The Riemann and Hurwitz zeta functions and the Mellin transform interpolation of the Bernoulli polynomials

This entry (expanding on the Bernoulli Appells entry) illustrates interpolation with the Mellin transform of the Bernoulli polynomials and their umbral inverses, the reciprocal polynomials, giving essentially the Hurwitz zeta function and the finite difference of x^{1-s}/(1-s), both of which can be umbrally inverted by the polynomials. It also elaborates on a set of generalized Bernoulli polynomials based on umbral composition of the Bernoulli polynomials with themselves and derives an “asymptotic” expression, or divergent series, for the Riemann zeta function noted in the Bernoulli Appells entry, which may be truncated to give very good approximations of the Hurwitz and Riemann zeta functions over ranges of parameters.

The Mellin transform interpolation of the Bernoulli polynomials gives the Hurwitz zeta function, essentially a sum of the integrally shifted x^{-1-s},

\displaystyle B_{-s}(x) = \int_{0}^{\infty} e^{-B.(x)t} \; \frac{t^{s-1}}{(s-1)!} \; dt = \int_{0}^{\infty} \frac{t}{1-e^{-t}} \; e^{-xt} \; \frac{t^{s-1}}{(s-1)!} \; dt

\displaystyle = \int_{0}^{\infty} \sum_{n \ge 0} e^{-(n+x)t} \; s \;\frac{t^s}{s!} \; dt = s \sum_{n \ge 0} \frac{1}{(x+n)^{s+1}}=s \; H(s,x) \; .

Since B_n(x+y) = (B.(x+y))^n = (B.(x)+y))^n by its Appell properties and for any umbral sequence (a.)^n = a_n the generalized shift theorem e^{a.\; D_y} \; y^n =(a. +y)^n = \sum_{k=0}^{n} \binom{n}{k} a_k \; y^{n-k} applies, this may also be expressed operationally as

\displaystyle B_{-s}(x+y) = e^{B.(x)D_y} \int_{0}^{\infty} e^{-yt} \; \frac{t^{s-1}}{(s-1)!} \; dt = e^{B.(x)D_y} \; y^{-s} = \frac{D_y}{e^{D_y}-1} \; e^{xD_y} \; y^{-s}

\displaystyle = \sum_{n \ge 0} e^{nD_y} \; s (x+y)^{-s-1} = s \; H(s, x+y) \; .

Likewise, the umbral inverses, the reciprocal polynomials, give a finite difference of x^{1-s} ,

\displaystyle \bar{B}_{-s}(x) = \int_{0}^{\infty} e^{-\bar{B}.(x)t} \; \frac{t^{s-1}}{(s-1)} \; dt = \int_{0}^{\infty} \frac{1-e^{-t}}{t} \; e^{-xt} \frac{t^{s-1}}{(s-1)!} \; dt

\displaystyle =\frac{(x+1)^{1-s}-x^{1-s}}{1-s} = \frac{e^{D_x}-1}{D_x}\; x^{-s} \;.

Since the Bernoulli polynomials and the reciprocal polynomials comprise an umbral inverse pair, i.e., B_n(\bar{B}.(x))=x^n= \bar{B}_n(B.(x)) (see Bernoulli Appells), umbrally composing either of the interpolated functions with the other gives x^{-s}, as can readily be seen by umbral subsitution in the integrand of the Mellin transforms, giving

\displaystyle e^{-B.(\bar{B}.(x))t}=e^{-xt}=e^{-\bar{B}.(B.(x)t)} \; .

This is consistent with the operational definition of the Bernoulli polynomials and the series relations:

\displaystyle \bar{B}_{-s}(B.(x))= \frac{(B.(x)+1)^{1-s}-(B.(x))^{1-s}}{1-s}

\displaystyle =\frac{(B.(x+1))^{1-s}-(B.(x))^{1-s}}{1-s} = D_x \; \frac{x^{1-s}}{1-s} = x^{-s}

\displaystyle = \frac{B_{1-s}(x+1)-B_{1-s}(x)}{1-s} = -[H(s-1,x+1)-H(s-1,x)] \; = x^{-s} \; .

Same for the converse using the operational definition of the reciprocal polynomials and series expansions,

\displaystyle B_{-s}(\bar{B}.(x))= \int_{x}^{x+1} B_{-s}(u) \; du = -[H(s-1,x+1)-H(s-1,x)] = x^{-s} \; .

\displaystyle = s \sum_{n \ge 0} (\bar{B}.(x)+n)^{-s-1} = s \sum_{n \ge 0} \bar{B}_{-s-1}(x+n) = x^{-s}  .

Now look at the double umbral composition \bar{B}_{-s}(B.(B.(x))) \; . From umbral substitution into the argument of the Mellin transform, this becomes B_{-s}(x)= (B.(x))^{-s}. Let’s evaluate this in other ways. Using the op definition of the Bernoulli polynomials,

\displaystyle \bar{B}_{-s}(B.(B.(x)))= \frac{(1+B.(B.(x)))^{1-s}-(B.(B.(x)))^{1-s}}{1-s} = \frac{(B.(B.(1+x)))^{1-s}-(B.(B.(x)))^{-1-s}}{1-s}

\displaystyle = D_x \; \frac{ B_{1-s}(x)}{1-s} = D_x \; [ -H(s-1,x)] = B_{-s}(x) \; .

