Organization of the Differentiation Process

Integer lists may be compared as in the Kronecker delta, 8 .,p, in the same way as two geometrical vectors. If every element in a is the same as the corresponding element in p, then the lists are the same. If there is a difference in any element, then = 0. 

The array n distinguishes first derivatives from second derivatives, and so on. For computer implementation, an integer list is constructed explicitly for all derivatives sought. This list depends on the number of parameters, the order of differentiation, and on whether differentiation is to be carried to the same level for each parameter. A routine is required that compares two vectors, i.e., that finds 8a,p. Another routine is needed for computing certain products of binomial coefficients. The following notation is helpful for these produas. 

N is the number of parameters. Factors that happen to be needed in an open ended differentiation procedure are designated here by t  

These factors enter derivative equations only if (a + P + 7) is found in the set of integer lists. This means it is sufficient to compute and store a table of values, a , for all a s, and then evaluate any t factor as needed by looking up just four values in that table and applying q. [15]. Instead of that, the t factors may be found with a stored table of binomial coefficients  

These are used to form / factors, which may be multiplied to yield t factors  

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