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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|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|Legendre transform]]; for  $  \alpha = \beta = \nu - 1/2 $
it is the [[Gegenbauer transform|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.J. Scott,   "Jacobi transforms"  ''Quart. J. Math.'' , '''4''' :  13  (1953)  pp. 36–40</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.A. Ditkin,   A.P. Prundnikov,   "Integral transforms"  ''Progress in Math.''  (1969)  pp. 1–85  ''Itogi Nauk. Mat. Anal. 1966''  (1967)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.G. Zemanian,   "Generalized integral transformations" , Interscience  (1968)</TD></TR></table>

====Comments====
See (the editorial comments to) [[Gegenbauer transform|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]].