Jacobi transform

The integral transforms

$$ J \{ F ( x) \} = \ f ^ { ( \alpha , \beta ) } ( n) = \ \int\limits _ { - } 1 ^ { 1 } P _ {n} ^ {( \alpha , \beta ) } ( x) F ( x) dx, $$

$$ n = 0, 1 \dots $$

where the $ P _ {n} ^ {( \alpha , \beta ) } ( x) $ are the Jacobi polynomials of degree $ n $, and $ \alpha > - 1 $ and $ \beta > - 1 $ are real numbers. The inversion formula has the form

$$ F ( x) = \ \sum _ {n = 0 } ^ \infty \frac{1}{\delta _ {n} } ( 1 - x) ^ \alpha ( 1 + x) ^ \beta P _ {n} ^ {( \alpha , \beta ) } ( x) f ^ { ( \alpha , \beta ) } ( n), $$

$$ - 1 < x < 1, $$

$$ \delta _ {n} = \frac{2 ^ {\alpha + \beta + 1 } \Gamma ( \alpha + n + 1) \Gamma ( \beta + n + 1) }{n! ( \alpha + \beta + 2n + 1) \Gamma ( \alpha + \beta + n + 1) } , $$

provided the series converges.

The Jacobi transform reduces the operation

$$ T [ F ( x)] = \ { \frac{d}{dx} } \left \{ ( 1 - x ^ {2} ) \frac{dF }{dx } + [( \alpha - \beta ) + ( \alpha + \beta ) x] \frac{dF }{dx } \right \} $$

to an algebraic one by the formula

$$ J \{ T [ F ( x)] \} = - ( n + 1) ( n + \alpha + \beta ) f ^ { ( \alpha , \beta ) } ( n) + $$

$$ + \left . \{ [( \alpha - \beta ) + ( \alpha + \beta ) x] P _ {n} ^ {( \alpha , \beta ) } ( x) F ( x) \} \right | _ {-} 1 ^ {1} . $$

When $ \alpha = \beta = 0 $ the Jacobi transform is the Legendre transform; for $ \alpha = \beta = \nu - 1/2 $ it is the Gegenbauer transform. Jacobi transforms are used in solving differential equations containing the operator $ T $. The Jacobi transform has also been defined for a special class of generalized functions.

References

[1]	E.J. Scott, "Jacobi transforms" Quart. J. Math. , 4 : 13 (1953) pp. 36–40
[2]	V.A. Ditkin, A.P. Prundnikov, "Integral transforms" Progress in Math. (1969) pp. 1–85 Itogi Nauk. Mat. Anal. 1966 (1967)
[3]	A.G. Zemanian, "Generalized integral transformations" , Interscience (1968)

Comments

See (the editorial comments to) Gegenbauer transform. Usually the Jacobi transform is written as

$$ \int\limits _ { - } 1 ^ { 1 } F ( x) P _ {n} ^ {( \alpha , \beta ) } ( x) ( 1 - x ) ^ \alpha ( 1 + x ) ^ \beta d x , $$

which generalizes the expression given in Gegenbauer transform.