Re: infinite product

In article <e09vm6$okq$1@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Boudewijn Moonen <B.Moonen@xxxxxxxxxxxxxxx> wrote:

Paul Abbott schrieb:

One approach is to note that

Product[(1 + a/k)^2/(1 + b/k), {k, 1, Infinity}]

is

Gamma[1 + b]/Gamma[1 + a]^2

if 2 Re[a] <= Re[b].

Could you be more explicit about this? What I know is

lim_{n->oo} Product((1+x/k),k=1..n)/n^x) = 1/GAMMA(1+x)

but how do you get rid of the denominator n^x?

The finite product

Product[(1 + a/k)^2/(1 + b/k), {k, 1, m}]

is just

Pochhammer[1 + a, m]^2/(m! Pochhammer[1 + b, m])

In terms of Gamma functions this is

(Gamma[1 + b]*Gamma[1 + a + m]^2)/
(Gamma[1 + a]^2*Gamma[1 + m]*Gamma[1 + b + m])

The leading term of the asymptotic series expansion is

m^(2 a - b) Gamma[b + 1] / Gamma[a + 1]^2

so, if 2 a == b (not 2 Re[a] <= Re[b] as I stated), then one obtains

Gamma[b + 1] / Gamma[a + 1]^2

Note that 2 a is the coefficient of k^(-1) in the (expanded) numerator
and b is the coefficient of k^(-1) in the denominator. This result
generalizes to products with an arbitrary number of numerator and
denominator terms. For example,

Product[ (1 + a[1]/k)(1 + a[2]/k) /
((1 + b[1]/k) (1 + b[2]/k) (1 + b[3]/k)), {k, 1, Infinity}]

is

Gamma[b[1]+1] Gamma[b[2]+1] Gamma[b[3]+1]
-----------------------------------------
Gamma[a[1]+1] Gamma[a[2]+1]

if a[1] + a[2] == b[1] + b[2] + b[3].

Now, both products

K3 = \prod (1 + 1/k + 1/k^2)^2/(1 + 2/k + 3/k^2)

K4 = \prod (1 + 1/k + 1/k^2 + 1/k^3)^2/(1 + 2/k + 3/k^2 + 4/k^3)

are such that the coefficient of k^(-1) in the (expanded) numerator is
the same as that of the denominator. Indeed, for finite n and m,
infinite products over k of terms of the form

1 + a[1]/k + a[2]/k^2 + ... + a[n]/k^n
--------------------------------------
1 + b[1]/k + b[2]/k^2 + ... + b[m]/k^m

where a[1] == b[1], can be computed in closed form, in terms of finite
products of gamma functions of the roots of the numerator and
denominator polynomials (in inverse powers of k).

Cheers,
Paul

_______________________________________________________________________
Paul Abbott Phone: 61 8 6488 2734
School of Physics, M013 Fax: +61 8 6488 1014
The University of Western Australia (CRICOS Provider No 00126G)
AUSTRALIA http://physics.uwa.edu.au/~paul
.

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