Difference between revisions of "Jacobi P"
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(Created page with "The Jacobi polynomial $P_n^{(\alpha,\beta)}$ is the coefficient of $t^n$ in the expansion of $$\dfrac{2^{\alpha+\beta}}{\sqrt{1-2xt+t^2}\left(1-t+ \sqrt{1-2xt+t^2} \right)^{\...") |
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| − | The Jacobi polynomial $P_n^{(\alpha,\beta)}$ | + | The Jacobi polynomial $P_n^{(\alpha,\beta)}$ are [[orthogonal polynomials]] defined to be coefficient of $t^n$ in the expansion of |
$$\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}}$$ | $$\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}}$$ | ||
in the sense that | in the sense that | ||
$$\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$$ | $$\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$$ | ||
holds. | holds. | ||
Revision as of 20:36, 7 October 2014
The Jacobi polynomial $P_n^{(\alpha,\beta)}$ are orthogonal polynomials defined to be coefficient of $t^n$ in the expansion of $$\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.
