A 'Natural Proof' of the Gauss Multiplication formula II: A Lemma

We continue with our treatment of the multiplication formula for the Gamma function following Kurokawa and Wakayama.

We will require the following:

Lemma

prodz_[n ∈ J] μ.(a_n + b) = [μ^Z(0,b)].prodz_[n ∈ J] (a_n + b)

for μ, b constants.

Z_[μ.a_n] (s, t) = sum_[n ∈ J] (μ.a_n + t)^(-s) = [μ^(-s)].[Z_[a_n] (s, t/μ)] , by taking μ^(-s) out of the sum and noting the resulting zeta function.

Then,

Z_[μ.a_n] '(s, t) = -(μ^(-s)).(log μ).[Z_[a_n] (s, t/μ)] + (μ^(-s)).[Z_[a_n] '(s, t/μ)] , by differention using the product rule.

With s=0,

-Z_[μ.a_n] (0, t) = log [μ^(Z_[a_n] (0, t/μ))] - Z_[a_n] '(0, t/μ)

Which gives,

exp (-Z_[μ.a_n] '(0, t)) = [μ^(Z_[a_n] (0, t/μ))]. exp (-Z_[a_n] '(0, t/μ))

t=μb gives the desired result.

We note that, in the classical case,

Z(-n, b) = -B_[n+1] (b) / (n+1) , where B_n (x) is the n th Bernoulli polynomial on x, for n a natural number and b a constant. Then, Z(0, b) = (1/2) - b

(a divergent series based proof of this will be given in a later post)

We finish the proof with the coming post!