[R] taylor expansions with real vectors

[bilelsan]

Dear list, 

I have a big deal concerning the development of a Taylor expansion. 

require(Matrix) 
e1 <- as.vector(1:5) 
e2 <- as.vector(6:10) 

in order to obtain all the combinations between these two vectors following
a Taylor expansion (or more simply through a Maclaurin series) for real
numbers. 
We have f(x) = f(0) + f'(0)(x-0) + f''(0)(x-0)^2/2! + … + f^(k)(0)(x-0)^k/k! 
with 
f(x) = e1 + e2 for Taylor expansion (r = 1) 
        + 1/2!*e1^2 + 1/2!*e2^2 + 1/2!*e1*e2 for Taylor expansion (r = 2)
excluding e2*e1 
        + 1/3!*e1^3 + 1/3!*e1^2*e2 + 1/3!*e2^2*e1 + 1/3!*e2^3 for Taylor
expansion (r = 3) excluding e2*e1^2 and e1*e2^2 
       ... 
I already write the number of possible combinations as : 
x <- as.vector(0) 
for (r in 1:r){x[r] <- 2*(sum(choose(2*q+r-1,r))-sum(choose(q+r-1,r)))}# q:
number of lag of e1 and e2; r: order of taylor expansion 
nstar   <- sum(x) # N* number of total combinations 
  
How to write f(x) in a general framework? 
Quid of this framework when e1 and e2 are completed with their lags if q >
1? 
Your help or advice would be greatly appreciated 

Bilel

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