A Fresh Approach to Classical Eisenstein Series and the Newer Hilbert–eisenstein Series
This paper is concerned with new results for the circular Eisenstein series as well as with a novel approach to Hilbert-Eisenstein series , introduced by Michael Hauss in 1995. The latter turn out to be the product of the hyperbolic sinh–function with an explicit closed form linear combination of digamma functions. The results, which include differentiability properties and integral representations, are established by independent and different argumentations. Highlights are new results on the Butzer–Flocke–Hauss Omega function, one basis for the study of Hilbert-Eisenstein series, which have been the subject of several recent papers.
Dedicated to the memory of Godfrey Harold Hardy, a Discoverer and Mentor of Srinivasa Aiyangar Ramanujan
Mathematics Subject Classification 2010: 11B68, 11M36, 33B15, 33E20, 40C10
1 Eisenstein series
In order to introduce his method for constructing elliptic functions, Ferdinand Gotthold Max Eisenstein (1823–1852) first considered the simpler case of trigonometric functions, specifically the series
originally discovered by Leonhard Euler in 1748, presented in [18, §178] 111It is worth mentioning that it is regarded by Konrad Knopp [27, p. 207] as the ”most remarkable expansion in partial fractions”. Also, J. Elstrodt  nominated this partial fraction expansion for the most interesting formula involving elementary functions, see also [2, p. 149].. Eisenstein introduced the series (later to be famously known as Eisenstein series, see e.g. Weil ,  and Iwaniec )
which are defined for and all , they being a normally convergent series of meromorphic functions in . Since these Eisenstein series of order do not exist for , one defines aesthetically
One observes that (Euler), and by differentiation
this results in the intriguing relation [34, p. 299]
There immediately arises the question: ”do there exist further such that is valid?” Our answer is the following result.
The unique solution in of the equation
Obviously has to be odd. Indeed, setting in (1), we can write in terms of the Dirichlet’s Lambda–function
in the form
we deduce from (3) the nonlinear second order ODE:
Moreover, as the Eisenstein series is 1–periodic in the sense that for all , we are looking for a 1–periodic particular solution of the ordinary differential equation (5). It is
where stand for integration constants. The function is –periodic, so
Now, we have
which coincides exactly for with the Riccati–type differential identity [34, p. 268, Eq. (1)]
Also, we observe that (6) becomes the Eisenstein series for .
for which the –th derivative [1, p. 260, Eq. 6.4.10], the so-called polygamma function reads
Special attention is given to the case , that is
where signifies the Euler–Mascheroni constant.
A first new, but simple result in this respect reads, noting (4),
For all we have
Our first more important result is a new integral representation of .
There holds the integral representation
for all and for all . Here stands for the integer part of .
By the -periodicity of Eisenstein’s functions , it is sufficient to consider it inside the vertical strip of the complex plane. Indeed, otherwise, assuming , by the relation , we have the same property. Therefore, letting , by the Gamma–function formula
The integral converges for , when , as the integrand’s behavior is controlled near to the origin and at infinity. The rest is clear.
