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 Hn(x) as kernels of the transform. The Hermite transform H{F(x)}fH(n) of a function F(x) is H{F(x)}fH(n)=ex2Hn(x)F(x)dx The inverse Hermite transform H1{fH(n)} is given by H1{fH(n)}F(x)=n=01π2nn!fH(n)Hn(x)

Some Hermite transform pairs

F(x) fH(n)
xm {m!π2mn(mn2)!,(mn) even and00,otherwise[1]
eax πanea2/4
e2xtt2,|t|<12 π(2t)n
Hm(x) π2nn!δnm
x2Hm(x) 2nn!π{1,n=m+2(n+12),n=m(n+1)(n+2),n=m20,otherwise
ex2Hm(x) (1)pm2p1/2Γ(p+1/2),m+n=2p,p
Hm2(x) {2m+n/2π(mn/2)m!n!(n/2)!,n even and2m0,otherwise[2]
Hm(x)Hp(x) {2kπm!n!p!(km)!(kn)!(kp)!,n+m+p=2k,k;|mp|nm+p0,otherwise[3]
Hn+p+q(x)Hp(x)Hq(x) π2n+p+q(n+p+q)!
dmdxmF(x) fH(n+m)
xdmdxmF(x) nfH(n+m1)+12fH(n+m+1)
ex2ddx[ex2ddxF(x)] 2nfH(n)
F(xx0) πk=0(x0)kk!fH(n+k)
F(x)*G(x) π(1)n[22n+1Γ(n+32)]1fH(n)gH(n)[4]
ez2sin(xz),|z|<12 {π(1)n2(2z)n,nodd0,neven
(1z2)1/2exp[2xyz(x2+y2)z2(1z2)] πznHn(y)[5][6]
Hm(y)Hm+1(x)Hm(x)Hm+1(y)2m+1m!(xy) {πHn(y)nm0n>m

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 français). 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 Deutsch) (66): 161–176, ISSN 0075-4102, ERAM 066.1720cj. See p. 174, eq. (18) and p. 173, eq. (13).

Sources