Mixing
Generating Function Differential Equation with a Catalytic Variable
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I have a bivariate generating function $H(x,y)=sum_{ngeq 0}sum_{kgeq 0}h(n,k)x^ny^k$ that satisfies $$(H(x,y)-H(x,0))left(xfrac{partial}{partial x}H(x,y)+H(x,y)-C(y)right)$$ $$=frac{H(x,y)-C(y)}{x}-frac{H(x,y)-H(x,0)}{y},$$ where $C(y)=H(0,y)=dfrac{1-sqrt{1-4y}}{2y}$ is the generating function of the Catalan numbers. I am interested in finding an asymptotic estimate (even just a numerical approximation of the exponential growth rate) for the coefficients $h(n,0)$ of the univariate generating function $H(x,0)$. In other words, $y$ is a "catalytic variable" that I would like to remove. I want to find (approximations of) the dominant singularities of $H(x,0)$.
If the partial derivative $dfrac{partial}{partial x}H(x,y)$ did not appear in the equation, then I could obtain a polynomial $P(a,b,x,y)$ such that $P(H(x,y),H(x,0),x,y)=0$. I then know methods that would allow me to obtain a polynomial $Q(b,x)$ such that $Q(H(x,0),x)=0$. There are then methods that would allow me to obtain fairly precise asymptotics for the coefficients of $H(x,0)$.
The problem is that the partial derivative $dfrac{partial}{partial x}H(x,y)$ does occur in the equation. I don't know any methods for dealing with this situation, so any help would be greatly appreciated.
combinatorics ordinary-differential-equations asymptotics generating-functions
asked Jan 10 at 20:49
Colin DefantColin Defant
687312
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add a comment |
$begingroup$
I have a bivariate generating function $H(x,y)=sum_{ngeq 0}sum_{kgeq 0}h(n,k)x^ny^k$ that satisfies $$(H(x,y)-H(x,0))left(xfrac{partial}{partial x}H(x,y)+H(x,y)-C(y)right)$$ $$=frac{H(x,y)-C(y)}{x}-frac{H(x,y)-H(x,0)}{y},$$ where $C(y)=H(0,y)=dfrac{1-sqrt{1-4y}}{2y}$ is the generating function of the Catalan numbers. I am interested in finding an asymptotic estimate (even just a numerical approximation of the exponential growth rate) for the coefficients $h(n,0)$ of the univariate generating function $H(x,0)$. In other words, $y$ is a "catalytic variable" that I would like to remove. I want to find (approximations of) the dominant singularities of $H(x,0)$.
If the partial derivative $dfrac{partial}{partial x}H(x,y)$ did not appear in the equation, then I could obtain a polynomial $P(a,b,x,y)$ such that $P(H(x,y),H(x,0),x,y)=0$. I then know methods that would allow me to obtain a polynomial $Q(b,x)$ such that $Q(H(x,0),x)=0$. There are then methods that would allow me to obtain fairly precise asymptotics for the coefficients of $H(x,0)$.
The problem is that the partial derivative $dfrac{partial}{partial x}H(x,y)$ does occur in the equation. I don't know any methods for dealing with this situation, so any help would be greatly appreciated.
combinatorics ordinary-differential-equations asymptotics generating-functions
asked Jan 10 at 20:49
Colin DefantColin Defant
687312
$endgroup$
add a comment |
$begingroup$
I have a bivariate generating function $H(x,y)=sum_{ngeq 0}sum_{kgeq 0}h(n,k)x^ny^k$ that satisfies $$(H(x,y)-H(x,0))left(xfrac{partial}{partial x}H(x,y)+H(x,y)-C(y)right)$$ $$=frac{H(x,y)-C(y)}{x}-frac{H(x,y)-H(x,0)}{y},$$ where $C(y)=H(0,y)=dfrac{1-sqrt{1-4y}}{2y}$ is the generating function of the Catalan numbers. I am interested in finding an asymptotic estimate (even just a numerical approximation of the exponential growth rate) for the coefficients $h(n,0)$ of the univariate generating function $H(x,0)$. In other words, $y$ is a "catalytic variable" that I would like to remove. I want to find (approximations of) the dominant singularities of $H(x,0)$.
If the partial derivative $dfrac{partial}{partial x}H(x,y)$ did not appear in the equation, then I could obtain a polynomial $P(a,b,x,y)$ such that $P(H(x,y),H(x,0),x,y)=0$. I then know methods that would allow me to obtain a polynomial $Q(b,x)$ such that $Q(H(x,0),x)=0$. There are then methods that would allow me to obtain fairly precise asymptotics for the coefficients of $H(x,0)$.
The problem is that the partial derivative $dfrac{partial}{partial x}H(x,y)$ does occur in the equation. I don't know any methods for dealing with this situation, so any help would be greatly appreciated.
combinatorics ordinary-differential-equations asymptotics generating-functions
asked Jan 10 at 20:49
Colin DefantColin Defant
687312
$endgroup$
I have a bivariate generating function $H(x,y)=sum_{ngeq 0}sum_{kgeq 0}h(n,k)x^ny^k$ that satisfies $$(H(x,y)-H(x,0))left(xfrac{partial}{partial x}H(x,y)+H(x,y)-C(y)right)$$ $$=frac{H(x,y)-C(y)}{x}-frac{H(x,y)-H(x,0)}{y},$$ where $C(y)=H(0,y)=dfrac{1-sqrt{1-4y}}{2y}$ is the generating function of the Catalan numbers. I am interested in finding an asymptotic estimate (even just a numerical approximation of the exponential growth rate) for the coefficients $h(n,0)$ of the univariate generating function $H(x,0)$. In other words, $y$ is a "catalytic variable" that I would like to remove. I want to find (approximations of) the dominant singularities of $H(x,0)$.
If the partial derivative $dfrac{partial}{partial x}H(x,y)$ did not appear in the equation, then I could obtain a polynomial $P(a,b,x,y)$ such that $P(H(x,y),H(x,0),x,y)=0$. I then know methods that would allow me to obtain a polynomial $Q(b,x)$ such that $Q(H(x,0),x)=0$. There are then methods that would allow me to obtain fairly precise asymptotics for the coefficients of $H(x,0)$.
The problem is that the partial derivative $dfrac{partial}{partial x}H(x,y)$ does occur in the equation. I don't know any methods for dealing with this situation, so any help would be greatly appreciated.
combinatorics ordinary-differential-equations asymptotics generating-functions
combinatorics ordinary-differential-equations asymptotics generating-functions
asked Jan 10 at 20:49
Colin DefantColin Defant
687312
asked Jan 10 at 20:49
Colin DefantColin Defant
687312
asked Jan 10 at 20:49
Colin DefantColin Defant
687312
asked Jan 10 at 20:49
Colin DefantColin Defant
687312
asked Jan 10 at 20:49
Colin DefantColin Defant
687312
687312
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