A 'Natural Proof' of the Gauss Multiplication formula I: Preliminaries

The following proof of the Gauss Multiplication formula for the Gamma function is an elaboration of the original that appears in the excellent paper 'Zeta Regularizations' by N Kurokawa and M Wakayama in Acta Applicandae Mathematicae Volume 81, Number 1 / March, 2004. We follow their exposition throughout.
http://www.springerlink.com/content/q185862551668217/

It is by far the most natural proof I have come across of this theorem even though the fundamental idea is from the theory of regularized determinants.

A few preliminaries

The notation introduced here will be used throughout.
Observe that for a given Hurwitz zeta function, Z_λ (s, t) = sum_[λ_n] (λ_n + t)^(-s),
∂[Z_λ (0, t)]/∂s = - sum_[λ_n] log (λ_n + t)

So we define the zeta-regularized product for the sequence (λ_n)_{n=0,...},
prodz_[λ_n] (λ_n + t) := exp (- ∂[Z_λ (0, t)]/∂s)

For convenience when λ_n = n we shall denote the resulting classical Hurwitz zeta function by Z.

The main result that will be employed is Lerch's formula,

∂[Z(0, t)]/∂s = -(1/2).log (2π) + log Γ(t)

where log is the natural logarithm and Γ is the Gamma function as defined at http://en.wikipedia.org/wiki/Gamma_function

Note that Z'(0) = -(1/2).log (2π) where Z is the Riemann zeta function.

For a proof the reader can refer to http://ocw.nctu.edu.tw/upload/fourier/supplement/Zeta-Function.pdf or the chapter on the Zeta and Gamma functions in S Lang's Complex Analysis (4th ed.), Springer Graduate Texts in Mathematics.

We now have the immediate corollary that,

prodz_[n=0 to +infinity] (n + t) = √(2π) / Γ(t) ------- (1)

(await latter half for proof)...