New Sigma and Pi functions

[sweety439]

SigmaArithmetic(a,d,n) = n/2 * (2*a+(n-1)*d)

which is the formula of the sum of members in an arithmetic sequence

In original, a and d can be any complex numbers, but n can only be nonnegative integer, however, since we already have the formula, we can also extend n to any complex numbers and create the new “SigmaArithmetic” function, especially, SigmaArithmetic(a,0,n) = a*n

SigmaGeometric(a,r,n) = a*(r^n-1)/(r-1) if r != 1, or a*n if r = 1

which is the formula of the sum of members in an geometric sequence

In original, a and r can be any complex numbers, but n can only be nonnegative integer, however, since we already have the formula, we can also extend n to any complex numbers and create the new “SigmaGeometric” function

PiArithmetic(a,d,n) = a*(a+d)*(a+2*d)*…*(a+(n-1)*d)

which is the formula of the product of members in an arithmetic sequence

If n is nonnegative integer, then we can directly use this formula (if n = 0, then this is empty product, and hence its value is 1)

We can extend n to all complex numbers by:

* If d = 0, then PiArithmetic(a,0,n) = a^n, and thus we can extend n to all complex numbers (in my world, 0^0 = 1)
* If d = 1, then PiArithmetic(a,1,n) = Gamma(a+n)/Gamma(a), where Gamma is the Gamma function, and thus we can extend n to all complex numbers (remember: Gamma(1) = 1, and Gamma(x) = infinity if x is nonpositive integer)
* If d = 2, then PiArithmetic(a,2,n) = (a+2*n-2)!!/(a-2)!!, and thus we can extend n to all complex numbers, see https://en.wikipedia.org/wiki/Double...ial#Extensions
etc.

PiGeometric(a,r,n) = a^n*r^(n*(n-1)/2)

which is the formula of the product of members in an geometric sequence

In original, a and r can be any complex numbers, but n can only be nonnegative integer, however, since we already have the formula, we can also extend n to any complex numbers and create the new “PiGeometric” function