- Jacobi polynomials
In
mathematics , Jacobi polynomials are a class oforthogonal polynomials . They are obtained fromhypergeometric series in cases where the series is in fact finite::
where is Pochhammer's symbol (for the rising factorial), (Abramowitz & Stegun [http://www.math.sfu.ca/~cbm/aands/page_561.htm p561] .) and thus have the explicit expression
:
from which the terminal value follows
:Here for integer :and is the usual
Gamma function , which has the property for . Thus,:They satisfy the orthogonality condition:for and .
The polynomials have the symmetry relation ; thus the other terminal value is
:
For real the Jacobi polynomial can alternatively bewritten as:where and .In the special case that the four quantities, , , and are nonnegative integers,the Jacobi polynomial can be written as:The sum on extends over all integer values for which the arguments of the factorials are nonnegative.
This form allows the expression of the Wigner d-matrix () in termsof Jacobi polynomials [L. C. Biedenharn and J. D. Louck, "Angular Momentum in Quantum Physics", Addison-Wesley, Reading, (1981)] :
Derivatives
The "k"th derivative of the explicit expression leads to
:
Differential equation
Jacobi polynomials are solution of
:
References
Cited referencesGeneral references
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*Citation | last1=Andrews | first1=George E. | last2=Askey | first2=Richard | last3=Roy | first3=Ranjan | title=Special functions | publisher=Cambridge University Press | series=Encyclopedia of Mathematics and its Applications | isbn=978-0-521-62321-6; 978-0-521-78988-2 | id=MathSciNet | id = 1688958 | year=1999 | volume=71
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