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Jacobi transform

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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 ) {\displaystyle P_{n}^{\alpha ,\beta }(x)} {\displaystyle P_{n}^{\alpha ,\beta }(x)} as kernels of the transform .[1] [2] [3] [4]

The Jacobi transform of a function F ( x ) {\displaystyle F(x)} {\displaystyle F(x)} is[5]

J { F ( x ) } = f α , β ( n ) = ∫ − 1 1 ( 1 − x ) α   ( 1 + x ) β   P n α , β ( x )   F ( x )   d x {\displaystyle J\{F(x)\}=f^{\alpha ,\beta }(n)=\int _{-1}^{1}(1-x)^{\alpha }\ (1+x)^{\beta }\ P_{n}^{\alpha ,\beta }(x)\ F(x)\ dx} {\displaystyle J\{F(x)\}=f^{\alpha ,\beta }(n)=\int _{-1}^{1}(1-x)^{\alpha }\ (1+x)^{\beta }\ P_{n}^{\alpha ,\beta }(x)\ F(x)\ dx}

The inverse Jacobi transform is given by

J − 1 { f α , β ( n ) } = F ( x ) = ∑ n = 0 ∞ 1 δ n f α , β ( n ) P n α , β ( x ) , where δ n = 2 α + β + 1 Γ ( n + α + 1 ) Γ ( n + β + 1 ) n ! ( α + β + 2 n + 1 ) Γ ( n + α + β + 1 ) {\displaystyle J^{-1}\{f^{\alpha ,\beta }(n)\}=F(x)=\sum _{n=0}^{\infty }{\frac {1}{\delta _{n}}}f^{\alpha ,\beta }(n)P_{n}^{\alpha ,\beta }(x),\quad {\text{where}}\quad \delta _{n}={\frac {2^{\alpha +\beta +1}\Gamma (n+\alpha +1)\Gamma (n+\beta +1)}{n!(\alpha +\beta +2n+1)\Gamma (n+\alpha +\beta +1)}}} {\displaystyle J^{-1}\{f^{\alpha ,\beta }(n)\}=F(x)=\sum _{n=0}^{\infty }{\frac {1}{\delta _{n}}}f^{\alpha ,\beta }(n)P_{n}^{\alpha ,\beta }(x),\quad {\text{where}}\quad \delta _{n}={\frac {2^{\alpha +\beta +1}\Gamma (n+\alpha +1)\Gamma (n+\beta +1)}{n!(\alpha +\beta +2n+1)\Gamma (n+\alpha +\beta +1)}}}

