Local compositional inversion of f(x) = x / (1+ax+bx^2)

Posted on March 23, 2022 by Tom Copeland

Graphical depiction of local compositional inverse curves of y=f(x) = x / (1+ax+bx^2), a bivariate generating function for OEIS A049310, the triangle of the coefficients of the Chebyshev’s S(n,x) := U(n,x/2) polynomials, for various values of a and b. The inverse is the curve x = f(y) = y / (1+ay+by^2), the reflection of y=f(x) through the diagonal line y=x, which can be characterized by a patchwork of functions involving the square root. The inverse of f(x) about x=0 and x=\infty are generating functions for A097610, enumerating certain types of Motzkin paths.

Local compositional inversion of f(x) = x / (1+ax+bx^2) (pdf)

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