'****************************************************
'*   Program to demonstrate Hermite coefficients    *
'* ------------------------------------------------ *
'* Reference: BASIC Scientific Subroutines, Vol. II *
'* By F.R. Ruckdeschel, BYTE/McGRAWW-HILL, 1981 [1].*
'*                                                  *
'* ------------------------------------------------ *
'* SAMPLE RUN:                                      *
'*                                                  *
'* Hermite polynomial coefficients for order 2      *
'*                                                  *
'*   A( 0) =        -2                              *
'*   A( 1) =         0                              *
'*   A( 2) =         4                              *
'*                                                  *
'* Hermite polynomial coefficients for order 3      *
'*                                                  *
'*   A( 0) =         0                              *
'*   A( 1) =       -12                              *
'*   A( 2) =         0                              *
'*   A( 3) =         8                              *
'*                                                  *
'* Hermite polynomial coefficients for order 4      *
'*                                                  *
'*   A( 0) =        12                              *
'*   A( 1) =         0                              *
'*   A( 2) =       -48                              *
'*   A( 3) =         0                              *
'*   A( 4) =        16                              *
'*                                                  *
'* Hermite polynomial coefficients for order 5      *
'*                                                  *
'*   A( 0) =         0                              *
'*   A( 1) =       120                              *
'*   A( 2) =         0                              *
'*   A( 3) =      -160                              *
'*   A( 4) =         0                              *
'*   A( 5) =        32                              *
'*                                                  *
'* Hermite polynomial coefficients for order 6      *
'*                                                  *
'*   A( 0) =      -120                              *
'*   A( 1) =         0                              *
'*   A( 2) =       720                              *
'*   A( 3) =         0                              *
'*   A( 4) =      -480                              *
'*   A( 5) =         0                              *
'*   A( 6) =        64                              *
'*                                                  *
'* Hermite polynomial coefficients for order 7      *
'*                                                  *
'*   A( 0) =         0                              *
'*   A( 1) =     -1680                              *
'*   A( 2) =         0                              *
'*   A( 3) =      3360                              *
'*   A( 4) =         0                              *
'*   A( 5) =     -1344                              *
'*   A( 6) =         0                              *
'*   A( 7) =       128                              *
'*                                                  *
'* Hermite polynomial coefficients for order 8      *
'*                                                  *
'*   A( 0) =      1680                              *
'*   A( 1) =         0                              *
'*   A( 2) =    -13440                              *
'*   A( 3) =         0                              *
'*   A( 4) =     13440                              *
'*   A( 5) =         0                              *
'*   A( 6) =     -3584                              *
'*   A( 7) =         0                              *
'*   A( 8) =       256                              *
'*                                                  *
'* Hermite polynomial coefficients for order 9      *
'*                                                  *
'*   A( 0) =         0                              *
'*   A( 1) =     30240                              *
'*   A( 2) =         0                              *
'*   A( 3) =    -80640                              *
'*   A( 4) =         0                              *
'*   A( 5) =     48384                              *
'*   A( 6) =         0                              *
'*   A( 7) =     -9216                              *
'*   A( 8) =         0                              *
'*   A( 9) =       512                              *
'*                                                  *
'* Hermite polynomial coefficients for order 10     *
'*                                                  *
'*   A( 0) =    -30240                              *
'*   A( 1) =         0                              *
'*   A( 2) =    302400                              *
'*   A( 3) =         0                              *
'*   A( 4) =   -403200                              *
'*   A( 5) =         0                              *
'*   A( 6) =    161280                              *
'*   A( 7) =         0                              *
'*   A( 8) =    -23040                              *
'*   A( 9) =         0                              *
'*   A(10) =      1024                              *
'*                                                  *
'****************************************************
' Explanations
' ------------
' 
' Hermite polynomials are defined over the range -inf. < x < inf. The weight- 
' ing function is w(x) = e^(-x^2): 
'
'    inf.
'    Sum   e^(-x^2) H (x) H (x) dx = 0     for n <> m         (3.9.8)
'    -inf.           n     m
'                                  = f(n)  for n = m
'
' The corresponding recursion relation is
' 
'    H   (x) = 2x H (x) - 2n H   (x)                          (3.9.9)
'     n+1          n          n-1
' 
' where H (x) = 1  and  H (x) = 2x
'        0               1
' 
' As with the other polynomials, a simple subroutine for evaluating the coefficients 
' be written (see program HERMITE). 
' 
' Note that Hermite polynomials are either even or odd, depending on N, and the co- 
' efficients are integers. 
'-----------------------------------------------------------------------------------
defint i-n
defdbl a-h,o-z
dim A(10),B(10,10)
  cls
  for n=2 to 10
    print
    print " Hermite polynomial coefficients for order ";n
    print
    gosub 1000
    for k=0 to n
      print "   A(";k;") = ";A(k)
    next k
    print
    if n<10 then input r$
  next n
  print
end

'**************************************************
'* Hermite polynomial coefficients evaluation by  *
'* means of recursion relation. The order of the  *
'* polynomial is n. The coefficients are returned *
'* in A(i).                                       *
'**************************************************
1000 'Establish h0 and h1 coefficients
  B(0,0)=1# : B(1,0)=0# : B(1,1)=2#
  'Return if order is less than two
  if n<2 then return
  for i=2 to n
    B(i,0)=-2*(i-1)*B(i-2,0)
    for j=1 to i
      'Basic recursion relation
      B(i,j)=2*B(i-1,j-1)-2*(i-1)*B(i-2,j)
    next j
  next i
  for i=0 to n
    A(i)=B(n,i)
  next i
return

'End of file Hermite.bas