Some Jacobi transform pairs

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Some Jacobi transform pairs
F ( x ) {\displaystyle F(x),円} {\displaystyle F(x),円} f α , β ( n ) {\displaystyle f^{\alpha ,\beta }(n),円} {\displaystyle f^{\alpha ,\beta }(n),円}
x m ,   m < n {\displaystyle x^{m},\ m<n,円} {\displaystyle x^{m},\ m<n,円} 0 {\displaystyle 0} {\displaystyle 0}
x n {\displaystyle x^{n},円} {\displaystyle x^{n},円} n ! ( α + β + 2 n + 1 ) δ n {\displaystyle n!(\alpha +\beta +2n+1)\delta _{n}} {\displaystyle n!(\alpha +\beta +2n+1)\delta _{n}}
P m α , β ( x ) {\displaystyle P_{m}^{\alpha ,\beta }(x),円} {\displaystyle P_{m}^{\alpha ,\beta }(x),円} δ n δ m , n {\displaystyle \delta _{n}\delta _{m,n}} {\displaystyle \delta _{n}\delta _{m,n}}
( 1 + x ) a − β {\displaystyle (1+x)^{a-\beta },円} {\displaystyle (1+x)^{a-\beta },円} ( n + α n ) 2 α + a + 1 Γ ( a + 1 ) Γ ( α + 1 ) Γ ( a − β + 1 ) Γ ( α + a + n + 2 ) Γ ( a − β + n + 1 ) {\displaystyle {\binom {n+\alpha }{n}}2^{\alpha +a+1}{\frac {\Gamma (a+1)\Gamma (\alpha +1)\Gamma (a-\beta +1)}{\Gamma (\alpha +a+n+2)\Gamma (a-\beta +n+1)}}} {\displaystyle {\binom {n+\alpha }{n}}2^{\alpha +a+1}{\frac {\Gamma (a+1)\Gamma (\alpha +1)\Gamma (a-\beta +1)}{\Gamma (\alpha +a+n+2)\Gamma (a-\beta +n+1)}}}
( 1 − x ) σ − α ,   ℜ σ > − 1 {\displaystyle (1-x)^{\sigma -\alpha },\ \Re \sigma >-1,円} {\displaystyle (1-x)^{\sigma -\alpha },\ \Re \sigma >-1,円} 2 σ + β + 1 n ! Γ ( α − σ ) Γ ( σ + 1 ) Γ ( n + β + 1 ) Γ ( α − σ + n ) Γ ( β + σ + n + 2 ) {\displaystyle {\frac {2^{\sigma +\beta +1}}{n!\Gamma (\alpha -\sigma )}}{\frac {\Gamma (\sigma +1)\Gamma (n+\beta +1)\Gamma (\alpha -\sigma +n)}{\Gamma (\beta +\sigma +n+2)}}} {\displaystyle {\frac {2^{\sigma +\beta +1}}{n!\Gamma (\alpha -\sigma )}}{\frac {\Gamma (\sigma +1)\Gamma (n+\beta +1)\Gamma (\alpha -\sigma +n)}{\Gamma (\beta +\sigma +n+2)}}}
( 1 − x ) σ − β P m α , σ ( x ) ,   ℜ σ > − 1 {\displaystyle (1-x)^{\sigma -\beta }P_{m}^{\alpha ,\sigma }(x),\ \Re \sigma >-1,円} {\displaystyle (1-x)^{\sigma -\beta }P_{m}^{\alpha ,\sigma }(x),\ \Re \sigma >-1,円} 2 α + σ + 1 m ! ( n − m ) ! 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 ) {\displaystyle {\frac {2^{\alpha +\sigma +1}}{m!(n-m)!}}{\frac {\Gamma (n+\alpha +1)\Gamma (\alpha +\beta +m+n+1)\Gamma (\sigma +m+1)\Gamma (\alpha -\beta +1)}{\Gamma (\alpha +\beta +n+1)\Gamma (\alpha +\sigma +m+n+2)\Gamma (\alpha -\beta +m+1)}}} {\displaystyle {\frac {2^{\alpha +\sigma +1}}{m!(n-m)!}}{\frac {\Gamma (n+\alpha +1)\Gamma (\alpha +\beta +m+n+1)\Gamma (\sigma +m+1)\Gamma (\alpha -\beta +1)}{\Gamma (\alpha +\beta +n+1)\Gamma (\alpha +\sigma +m+n+2)\Gamma (\alpha -\beta +m+1)}}}
Some more Jacobi transform pairs
F ( x ) {\displaystyle F(x),円} {\displaystyle F(x),円} f α , β ( n ) {\displaystyle f^{\alpha ,\beta }(n),円} {\displaystyle f^{\alpha ,\beta }(n),円}
2 α + β Q − 1 ( 1 − z + Q ) − α ( 1 + z + Q ) − β ,   Q = ( 1 − 2 x z + z 2 ) 1 / 2 ,   | z | < 1 {\displaystyle 2^{\alpha +\beta }Q^{-1}(1-z+Q)^{-\alpha }(1+z+Q)^{-\beta },\ Q=(1-2xz+z^{2})^{1/2},\ |z|<1,円} {\displaystyle 2^{\alpha +\beta }Q^{-1}(1-z+Q)^{-\alpha }(1+z+Q)^{-\beta },\ Q=(1-2xz+z^{2})^{1/2},\ |z|<1,円} ∑ n = 0 ∞ δ n z n {\displaystyle \sum _{n=0}^{\infty }\delta _{n}z^{n}} {\displaystyle \sum _{n=0}^{\infty }\delta _{n}z^{n}}
( 1 − x ) − α ( 1 + x ) − β d d x [ ( 1 − x ) α + 1 ( 1 + x ) β + 1 d d x ] F ( x ) {\displaystyle (1-x)^{-\alpha }(1+x)^{-\beta }{\frac {d}{dx}}\left[(1-x)^{\alpha +1}(1+x)^{\beta +1}{\frac {d}{dx}}\right]F(x),円} {\displaystyle (1-x)^{-\alpha }(1+x)^{-\beta }{\frac {d}{dx}}\left[(1-x)^{\alpha +1}(1+x)^{\beta +1}{\frac {d}{dx}}\right]F(x),円} − n ( n + α + β + 1 ) f α , β ( n ) {\displaystyle -n(n+\alpha +\beta +1)f^{\alpha ,\beta }(n)} {\displaystyle -n(n+\alpha +\beta +1)f^{\alpha ,\beta }(n)}
{ ( 1 − x ) − α ( 1 + x ) − β d d x [ ( 1 − x ) α + 1 ( 1 + x ) β + 1 d d x ] } k F ( x ) {\displaystyle \left\{(1-x)^{-\alpha }(1+x)^{-\beta }{\frac {d}{dx}}\left[(1-x)^{\alpha +1}(1+x)^{\beta +1}{\frac {d}{dx}}\right]\right\}^{k}F(x),円} {\displaystyle \left\{(1-x)^{-\alpha }(1+x)^{-\beta }{\frac {d}{dx}}\left[(1-x)^{\alpha +1}(1+x)^{\beta +1}{\frac {d}{dx}}\right]\right\}^{k}F(x),円} ( − 1 ) k n k ( n + α + β + 1 ) k f α , β ( n ) {\displaystyle (-1)^{k}n^{k}(n+\alpha +\beta +1)^{k}f^{\alpha ,\beta }(n)} {\displaystyle (-1)^{k}n^{k}(n+\alpha +\beta +1)^{k}f^{\alpha ,\beta }(n)}

References

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  1. ^ Debnath, L. "On Jacobi Transform." Bull. Cal. Math. Soc 55.3 (1963): 113-120.
  2. ^ Debnath, L. "SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS BY JACOBI TRANSFORM." BULLETIN OF THE CALCUTTA MATHEMATICAL SOCIETY 59.3-4 (1967): 155.
  3. ^ Scott, E. J. "Jacobi transforms." (1953).
  4. ^ Shen, Jie; Wang, Yingwei; Xia, Jianlin (2019). "Fast structured Jacobi-Jacobi transforms". Math. Comp. 88 (318): 1743–1772. doi:10.1090/mcom/3377 .
  5. ^ Debnath, Lokenath, and Dambaru Bhatta. Integral transforms and their applications. CRC press, 2014.


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