The Special Function {P_{n}^{(a,b)}\left(x\right)}

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1. Differential Equation[-]

The function {P_{n}^{(a,b)}\left(x\right)} satisfies the differential equation with initial values y \left( 0 \right) =\frac{\Gamma \left( 1+a+n \right) \,{{}_{2}F_{1}\!\left(-n,-b-n;\,a+1;\,-1\right)}}{{2}^{n}\,\Gamma \left( a+1 \right) \,\Gamma \left( n+1 \right) } and y' \left( 0 \right) =\frac{n\left(n+a+b+1\right)\Gamma \left( 1+a+n \right) \,{{}_{2}F_{1}\!\left(-b-n,1-n;\,a+2;\,-1\right)}}{{2}^{n}\left(a+1\right)\Gamma \left( a+1 \right) \,\Gamma \left( n+1 \right) }.
All formulas on this page are valid under the condition that a, b, and a+b+2\,n are not integers (special values for parameters can be entered at the bottom).

2. Derivative in Terms of Lower-Order Derivatives[+]

3. Expansion at 0[+]

4. Local Expansions at Singularities and at Infinity[-]

The differential equation above has 2 non-zero finite singular pointsA complex numberĀ z_0 is a singular point (or singularity of a linear ordinary differential equation with polynomial coefficients, if the leading coefficient of the equation vanishes atĀ z_0..

4.1. Expansion at -1[+]

4.2. Expansion at 1[+]

4.3. Expansion at \infty [+]

5. Hypergeometric Representation[+]

6. Chebyshev Expansion over [-1,1][+]

7. Parameters[-]

The Jacobi function {P_{n}^{(a,b)}\left(x\right)} depends on the parameters n, a, and b. The boxes below can be used to rename or instantiate these parameters.
p1 =  p2 =  p3 =