Mixing
generating function for recursive formula of two variables
Clash Royale CLAN TAG#URR8PPP
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I have a pice-wise function which is divided into even and odd parts as you can see in the following. I hope to know if there's a generating function that fit this recursive relation.
$$
f(x) =
left{
beginarrayll
2^j+1 sum_n=0^j alpha_ell,n x^2n+1 &ell=2j+1,\
2^j sum_n=0^j alpha_ell,n x^2n &ell=2j,\
endarray
right.
$$
$ell=0,1,2,ldots$, the coefficient $alpha_ell,n$ has the recursive relation
$$
left{
beginarrayll
alpha_2j+1,n = -(n+1)alpha_2j,n + 1 + alpha_2j,n &n=0,1,2,ldots,j-1\
alpha_2j,n = -( 2n + m)alpha_2j - 1,n + 2alpha_2j-1,n-1 &n=1,2,ldots,j-1\
endarray
right.
$$
(should mention here that the maximum of $n$ is $j-1$, so it doesn't break the boundary) and $alpha_0,0 = 1$, $alpha_1,0 = 1$, $alpha_2j,0 = -malpha_2j - 1,0$, $alpha_2j,j = 2alpha_2j-1,j-1$ and $alpha_2j+1,j = alpha_2j,j$.
and the value format of recurisve formula is like (only to show the format, the explicit formula is too long to show).
Any suggestion or advice is welcome. Thank you!
recurrence-relations generating-functions
asked Jul 23 at 13:42
Ren Hu
12
add a comment |Â
up vote
-1
down vote
favorite
I have a pice-wise function which is divided into even and odd parts as you can see in the following. I hope to know if there's a generating function that fit this recursive relation.
$$
f(x) =
left{
beginarrayll
2^j+1 sum_n=0^j alpha_ell,n x^2n+1 &ell=2j+1,\
2^j sum_n=0^j alpha_ell,n x^2n &ell=2j,\
endarray
right.
$$
$ell=0,1,2,ldots$, the coefficient $alpha_ell,n$ has the recursive relation
$$
left{
beginarrayll
alpha_2j+1,n = -(n+1)alpha_2j,n + 1 + alpha_2j,n &n=0,1,2,ldots,j-1\
alpha_2j,n = -( 2n + m)alpha_2j - 1,n + 2alpha_2j-1,n-1 &n=1,2,ldots,j-1\
endarray
right.
$$
(should mention here that the maximum of $n$ is $j-1$, so it doesn't break the boundary) and $alpha_0,0 = 1$, $alpha_1,0 = 1$, $alpha_2j,0 = -malpha_2j - 1,0$, $alpha_2j,j = 2alpha_2j-1,j-1$ and $alpha_2j+1,j = alpha_2j,j$.
and the value format of recurisve formula is like (only to show the format, the explicit formula is too long to show).
Any suggestion or advice is welcome. Thank you!
recurrence-relations generating-functions
asked Jul 23 at 13:42
Ren Hu
12
add a comment |Â
up vote
-1
down vote
favorite
up vote
-1
down vote
favorite
I have a pice-wise function which is divided into even and odd parts as you can see in the following. I hope to know if there's a generating function that fit this recursive relation.
$$
f(x) =
left{
beginarrayll
2^j+1 sum_n=0^j alpha_ell,n x^2n+1 &ell=2j+1,\
2^j sum_n=0^j alpha_ell,n x^2n &ell=2j,\
endarray
right.
$$
$ell=0,1,2,ldots$, the coefficient $alpha_ell,n$ has the recursive relation
$$
left{
beginarrayll
alpha_2j+1,n = -(n+1)alpha_2j,n + 1 + alpha_2j,n &n=0,1,2,ldots,j-1\
alpha_2j,n = -( 2n + m)alpha_2j - 1,n + 2alpha_2j-1,n-1 &n=1,2,ldots,j-1\
endarray
right.
$$
(should mention here that the maximum of $n$ is $j-1$, so it doesn't break the boundary) and $alpha_0,0 = 1$, $alpha_1,0 = 1$, $alpha_2j,0 = -malpha_2j - 1,0$, $alpha_2j,j = 2alpha_2j-1,j-1$ and $alpha_2j+1,j = alpha_2j,j$.
and the value format of recurisve formula is like (only to show the format, the explicit formula is too long to show).
Any suggestion or advice is welcome. Thank you!
recurrence-relations generating-functions
asked Jul 23 at 13:42
Ren Hu
12
I have a pice-wise function which is divided into even and odd parts as you can see in the following. I hope to know if there's a generating function that fit this recursive relation.
$$
f(x) =
left{
beginarrayll
2^j+1 sum_n=0^j alpha_ell,n x^2n+1 &ell=2j+1,\
2^j sum_n=0^j alpha_ell,n x^2n &ell=2j,\
endarray
right.
$$
$ell=0,1,2,ldots$, the coefficient $alpha_ell,n$ has the recursive relation
$$
left{
beginarrayll
alpha_2j+1,n = -(n+1)alpha_2j,n + 1 + alpha_2j,n &n=0,1,2,ldots,j-1\
alpha_2j,n = -( 2n + m)alpha_2j - 1,n + 2alpha_2j-1,n-1 &n=1,2,ldots,j-1\
endarray
right.
$$
(should mention here that the maximum of $n$ is $j-1$, so it doesn't break the boundary) and $alpha_0,0 = 1$, $alpha_1,0 = 1$, $alpha_2j,0 = -malpha_2j - 1,0$, $alpha_2j,j = 2alpha_2j-1,j-1$ and $alpha_2j+1,j = alpha_2j,j$.
and the value format of recurisve formula is like (only to show the format, the explicit formula is too long to show).
Any suggestion or advice is welcome. Thank you!
recurrence-relations generating-functions
asked Jul 23 at 13:42
Ren Hu
12
asked Jul 23 at 13:42
Ren Hu
12
asked Jul 23 at 13:42
Ren Hu
12
asked Jul 23 at 13:42
Ren Hu
12
12
add a comment |Â
add a comment |Â
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