Or, define the generalized Bernoulli polynomials of second order through double umbral composition

\displaystyle e^{{B}.^{(2)}(x)t}= e^{B.(B.(x))t} = \frac{t}{e^t-1} \; e^{B.(x)t} = (\frac{t}{e^t-1})^{2} \; e^{xt} \; .

Then these are Appell polynomials and

\displaystyle \bar{B}_{-s}(B.(B.(x))) = \bar{B}_{-s}(B.^{(2)}(x)) = \frac{B_{1-s}^{(2)}(x+1)-B_{1-s}^{(2)}(x)}{1-s}

\displaystyle = \int_{0}^{\infty} (\frac{t}{1-e^{-t}})^2 (e^{-(x+1)t} - e^{-xt}) \; \frac{1}{1-s} \; \frac{t^{s-2}}{(s-2)!} \; dt

\displaystyle = \int_{0}^{\infty} \frac{t}{1-e^{-t}} \; e^{-xt} \; \frac{t^{s-1}}{(s-1)!} \; dt = B_{-s}(x) .

 These lead to another series rep for the Hurwitz and Riemann zeta functions (the equality for the general binomial series expressions is only formal below):

\displaystyle B_{-s}(x) = \bar{B}_{-s}(B.(B.(x))) = \bar{B}_{-s}(B.^{(2)}(x)) = \frac{(1+B.^{(2)}(x))^{1-s}-(B.^{(2)}(x))^{1-s}}{1-s}

\displaystyle = \frac{1}{1-s} \; [(1+1-(1-B.^{(2)}(x)))^{1-s}-(1-(1-B.^{(2)}(x))^{1-s}]

\displaystyle =\frac{1}{1-s} [(2+ B.^{(2)}(x-1))^{1-s}-(1+B.^{(2)}(x-1))^{1-s}]

\displaystyle = \frac{1}{1-s} \; \sum_{n \ge 0} \binom{1-s}{n} [2^{1-s-n}-1] B_n^{(2)}(x-1)

\displaystyle = \sum_{n \ge 0} \binom{-s}{n} \; \frac{2^{1-s-n}-1}{1-s-n} \; B_n^{(2)}(x-1)= \sum_{n \ge 0} \binom{-s}{n} \; \bar{B}_{-(s+n)}(1) \; B_n^{(2)}(x-1)

\displaystyle = \bar{B}.^{-s}(1) \; (1+\frac{B.^{(2)}(x-1)}{\bar{B}.(1))})^{-s} =(\bar{B}.(1)+B.^{(2)}(x-1))^{-s}

\displaystyle =(B.^{(2)}(\bar{B}.(x))^{-s}=(B.(B.(\bar{B}.(x))))^{-s}=(B.(x))^{-s}

The formalism is internally consistent, and the formulas easily generalize to

\displaystyle B_{-s}(x) = (\bar{B}.(y) + B.^{(2)}(x-y))^{-s}= \sum_{n \ge 0} \binom{-s}{n} \; \bar{B}_{-(s+n)}(y) \; B_n^{(2)}(x-y) .

Evaluation of these expressions at x=y=1 gives

\displaystyle B_{-s}(1)= s \: \zeta(s+1) = \sum_{n \ge 0} \binom{-s}{n} \frac{2^{1-s-n}-1}{1-s-n} \; B_n^{(2)}(0) = \sum_{n \ge 0} \binom{-s}{n} \bar{B}_{-(s+n)}(1) \; B_n^{(2)}(0) .

From the asymptotics of the generalized Bernoulli polynomials, these series appear to be divergent, yet give very good agreement with the Riemann zeta about the real line for s between -9 and .5 with the series truncated at n=10, leaving  enough terms to give polynomials interpolating between the simple zeros in that range. So, what we have arrived at is a method of umbral summation of divergent series that is tantamount to a Borel-Laplace-Mellin summation, and a simple truncated series for approximating the Riemann and Hurwitz zetas about the real line in a window about s=0.

Compare this with the ultimately divergent expansion

\displaystyle sH(s,x+y) = B_{-s}(x+y) = (B.(x+y))^{-s} = (B.(x)+y)^{-s} \sim \sum_{n \ge 0} \binom{-s}{n} B_n(x)\; y^{-s-n}\;

specialized to

\displaystyle s \; H(s,1) = s \; \zeta(s+1) \sim \sum_{n \ge 0} \binom{-s}{n} \; B_n(1-x) \; x^{-s-n} \; ,

which gives very good agreement with x = 1.5 for -10 \le s \le 3 when truncated at n = 10 . And, compare these with the polygamma function and its asymptotic expansion as given in Wikipedia:

\displaystyle (-1)^{m+1} \frac{\psi^{m}(z)}{(m-1)!}=m \; H(m,z) \sim \sum_{n \ge 0} \binom{n+m-1}{n} B_{n}(1) \; z^{-m-n} = \sum_{n \ge 0} \binom{-m}{n} B_n(0) \; z^{-m-n}.

