Jacobi transform

In mathematics, Jacobi transform is an integral transform named after the mathematician Carl Gustav Jacob Jacobi, which uses Jacobi polynomials $P_{n}^{α, β} (x)$ as kernels of the transform .^[1]^[2]^[3]^[4]

The Jacobi transform of a function $F (x)$ is^[5]

J {F (x)} = f^{α, β} (n) = \int_{- 1}^{1} (1 - x)^{α} (1 + x)^{β} P_{n}^{α, β} (x) F (x) d x

The inverse Jacobi transform is given by

J^{- 1} {f^{α, β} (n)} = F (x) = \sum_{n = 0}^{\infty} \frac{1}{δ_{n}} f^{α, β} (n) P_{n}^{α, β} (x), where δ_{n} = \frac{2^{α + β + 1} Γ (n + α + 1) Γ (n + β + 1)}{n! (α + β + 2 n + 1) Γ (n + α + β + 1)}

Some Jacobi transform pairs

Some Jacobi transform pairs
$F (x)$	$f^{α, β} (n)$
$x^{m}, m < n$	$0$
$x^{n}$	$n! (α + β + 2 n + 1) δ_{n}$
$P_{m}^{α, β} (x)$	$δ_{n} δ_{m, n}$
$(1 + x)^{a - β}$	$(\binom{n + α}{n}) 2^{α + a + 1} \frac{Γ (a + 1) Γ (α + 1) Γ (a - β + 1)}{Γ (α + a + n + 2) Γ (a - β + n + 1)}$
$(1 - x)^{σ - α}, ℜ σ > - 1$	$\frac{2^{σ + β + 1}}{n! Γ (α - σ)} \frac{Γ (σ + 1) Γ (n + β + 1) Γ (α - σ + n)}{Γ (β + σ + n + 2)}$
$(1 - x)^{σ - β} P_{m}^{α, σ} (x), ℜ σ > - 1$	$\frac{2^{α + σ + 1}}{m! (n - m)!} \frac{Γ (n + α + 1) Γ (α + β + m + n + 1) Γ (σ + m + 1) Γ (α - β + 1)}{Γ (α + β + n + 1) Γ (α + σ + m + n + 2) Γ (α - β + m + 1)}$

Some more Jacobi transform pairs
$F (x)$	$f^{α, β} (n)$
$2^{α + β} Q^{- 1} (1 - z + Q)^{- α} (1 + z + Q)^{- β}, Q = (1 - 2 x z + z^{2})^{1 / 2}, \| z \| < 1$	$\sum_{n = 0}^{\infty} δ_{n} z^{n}$
$(1 - x)^{- α} (1 + x)^{- β} \frac{d}{d x} [(1 - x)^{α + 1} (1 + x)^{β + 1} \frac{d}{d x}] F (x)$	$- n (n + α + β + 1) f^{α, β} (n)$
${(1 - x)^{- α} (1 + x)^{- β} \frac{d}{d x} [(1 - x)^{α + 1} (1 + x)^{β + 1} \frac{d}{d x}]}^{k} F (x)$	$(- 1)^{k} n^{k} (n + α + β + 1)^{k} f^{α, β} (n)$

↑ Debnath, L. "On Jacobi Transform." Bull. Cal. Math. Soc 55.3 (1963): 113-120.
↑ Debnath, L. "SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS BY JACOBI TRANSFORM." BULLETIN OF THE CALCUTTA MATHEMATICAL SOCIETY 59.3-4 (1967): 155.
↑ Scott, E. J. "Jacobi transforms." (1953).
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↑ Debnath, Lokenath, and Dambaru Bhatta. Integral transforms and their applications. CRC press, 2014.