Srivastava's Functions of Three Variables for the HP-41

Overview
 

-Srivastava's Functions generalize the hypergeometric functions of 3 variables  x , y , z.

-Given 14 non-negative integers  A  B  C  D  E  F  G  H  I  J  K  L  M  N   and

 A numbers  a₁ ..... a_A   B  numbers  b₁ ..... b_B  C  numbers  c₁ ... c_C  D  numbers  d₁ .... d_D  E numbers  e₁ .... e_E  F numbers  f₁ ....  f_F    G numbers g₁....g_G
 H numbers  h₁ ..... h_H    I  numbers   i₁  .....  i_I   J   numbers   j₁ ...  j_J   K numbers   k₁ .... k_K  L numbers  l₁ ....  l_L  M numbers m₁ ... m_M N numbers n₁ ... n_N

 the following program computes

      F ^A_H^B_I^C_J^D_K^E_L^F_M^G_N(^{a1 ... aA ;}_{h1 ... hH ;} ^{b1 ... bB ;}_{i1 ... iI ;} ^{c1 ... cC ;}_{j1 ... jJ ;} ^{d1 ... dD ;}_{k1 ... kK ;} ^{e1 ... eE ;}_{l1 ...lL ;} ^{f1 ... fF ;}_{m1 ... mM ;} ^{g1... gG ;}_{n1 ... nN}  ;  x , y , z  )

      =  SUM_{p,q,r=0 to infinity}    u_p,q,r  x^p y^q z^r / ( p! q! r! )

where   u_p,q,r  = [ (a₁)_p+q+r ...... (a_A)_p+q+r (b₁)_p+q ...... (b_B)_p+q (c₁)_q+r ...... (c_C)_q+r (d₁)_p+r ...... (d_D)_p+r (e₁)_p ...... (e_E)_p (f₁)_q ......   (f_F)_q  (g₁)_r ...... (g_G)_r ] /
                         [ (h₁)_p+q+r ...... (h_H)_p+q+r  (i₁)_p+q ......  (i_I)_p+q   (j₁)_q+r ......  (j_J)_q+r  (k₁)_p+r ...... (k_K)_p+r  (l₁)_p ......  (l_L)_p (m₁)_q ...... (m_M)_q (n₁)_r ...... (n_N)_r ]

   and    (t)_n = Pochhammer's symbol:

                      (t)₀ = 1
                      (t)_n = t(t+1)(t+2) ...... (t+n-1)    if  n > 0
 

Program Listing
 

Data Registers:              R00: temp                     ( Registers R01 thru R14 & R37 thru .... are to be initialized before executing "SRI" )

                                      •  R01 = A    •  R02 = B    •  R03 = C    •  R04 = D    •  R05 = E    •  R06 = F    •  R07 = G
                                      •  R08 = H    •  R09 = I     •  R10 =  J    •  R11 = K    •  R12 = L    •  R13 = M   •  R14 = N

                                         R15 = x   R16 = y   R17 = z      R18 = p   R19 = q   R20 = r       R21 to R36: temp

                                      •  R37 = a₁    ...........................................................................   •  R36+A = a_A
                                      •  R37+A =  b₁ ........................................................................   •  R36+A+B = b_B
                                      •  R37+A+B =  c₁ .....................................................................  •  R36+A+B+C = c_C
                                      •  R37+A+B+C =  d₁ ................................................................  •  R36+A+B+C+D = d_D
                                      •  R37+A+B+C+D =  e₁ ............................................................  •  R36+A+B+C+D+E = e_E
                                      •  R37+A+B+C+D+E = f₁ .........................................................  •  R36+A+B+C+D+E+F = f_F
                                      •  R37+A+B+C+D+E+F = g₁ ....................................................  •  R36+A+B+C+D+E+F+G = g_G
                                      •  R37+A+B+C+D+E+F+G = h₁ ...............................................  •  R36+A+B+C+D+E+F+G+H = h_H
                                      •  R37+A+B+C+D+E+F+G+H = i₁ ...........................................  •  R36+A+B+C+D+E+F+G+H+I = i_I
                                      •  R37+A+B+C+D+E+F+G+H+I = j₁ .......................................  •  R36+A+B+C+D+E+F+G+H+I+J = j_J
                                      •  R37+A+B+C+D+E+F+G+H+I+J = k₁ ..................................  •  R36+A+B+C+D+E+F+G+H+I+J+K = k_K
                                      •  R37+A+B+C+D+E+F+G+H+I+J+K = l₁ ..............................  •  R36+A+B+C+D+E+F+G+H+I+J+K+L = l_L
                                      •  R37+A+B+C+D+E+F+G+H+I+J+K+L = m₁ ........................  •  R36+A+B+C+D+E+F+G+H+I+J+K+L+M = m_M
                                      •  R37+A+B+C+D+E+F+G+H+I+J+K+L+M = n₁ ...................  •  R36+A+B+C+D+E+F+G+H+I+J+K+L+M+N = n_N

Flag:  F10 is cleared at the end
Subroutines: /
 

-I've used an M-Code function   X+1  to add 1 to register X
-It may be replaced by  ISG X  CLX

-Lines 02-03 are only useful to remind that the cefficients are to be stored into R37 , R38 , ... etc ...
-Otherwise, they may be deleted.

-Lines 35-222-279  are three-byte  GTOs
 
 

 
   ( 489 bytes / SIZE 037+A+B+C+D+E+F+G+H+I+J+K+L+M+N )
 
 

 
Example:      With  A = B = ........................... = N = 1   ( to be stored into R01 thru R14 )

          a₁ = sqrt(2)   b₁ = sqrt(3)   c₁ = sqrt(5)   d₁ = sqrt(6)    e₁ = sqrt(7)    f₁  = sqrt(8)    g₁ = sqrt(10)                ( to be stored into
          h₁ = sqrt(11)  i₁ = sqrt(12)  j₁ = sqrt(13)  k₁ = sqrt(14)  l₁ = sqrt(15)  m₁ = sqrt(17)  n₁ = sqrt(18)                  R37 thru R50 )

  and   x = 0.1   y = 0.2   z = 0.3

  0.3   ENTER^
  0.2   ENTER^
  0.1   XEQ "SRI"  >>>>>   F( 0.1 , 0.2 , 0.3 ) = 1.070158801            ---Execution time = 7m18s---
 

Notes:

-The routine is obviously very slow and a good emulator like V41 is probably preferable.
-Execution time may be saved if you add 7 other subroutines similar to  LBL 01 .....  LBL 07  with  /  instead of  FC? 10  1/X  *

-This program does not check if the series is convergent or not.
-If A = B = ..... = N = 0 , "SRI"  returns  exp(x+y+z)

-The termination criterion may be doubtful for certain values of the parameters:
-Press  XEQ 10  without changing the stack to check.