Solving 2-dimensional recurrence matrix of homogenous polynomials
$begingroup$
In $2012$, Rajkumar presented an interesting simplification of Apéry's theorem, i.e., that $zeta(3)$ is irrational, where $zeta$ denotes the Riemann zeta function. I am trying to understand his proof, but I severely lack knowledge on solving 2-dimensional recurrence relations in multiple homogeneous polynomials. In particular, he begain by defining
$$
f(i,j)=i^3+2i^2j+2ij^2+j^3, \
g(i,j)=i^3-2i^2j+2ij^2-j^3.
$$
He then constructed the recurrence relation
$$
left(begin{array}{cc}
f(i,j) & g(0,j) \
f(0,j) & g(i,j)
end{array}right)
left(begin{array}{c}
u_{i-1,j}\
u_{i-1,j-1}
end{array}right)
=
f(i,0)
left(begin{array}{c}
u_{i,j}\
u_{i,j-1}
end{array}right)
$$
and showed that for integers $i, j geq 1$ it has a rational valued solution $u_{i,j}$ for certain boundary conditions.
I would really like to understand how one solves such recurrence relations, but unfortunately after searching extensively on the internet for related articles or tutorials I have found nothing. I understand how basic recurrence relations work, but in this case I'm not sure what I'm supposed to do and what theory I can apply.
Can anyone give me any pointers?
polynomials recurrence-relations
asked Oct 12 '18 at 14:18
KlangenKlangen
1,69811334
$endgroup$
add a comment |
$begingroup$
In $2012$, Rajkumar presented an interesting simplification of Apéry's theorem, i.e., that $zeta(3)$ is irrational, where $zeta$ denotes the Riemann zeta function. I am trying to understand his proof, but I severely lack knowledge on solving 2-dimensional recurrence relations in multiple homogeneous polynomials. In particular, he begain by defining
$$
f(i,j)=i^3+2i^2j+2ij^2+j^3, \
g(i,j)=i^3-2i^2j+2ij^2-j^3.
$$
He then constructed the recurrence relation
$$
left(begin{array}{cc}
f(i,j) & g(0,j) \
f(0,j) & g(i,j)
end{array}right)
left(begin{array}{c}
u_{i-1,j}\
u_{i-1,j-1}
end{array}right)
=
f(i,0)
left(begin{array}{c}
u_{i,j}\
u_{i,j-1}
end{array}right)
$$
and showed that for integers $i, j geq 1$ it has a rational valued solution $u_{i,j}$ for certain boundary conditions.
I would really like to understand how one solves such recurrence relations, but unfortunately after searching extensively on the internet for related articles or tutorials I have found nothing. I understand how basic recurrence relations work, but in this case I'm not sure what I'm supposed to do and what theory I can apply.
Can anyone give me any pointers?
polynomials recurrence-relations
asked Oct 12 '18 at 14:18
KlangenKlangen
1,69811334
$endgroup$
add a comment |
$begingroup$
In $2012$, Rajkumar presented an interesting simplification of Apéry's theorem, i.e., that $zeta(3)$ is irrational, where $zeta$ denotes the Riemann zeta function. I am trying to understand his proof, but I severely lack knowledge on solving 2-dimensional recurrence relations in multiple homogeneous polynomials. In particular, he begain by defining
$$
f(i,j)=i^3+2i^2j+2ij^2+j^3, \
g(i,j)=i^3-2i^2j+2ij^2-j^3.
$$
He then constructed the recurrence relation
$$
left(begin{array}{cc}
f(i,j) & g(0,j) \
f(0,j) & g(i,j)
end{array}right)
left(begin{array}{c}
u_{i-1,j}\
u_{i-1,j-1}
end{array}right)
=
f(i,0)
left(begin{array}{c}
u_{i,j}\
u_{i,j-1}
end{array}right)
$$
and showed that for integers $i, j geq 1$ it has a rational valued solution $u_{i,j}$ for certain boundary conditions.
I would really like to understand how one solves such recurrence relations, but unfortunately after searching extensively on the internet for related articles or tutorials I have found nothing. I understand how basic recurrence relations work, but in this case I'm not sure what I'm supposed to do and what theory I can apply.
Can anyone give me any pointers?
polynomials recurrence-relations
asked Oct 12 '18 at 14:18
KlangenKlangen
1,69811334
$endgroup$
In $2012$, Rajkumar presented an interesting simplification of Apéry's theorem, i.e., that $zeta(3)$ is irrational, where $zeta$ denotes the Riemann zeta function. I am trying to understand his proof, but I severely lack knowledge on solving 2-dimensional recurrence relations in multiple homogeneous polynomials. In particular, he begain by defining
$$
f(i,j)=i^3+2i^2j+2ij^2+j^3, \
g(i,j)=i^3-2i^2j+2ij^2-j^3.
$$
He then constructed the recurrence relation
$$
left(begin{array}{cc}
f(i,j) & g(0,j) \
f(0,j) & g(i,j)
end{array}right)
left(begin{array}{c}
u_{i-1,j}\
u_{i-1,j-1}
end{array}right)
=
f(i,0)
left(begin{array}{c}
u_{i,j}\
u_{i,j-1}
end{array}right)
$$
and showed that for integers $i, j geq 1$ it has a rational valued solution $u_{i,j}$ for certain boundary conditions.
I would really like to understand how one solves such recurrence relations, but unfortunately after searching extensively on the internet for related articles or tutorials I have found nothing. I understand how basic recurrence relations work, but in this case I'm not sure what I'm supposed to do and what theory I can apply.
Can anyone give me any pointers?
polynomials recurrence-relations
polynomials recurrence-relations
asked Oct 12 '18 at 14:18
KlangenKlangen
1,69811334
asked Oct 12 '18 at 14:18
KlangenKlangen
1,69811334
edited Dec 13 '18 at 8:50
Klangen
asked Oct 12 '18 at 14:18
KlangenKlangen
1,69811334
asked Oct 12 '18 at 14:18
KlangenKlangen
1,69811334
asked Oct 12 '18 at 14:18
KlangenKlangen
1,69811334
1,69811334
add a comment |
add a comment |
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