[seqfan] Re: A series for exp(Pi)
mathar at strw.leidenuniv.nl
Sat Oct 24 21:41:34 CEST 2009
let's assume that exp(Pi) is (i) not the expatriate pi, (ii) not the expectation
value of pi -- pi is a constant, not fluctuating much unless used in
quantum mechanics with some uncertainty relation, or (iii) saying that its
expensive to buy a pie, but that it deals with the exponential function
evaluated at 3.14159...
Then a(n)=(40-4*n+n^2)*a(n-2) splits into 2 non-coupled subsequences,
one at the even and one at the odd indices n. Then we split the
2-nd order polynomial n^2-4*n+40 into (n-alpha)*(n-beta), where
alpha,beta = 2+- sqrt(-36) = 2+-6i follow from finding the roots of the polynomial.
The we note that building the products while telescoping the terms a(n),
a(n+2), a(n+4), a(n+6) etc accumulates factors of the form (n-alpha)*(n-beta)
*(n+2-alpha)*(n+2-beta)*(n+4-alpha)*(n+4-beta) which is obviously a
Pochhammer symbol and which can be written as a fraction of two Gamma functions.
This is "well known", see for example Remark 1 on page 3 of my
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