# How to define function with indexed variables and sum

I would like to define a function f(n)=sum(x[k]*x[k+1], k= 1..n) so that it produces expressions like this:

f(1)= x[1]*x[2],
f(2)=x[1]*x[2]+x[2]*x[3]
f(3)=x[1]*x[2]+x[2]*x[3]+x[3]*x[4]


How to do this in sage? I tried to generate indexed variable with the following:

class VariableGenerator(object):
def __init__(self, prefix):
self.__prefix = prefix
@cached_method
def __getitem__(self, key):
return SR.var("%s%s"%(self.__prefix,key))

x=VariableGenerator('x')
k, n = var('k,n')
f(n)=sum(x[k], k, 1, n)


Unfortunately this does not work. Any ideas?

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Would defining f as a Python function work for you?

f = lambda n: sum( x[k] for k in range(1,n+1) )

( 2023-03-14 03:24:00 +0200 )edit

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Would something like the following help ?

def f(n):
return sum(SR.var('x_{}'.format(k))*SR.var('x_{}'.format(k+1)) for k in range(1,n+1))

f(3)
x_1*x_2 + x_2*x_3 + x_3*x_4

more

This solution does create the necessary variables, but does not inject them in the global namespace (which can be done later with sage.misc.misc.inject_variable).

Alternatively, one can use var rather than SR.var to the same effect...

( 2023-03-14 10:13:49 +0200 )edit