This function returns a vector with *n + 1* elements containing the inner product of
an order *k* Jacobi polynomial, *P_k^{≤ft( {α ,β } \right)} ≤ft( x \right)*,
with itself (i.e. the norm squared) for orders *k = 0,\;1,\; … ,\;n *.

1 | ```
jacobi.p.inner.products(n,alpha,beta)
``` |

`n` |
integer value for the highest polynomial order |

`alpha` |
numeric value for the first polynomial parameter |

`beta` |
numeric value for the first polynomial parameter |

The formula used to compute the innser products is as follows.

*h_n = ≤ft\langle {P_n^{≤ft( {α ,β } \right)} |P_n^{≤ft( {α ,β } \right)} } \right\rangle = \frac{{2^{α + β + 1} }}
{{2\,n + α + β + 1}}\frac{{Γ ≤ft( {n + α + 1} \right)\,Γ ≤ft( {n + β + 1} \right)}}
{{n!\;Γ ≤ft( {n + α + β + 1} \right)}}*.

A vector with *n + 1* elements

`1 ` |
inner product of order 0 orthogonal polynomial |

`2 ` |
inner product of order 1 orthogonal polynomial |

...

`n+1 ` |
inner product of order |

Frederick Novomestky fnovomes@poly.edu

Abramowitz, M. and I. A. Stegun, 1968. *Handbook of Mathematical Functions with
Formulas, Graphs, and Mathematical Tables*, Dover Publications, Inc., New York.

Courant, R., and D. Hilbert, 1989. *Methods of Mathematical Physics*,
John Wiley, New York, NY.

Szego, G., 1939. *Orthogonal Polynomials*, 23, American Mathematical Society
Colloquium Publications, Providence, RI.

1 2 3 4 5 | ```
###
### generate the inner product vector for the P Jacobi polynomials of orders 0 to 10
###
h <- jacobi.p.inner.products( 10, 2, 2 )
print( h )
``` |

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