Jacobi P

Let $\alpha > -1$ and $\beta > -1$. The Jacobi polynomial $P_n^{(\alpha,\beta)}$ are orthogonal polynomials with weight function $w(x)=(1-x)^{\alpha}(1-x)^{\beta}$ on the interval $[-1,1]$ that obey $P_n^{(\alpha,\beta)}(1) = {{n + \alpha} \choose n}$. $$P_n^{(\alpha,\beta)}(z)=\dfrac{(\alpha+1)^{\overline{n}}}{n!} {}_2F_1 \left(-n, 1+\alpha+\beta+n;\alpha+1; \dfrac{1}{2}(1-z) \right),$$ where ${}_2F_1$ is the generalized hypergeometries series.

Properties

Relationship between the Gegenbauer C polynomials and the Jacobi P polynomials
Differential equation for Jacobi P

References

• 1975: Gabor Szegő: Orthogonal Polynomials ... (previous) ... (next): page 58

Orthogonal polynomials

Laguerre $L_n^{(\alpha)}$

Bateman $F_n$

Bernoulli $B$

Bessel poly

Chebyshev T

Chebyshev U

Dickson

Euler $E$

Gegenbauer $C$

Hermite (phys)

Hermite (prob)

Jacobi $P^{(\alpha,\beta)}$

Laguerre $L$

Legendre $P$

Bernoulli $B$

Chebyshev $T$

Fibonacci

Lucas