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Hermite transform

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In mathematics, the Hermite transform is an integral transform named after the mathematician Charles Hermite that uses Hermite polynomials H n ( x ) {\displaystyle H_{n}(x)} as kernels of the transform.

The Hermite transform H { F ( x ) } ≡ f H ( n ) {\displaystyle H\{F(x)\}\equiv f_{H}(n)} of a function F ( x ) {\displaystyle F(x)} is H { F ( x ) } ≡ f H ( n ) = ∫ − ∞ ∞ e − x 2   H n ( x )   F ( x )   d x {\displaystyle H\{F(x)\}\equiv f_{H}(n)=\int _{-\infty }^{\infty }e^{-x^{2}}\ H_{n}(x)\ F(x)\ dx}

The inverse Hermite transform H − 1 { f H ( n ) } {\displaystyle H^{-1}\{f_{H}(n)\}} is given by H − 1 { f H ( n ) } ≡ F ( x ) = ∑ n = 0 ∞ 1 π 2 n n ! f H ( n ) H n ( x ) {\displaystyle H^{-1}\{f_{H}(n)\}\equiv F(x)=\sum _{n=0}^{\infty }{\frac {1}{{\sqrt {\pi }}2^{n}n!}}f_{H}(n)H_{n}(x)}

Some Hermite transform pairs

F ( x ) {\displaystyle F(x)\,} f H ( n ) {\displaystyle f_{H}(n)\,}
x m {\displaystyle x^{m}} { m ! π 2 m − n ( m − n 2 ) ! , ( m − n )  even and ≥ 0 0 , otherwise {\displaystyle {\begin{cases}{\frac {m!{\sqrt {\pi }}}{2^{m-n}\left({\frac {m-n}{2}}\right)!}},&(m-n){\text{ even and}}\geq 0\\0,&{\text{otherwise}}\end{cases}}}
e a x {\displaystyle e^{ax}\,} π a n e a 2 / 4 {\displaystyle {\sqrt {\pi }}a^{n}e^{a^{2}/4}\,}
e 2 x t − t 2 ,   | t | < 1 2 {\displaystyle e^{2xt-t^{2}},\ |t|<{\frac {1}{2}}\,} π ( 2 t ) n {\displaystyle {\sqrt {\pi }}(2t)^{n}}
H m ( x ) {\displaystyle H_{m}(x)\,} π 2 n n ! δ n m {\displaystyle {\sqrt {\pi }}2^{n}n!\delta _{nm}\,}
x 2 H m ( x ) {\displaystyle x^{2}H_{m}(x)\,} 2 n n ! π { 1 , n = m + 2 ( n + 1 2 ) , n = m ( n + 1 ) ( n + 2 ) , n = m − 2 0 , otherwise {\displaystyle 2^{n}n!{\sqrt {\pi }}{\begin{cases}1,&n=m+2\\\left(n+{\frac {1}{2}}\right),&n=m\\(n+1)(n+2),&n=m-2\\0,&{\text{otherwise}}\end{cases}}}
e − x 2 H m ( x ) {\displaystyle e^{-x^{2}}H_{m}(x)\,} ( − 1 ) p − m 2 p − 1 / 2 Γ ( p + 1 / 2 ) ,   m + n = 2 p ,   p ∈ Z {\displaystyle \left(-1\right)^{p-m}2^{p-1/2}\Gamma (p+1/2),\ m+n=2p,\ p\in \mathbb {Z} }
H m 2 ( x ) {\displaystyle H_{m}^{2}(x)\,} { 2 m + n / 2 π ( m n / 2 ) m ! n ! ( n / 2 ) ! , n  even and ≤ 2 m 0 , otherwise {\displaystyle {\begin{cases}2^{m+n/2}{\sqrt {\pi }}{\binom {m}{n/2}}{\frac {m!n!}{(n/2)!}},&n{\text{ even and}}\leq 2m\\0,&{\text{otherwise}}\end{cases}}}
H m ( x ) H p ( x ) {\displaystyle H_{m}(x)H_{p}(x)\,} { 2 k π m ! n ! p ! ( k − m ) ! ( k − n ) ! ( k − p ) ! , n + m + p = 2 k ,   k ∈ Z ;   | m − p | ≤ n ≤ m + p 0 , otherwise {\displaystyle {\begin{cases}{\frac {2^{k}{\sqrt {\pi }}m!n!p!}{(k-m)!(k-n)!(k-p)!}},&n+m+p=2k,\ k\in \mathbb {Z} ;\ |m-p|\leq n\leq m+p\\0,&{\text{otherwise}}\end{cases}}\,}
H n + p + q ( x ) H p ( x ) H q ( x ) {\displaystyle H_{n+p+q}(x)H_{p}(x)H_{q}(x)\,} π 2 n + p + q ( n + p + q ) ! {\displaystyle {\sqrt {\pi }}2^{n+p+q}(n+p+q)!\,}
d m d x m F ( x ) {\displaystyle {\frac {d^{m}}{dx^{m}}}F(x)\,} f H ( n + m ) {\displaystyle f_{H}(n+m)\,}
x d m d x m F ( x ) {\displaystyle x{\frac {d^{m}}{dx^{m}}}F(x)\,} n f H ( n + m − 1 ) + 1 2 f H ( n + m + 1 ) {\displaystyle nf_{H}(n+m-1)+{\frac {1}{2}}f_{H}(n+m+1)\,}
e x 2 d d x [ e − x 2 d d x F ( x ) ] {\displaystyle e^{x^{2}}{\frac {d}{dx}}\left\,} − 2 n f H ( n ) {\displaystyle -2nf_{H}(n)\,}
F ( x − x 0 ) {\displaystyle F(x-x_{0})} π ∑ k = 0 ∞ ( − x 0 ) k k ! f H ( n + k ) {\displaystyle {\sqrt {\pi }}\sum _{k=0}^{\infty }{\frac {(-x_{0})^{k}}{k!}}f_{H}(n+k)}
F ( x ) ∗ G ( x ) {\displaystyle F(x)*G(x)\,} π ( − 1 ) n [ 2 2 n + 1 Γ ( n + 3 2 ) ] − 1 f H ( n ) g H ( n ) {\displaystyle {\sqrt {\pi }}(-1)^{n}\left^{-1}f_{H}(n)g_{H}(n)\,}
e z 2 sin ⁡ ( x z ) ,   | z | < 1 2   {\displaystyle e^{z^{2}}\sin(xz),\ |z|<{\frac {1}{2}}\ \,} { π ( − 1 ) ⌊ n 2 ⌋ ( 2 z ) n , n o d d 0 , n e v e n {\displaystyle {\begin{cases}{\sqrt {\pi }}(-1)^{\lfloor {\frac {n}{2}}\rfloor }(2z)^{n},&n\,\mathrm {odd} \\0,&n\,\mathrm {even} \end{cases}}\,}
( 1 − z 2 ) − 1 / 2 exp ⁡ [ 2 x y z − ( x 2 + y 2 ) z 2 ( 1 − z 2 ) ] {\displaystyle (1-z^{2})^{-1/2}\exp \left\,} π z n H n ( y ) {\displaystyle {\sqrt {\pi }}z^{n}H_{n}(y)}
H m ( y ) H m + 1 ( x ) − H m ( x ) H m + 1 ( y ) 2 m + 1 m ! ( x − y ) {\displaystyle {\frac {H_{m}(y)H_{m+1}(x)-H_{m}(x)H_{m+1}(y)}{2^{m+1}m!(x-y)}}} { π H n ( y ) n ≤ m 0 n > m {\displaystyle {\begin{cases}{\sqrt {\pi }}H_{n}(y)&n\leq m\\0&n>m\end{cases}}}

