Difference between revisions of "Jacobi P"

Revision as of 09:38, 19 January 2015

The Jacobi polynomial $P_n^{(\alpha,\beta)}$ are orthogonal polynomials defined to be coefficient of $t^n$ in the expansion of $$\dfrac{2^{\alpha+\beta}}{\sqrt{1-2xt+t^2}\left(1-t+ \sqrt{1-2xt+t^2} \right)^{\alpha} \left(1+t+\sqrt{1-2xt+t^2} \right)^{\beta}}$$ in the sense that $$\dfrac{2^{\alpha+\beta}}{\sqrt{1-2xt+t^2}\left(1-t+ \sqrt{1-2xt+t^2} \right)^{\alpha} \left(1+t+\sqrt{1-2xt+t^2} \right)^{\beta}} = \sum_{k=0}^{\infty} P_k^{(\alpha,\beta)}(x)t^k$$ holds.