A formal summation formula

Define the Bernoulli numbers B_n by,

t / (e^t - 1) = sum_[n=0 to +infinity] (B_n).t^n / n!

Let ∂ f(t) = d[f(t)]/dt and put ∂^(-1) f(t) = integral f(t) dt

Note that,

exp [n∂ f(t)] = sum_[k=0 to +infinity] (n^k).∂^(n) f(t) / k! = f(t+n)

where the last inequality is by taking Taylor's theorem as a formal identity.

Then,

sum_[n=0 to +infinity] f(t+n) = sum_[n=0 to +infinity] exp [n∂ f(t)] = (1 - e^∂)^(-1) f(t) , by summing the geometric series.

Then,

sum_[n=0 to +infinity] f(t+n) = -sum_[n=0 to +infinity] (B_n)(∂^(n-1) f(t)) / n!

Let t->0 and we have,

sum_[n=0 to +infinity] f(n) = -sum_[n=0 to +infinity] (B_n)(∂^(n-1) f(0)) / n! -------- (1)

Put f(n) = n^k for k ∈ Z+ and we have all terms vanishing on the right hand side except that at n=k+1

For n=0 on the left there is no term so the left hand side is just the Riemann zeta function at -k, and we have,

Z(-k) = -(B_(k+1)).k! / (k+1)! = -(B_(k+1)) / (k+1)

We define the Bernoulli polynomials B_n (x) = sum_[k=0 to n] (nCk).(B_k).x^(n-k)

These have generating function,

t.e^(xt) / (e^t - 1) = sum_[n=0 to +infinity] (B_n (x)).t^n / n!

Put f(n) = (n+μ)^k for k ∈ Z+ and μ a constant in (1) and we have the Hurwitz zeta function Z on the left hand side and,

Z(k, μ) = -sum_[r=0 to k+1] (B_r)(k!).μ^(k-r+1) / (r!)(k-r+1)!

then,

Z(k, μ) = -(1/(k+1)) sum_[r=0 to k+1] ((k+1)Cr).(B_r).μ^(k-r+1) = -(B_(k+1) (μ)) / (k+1)

Which is an identity we needed in our proof of the Gauss multiplication theorem for the Gamma function.

We also remark that a two-variable generalisation is possible by considering f(t+y) rather than f(t) and deriving,

sum_[k=0 to n-2] f(t+y+k) = [sum_[k=0 to n-1] e^((y+k)D)] f(t) = [e^(yD) / (e^D - 1)] (f(t+n-1) - f(t))

(by summing the geometric series as before).

Whence by using the definition of the Bernoulli polynomials the operator in square brackets can be expanded as an infinite series. Putting f(t) = log (t) we can obtain an asymptotic expansion for log Γ(n+y) in powers of the reciprocal of n, which is Stirling's approximation (for y=0) with the constant involved expressed in series form!

Our formal manipulations can be shown to match up with the analytic extensions of these functions.

See G H Hardy, Divergent Series, 2nd ed. AMS Chelsea, 1991 (cf. sec.13.12) for more on the Euler-Maclaurin summation formula.