References

  1. McCully, Joseph Courtney; Churchill, Ruel Vance (1953), Hermite and Laguerre integral transforms : preliminary report
  2. Feldheim, Ervin (1938). "Quelques nouvelles relations pour les polynomes d'Hermite". Journal of the London Mathematical Society (in French). s1-13: 22–29. doi:10.1112/jlms/s1-13.1.22.
  3. Bailey, W. N. (1939). "On Hermite polynomials and associated Legendre functions". Journal of the London Mathematical Society. s1-14 (4): 281–286. doi:10.1112/jlms/s1-14.4.281.
  4. Glaeske, Hans-Jürgen (1983). "On a convolution structure of a generalized Hermite transformation" (PDF). Serdica Bulgariacae Mathematicae Publicationes. 9 (2): 223–229.
  5. Erdélyi et al. 1955, p. 194, 10.13 (22).
  6. Mehler, F. G. (1866), "Ueber die Entwicklung einer Function von beliebig vielen Variabeln nach Laplaceschen Functionen höherer Ordnung" [On the development of a function of arbitrarily many variables according to higher-order Laplace functions], Journal für die Reine und Angewandte Mathematik (in German) (66): 161–176, ISSN 0075-4102, ERAM 066.1720cj. See p. 174, eq. (18) and p. 173, eq. (13).

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