}^{3}+ left( {a}^{2}+1+{a}^{-2} right) {x}^{2}+ left( -{a}^{2}-
1-{a}^{-2} right) x
end{equation*}
begin{equation}
F(a,3,x)= 1+{x}^{4}+ left( -{a}^{3}-a-{a}^{-3}-{a}^{-1} right) {x}^{3}+
left( {a}^{4}+2+{a}^{-2}+{a}^{2}+{a}^{-4} right) {x}^{2}+ left( -{
a}^{3}-a-{a}^{-3}-{a}^{-1} right) x
end{equation}
The coefficients of $a$ gets more interesting as $n$ grows. I am interested in the coefficient of $a.$ Does anyone know how to expand product of $n$ factors.
binomial-coefficients products multinomial-coefficients
binomial-coefficients products multinomial-coefficients
asked Nov 11 at 23:31
Learner
404310
asked Nov 11 at 23:31
Learner
404310
edited Nov 12 at 16:34
asked Nov 11 at 23:31
Learner
404310
asked Nov 11 at 23:31
Learner
404310
asked Nov 11 at 23:31
Learner
404310
404310
By "coefficients of $a$" do you mean "coefficients of $x^k$ in terms of $a$"?
– YiFan
Nov 12 at 1:06
@ YiFan yes you can say that.
– Learner
Nov 12 at 1:23
There should be a development of this product involving the $a^2$-binomial coefficients. See here.
– René Gy
Nov 12 at 18:38
@Thanks René Gy
– Learner
Nov 13 at 2:52
Apply the $q$-binomial formula $prodlimits_{k=0}^{n-1} left(1+q^k tright) = sumlimits_{k=0}^n q^{kleft(k-1right)/2} dbinom{n}{k}_q t^k$ to $n+1$, $a^n x$ and $a^{-2}$ instead of $n$, $t$ and $q$.
– darij grinberg
Nov 14 at 0:44
add a comment |
By "coefficients of $a$" do you mean "coefficients of $x^k$ in terms of $a$"?
– YiFan
Nov 12 at 1:06
@ YiFan yes you can say that.
– Learner
Nov 12 at 1:23
There should be a development of this product involving the $a^2$-binomial coefficients. See here.
– René Gy
Nov 12 at 18:38
@Thanks René Gy
– Learner
Nov 13 at 2:52
Apply the $q$-binomial formula $prodlimits_{k=0}^{n-1} left(1+q^k tright) = sumlimits_{k=0}^n q^{kleft(k-1right)/2} dbinom{n}{k}_q t^k$ to $n+1$, $a^n x$ and $a^{-2}$ instead of $n$, $t$ and $q$.
– darij grinberg
Nov 14 at 0:44
By "coefficients of $a$" do you mean "coefficients of $x^k$ in terms of $a$"?
– YiFan
Nov 12 at 1:06
By "coefficients of $a$" do you mean "coefficients of $x^k$ in terms of $a$"?
– YiFan
Nov 12 at 1:06
@ YiFan yes you can say that.
– Learner
Nov 12 at 1:23
@ YiFan yes you can say that.
– Learner
Nov 12 at 1:23
There should be a development of this product involving the $a^2$-binomial coefficients. See here.
– René Gy
Nov 12 at 18:38
There should be a development of this product involving the $a^2$-binomial coefficients. See here.
– René Gy
Nov 12 at 18:38
@Thanks René Gy
– Learner
Nov 13 at 2:52
@Thanks René Gy
– Learner
Nov 13 at 2:52
Apply the $q$-binomial formula $prodlimits_{k=0}^{n-1} left(1+q^k tright) = sumlimits_{k=0}^n q^{kleft(k-1right)/2} dbinom{n}{k}_q t^k$ to $n+1$, $a^n x$ and $a^{-2}$ instead of $n$, $t$ and $q$.
– darij grinberg
Nov 14 at 0:44
Apply the $q$-binomial formula $prodlimits_{k=0}^{n-1} left(1+q^k tright) = sumlimits_{k=0}^n q^{kleft(k-1right)/2} dbinom{n}{k}_q t^k$ to $n+1$, $a^n x$ and $a^{-2}$ instead of $n$, $t$ and $q$.
– darij grinberg
Nov 14 at 0:44
add a comment |
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By "coefficients of $a$" do you mean "coefficients of $x^k$ in terms of $a$"?
– YiFan
Nov 12 at 1:06
@ YiFan yes you can say that.
– Learner
Nov 12 at 1:23
There should be a development of this product involving the $a^2$-binomial coefficients. See here.
– René Gy
Nov 12 at 18:38
@Thanks René Gy
– Learner
Nov 13 at 2:52
Apply the $q$-binomial formula $prodlimits_{k=0}^{n-1} left(1+q^k tright) = sumlimits_{k=0}^n q^{kleft(k-1right)/2} dbinom{n}{k}_q t^k$ to $n+1$, $a^n x$ and $a^{-2}$ instead of $n$, $t$ and $q$.
– darij grinberg
Nov 14 at 0:44