Lagrange inversion formula in 2 variables?

Question

I have the following implicit equation for the function $G(x,y)$ in 2 variables:

$$G \;=\; (1 + x\,G^2)\,(1 + y\,G^2)$$

I want to use a form of a multivariable Lagrange inversion formula but I'm not sure that it's possible. Some ideas?

Thanks for help. Gianfranco

Answer 1

The coefficients of $G(x,y)=\sum_{i,j=0}^\infty c_{ij}x^i y^j$ are given by $$c_{ij}=\frac{1}{2i+2j+1}\binom{2i+2j+1}{i} \binom{2i+2j+1}{j}$$ Very Catalan-like, indeed. I did not try to write down a proof -- my computer is a much better combinatorialist than me.

Suspiciously similar expressions appear at the end of these slides.

The rest of the post refers to experiments that led to the above formula.

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You can find any number of coefficients of the Taylor expansion at the origin just by solving a system for the coefficients. I used Maple to get coefficients up to 9th order; I'm sure it can do more. The system is actually trivial to solve if one does it in the right order, which means Maple could be replaced by a Python script or something.

n:=9:
c[0,0]:=1: 
G:=sum(sum(c[i,j]*x^i*y^j,j=0..n-i),i=0..n):
H:=(1+x*G^2)*(1+y*G^2):
eqs:=[seq(c[0,j]=coeff(eval(H,x=0),y^j),j=1..n),seq(c[i,0]=coeff(eval(H,y=0),x^i),i=1..n),seq(seq(c[i,j]=coeff(coeff(H,x^i),y^j),j=1..n-i),i=1..n)]: 
solve(eqs);

The output was

{c[0, 1] = 1, c[0, 2] = 2, c[0, 3] = 5, c[0, 4] = 14, c[0, 5] = 42, c[0, 6] = 132, c[0, 7] = 429, c[0, 8] = 1430, c[0, 9] = 4862, c[1, 0] = 1, c[1, 1] = 5, c[1, 2] = 21, c[1, 3] = 84, c[1, 4] = 330, c[1, 5] = 1287, c[1, 6] = 5005, c[1, 7] = 19448, c[1, 8] = 75582, c[2, 0] = 2, c[2, 1] = 21, c[2, 2] = 144, c[2, 3] = 825, c[2, 4] = 4290, c[2, 5] = 21021, c[2, 6] = 99008, c[2, 7] = 453492, c[3, 0] = 5, c[3, 1] = 84, c[3, 2] = 825, c[3, 3] = 6292, c[3, 4] = 41405, c[3, 5] = 247520, c[3, 6] = 1383732, c[4, 0] = 14, c[4, 1] = 330, c[4, 2] = 4290, c[4, 3] = 41405, c[4, 4] = 333200, c[4, 5] = 2372112, c[5, 0] = 42, c[5, 1] = 1287, c[5, 2] = 21021, c[5, 3] = 247520, c[5, 4] = 2372112, c[6, 0] = 132, c[6, 1] = 5005, c[6, 2] = 99008, c[6, 3] = 1383732, c[7, 0] = 429, c[7, 1] = 19448, c[7, 2] = 453492, c[8, 0] = 1430, c[8, 1] = 75582, c[9, 0] = 4862}