# 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)

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