These maneuvers, of course, suggest other expansions by generalizing the reciprocal polynomials in the same manner as the Bernoulli polynomials through umbral composition and forming similar binomial convolutions. These generalized reciprocal polynomials would be umbral inverses of the corresponding generalized Bernoulli polynomials, giving

\displaystyle B_{-s}(x) = (\bar{B}.^{(p)}(y) + B.^{(p+1)}(x-y))^{-s} .

From above, the Riemann zeta can be expressed as

\displaystyle \zeta(s) = -\frac{B_{1-s}(1)}{1-s} = - \bar{B}_{-s}(0) \; B_{1-s}(1) ,

and the Dirichlet eta function as

\displaystyle \eta(s) = \sum_{n \ge 1} \frac{(-1)^{n+1}}{n^s} = (1-2^{1-s}) \; \zeta(s) = \bar{B}_{-s}(1) \; B_{1-s}(1) .

Let’s look more closely at the generalized Bernoulli polynomials of order two. From the defining relation above (caution here–can’t let x=0, must remain greater than zero, as in the Mellin transforms, but we may let x=1 and substitute the relation B_n(1)=(-1)^n B_n(0) once the final umbral evaluation is made; i.e. the convolved umbrae must be treated as independent of each other as discussed in the entry on the Hirzebruch criterion),

\displaystyle B_n^{(2)}(x) = B_n(B.(x)) = (B.(0)+B.(x))^n = \sum_{j=0}^{n} \binom{n}{j} B_{n-j}(0) \; B_{j}(x) ,

and the e.g.f. is

\displaystyle e^{B.^{(2)}(x)t} = e^{B.(B.(x))t}= \frac{t}{e^t-1} \; e^{B.(x)t}= (\frac{t}{e^t-1})^2 \; e^{xt} .

Using the Mellin transform to interpolate the coefficients of the e.g.f., we have

\displaystyle B^{(2)}_{-s}(x) = \int_{0}^{\infty} (\frac{t}{1-e^{-t}})^2 \; e^{-xt} \frac{t^{s-1}}{(s-1)!} \; dt

\displaystyle = \int_{0}^{\infty} \sum_{n \ge 0} (n+1) \; e^{-(n+x)t} \; s(s+1)\; \frac{t^{s+1}}{(s+1)!} \; dt = s\; (s+1) \sum_{n \ge 0} \frac{n+1}{(n+x)^{s+1}}  , so

B_{-s}^{(2)}(1) = s \; (s+1) \; \zeta(s+1) = (s+1) \; B_{-s}(1) .

Then

\; \; \; B_n^{(2)}(1) = (1-n) \; B_n(1) = (-1)^n (1-n) \; B_n(0) .

The Riemann zeta, the Dirichlet eta, and the Hurwitz zeta functions (and therefore the interpolated Bernoulli polynomials) are related by Witten  to the volumes of moduli spaces of certain Riemann surfaces in “On quantum guage theories in two dimensions”.

The above products also suggest looking at the bivariate function \bar{B}_{-s}(y) \; B_{1-s}(x)  .

For an Appell sequence, there are other simple, formally consistent operator relations equivalent to Newton interpolation, but eventually you must check for convergence in particular applications. For example,

A_{zD_z}(x+y) = (A.(x+y))^{zD_z} = [1-(1-A.(x+y))]^{zD_z} = \triangledown_n^{zDz} \triangledown_k^{n} A_k(x+y)

with

\displaystyle \triangledown^{s}_n \; c_n=\sum_{n=0}^{\infty}(-1)^n \binom{s}{n} \; c_n \; ,

which reduces to Newton interpolation for A_{-s}(x+y) when acting on z^{-s}. The Newton series and the associated Mellin transform may or may not be be convergent for particular values or ranges of s, but the Mellin transform can be regularized often in these cases as it can for the simplest e.g.f. e^{xt} for the prototypical Appell sequence x^n . The action of this operator on functions described by the inverse Mellin transform could also be used to define umbral composition or substitution of the Appell sequence into the argument of these functions by action of the operator on the kernel of the inverse transform, containing x^{-s} .  The inverse Mellin transform might be used also to indicate how to regularize the integrand of the Mellin transform and therefore possibly the Newton interpolator to extend them to other ranges, again in analogy with the gamma function, say, for extending the associated Euler integral to Real(s)<0 . And, of course, you could also invoke Hankel contours (or even Barnes integrals), as Riemann did for his zeta function.

Related stuff:

“On quantum gauge theories in two dimensions” by Witten

“Bernoulli polynomials old and new: Problems in complex analysis and asymptotics” by N. Temme

“Large degree asymptotics of generalized Bernoulli and Euler polynomials” by Lopez and Temme

“A generalization  of the Bernoulli polynomials of order one” by Kimball

“Reduction and duality of the Hurwitz-Lerch zetas” by Bayad and Chikhi

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