For all and , we have
2 Backgrounds to Hilbert–Eisenstein series
A basis to the Hilbert–Eisenstein series includes the classical Bernoulli numbers , defined in terms of the Bernoulli polynomials , defined, for example, via their exponential generating function
We need some facts concerned with . Starting with the –periodic Bernoulli polynomials defined as the periodic extension of , , we need to introduce the –periodic conjugate functions , ( if ) by
Here is the (periodic) Hilbert transform of the –periodic function defined by
with for all , since . Written as a Fourier series, they then are to be 
These conjugate periodic functions are used to define the non–periodic functions , which can be regarded as conjugate Bernoulli ”polynomials” in a form such that their properties are similar to those of the classical Bernoulli polynomials . For details see Butzer and Hauss [11, p. 22] and Butzer [10, pp. 37-56]. The conjugate Bernoulli numbers needed, the , are the for which
Some values of the conjugate Bernoulli numbers are (see [10, p. 69])
Of basic importance is also the exponential generating function of , given for by
first established by M. Hauss [23, p. 91–95],  (see also [11, pp. 21–29.] and [10, pp. 37–38, 78–80]). Above, is the so–called Butzer–Flocke-Hauss (complete) Omega function introduced in  in the form
It is the Hilbert transform at zero of the –periodic function , defined by the periodic extension of the exponential function , , thus
As to the Omega function, we further need its basic partial fraction development for , namely
A useful formula which will link Hilbert–Eisenstein series, Hilbert transform versions of the Bernoulli numbers and the Riemann Zeta function is given by (see [17, Eq. 1.17(7)] and [22, Eq. (54.10.3)])
or, replacing , then
3 Hilbert–Eisenstein series
Now, we come to the main sections of this article, dealing with Hilbert–Eisenstein series. A Hilbert (conjugate function) – type version of the Eisenstein series was first studied by Michael Hauss in his doctoral thesis .
The Hilbert–Eisenstein series are defined for and by
and, for recalling (14), by
with , noting .
In this respect recall that
The basic properties of for and , are
stands for the Pochhammer symbol (or shifted, rising factorial). Note that it being understood conventionally that .
For , one has
where stands for the Dirichlet’s Eta function.
the interchange of the summation order being possible on account of the Weierstraß double series theorem (see e.g. [27, p. 428]). This proves part a).
As to part b), on account of (13)
This finishes the proof of proposition.
Now, the generating function of can be expressed in terms of the digamma function. In fact,
For with there holds
Now, the can also be represented in terms of the classical digamma function.
First proof. According to (19) of Proposition 3.2 and Definition 3.1, we have, noting , ,
We now express the sum via the linear combination of digamma functions, recalling (16), that means
On the other hand
Now, obvious steps lead to the assertion of Theorem 3.4.
Observe that the real parts of Theorem 3.4 can also be expressed as integrals, noting
Although Theorem 3.4 is to be found in [10, Eq. (7.8)], the above proof is a new approach to Hilbert–Eisenstein series.
Second proof. According to (13), we have on the one hand
and, on the other hand
Thus, following the argument along the lines of the proof of Theorem 3.3,
This completes the proof of the first formula of Theorem 3.4. The second one follows immediately by the mirror symmetry formula .
The Omega function has the representation
The proof is immediate from Theorem 3.3 in view of
so it is omitted. However, we remark that this corollary could also be derived, just as simply, via Theorem 3.4.
Although, as observed in [10, p. 67], the Omega function is not an ”elementary function”, it is nevertheless expressible in terms of the hyperbolic sine function multiplied by a (simple) linear combination of digamma functions.
4 A novel alternative approach to Theorem 3.4
The representation of in terms of certain combinations of –functions with the constant (Theorem 3.4), was established via the generating function of the conjugate Bernoulli numbers , equation (13), plus arguments used in the proof of Theorem 3.3, which in turn were based fundamentally upon the representation of in terms of , thus
as well as the delicate formulae (16) and (17) given in tables by Hansen  and Abramowitz–Stegun . But the last four formulae are only to be found in tables. Thus one does not know of the possible difficulties of their proofs.
A further aim of this article is to present alternative proofs which are fully independent of these two formulae (16) and (17). Moreover, since two proofs of Theorem 3.4 presented are based on power series expansions in the open unit disc , we can extend the validity range to the whole by our present approach.
There exists a vast literature concerning Mathieu series and the more recent alternating Mathieu series , both of which are defined by
see among others , ,  and the references therein. These articles are of interest, in particular since the series is connected to HE series . We now come to a new third proof of Theorem 3.4, at the same time establishing the representation not only for but also for higher order on the extended range.
For all there holds
Moreover, for the same –domain we have for
To perform the proof of the representation formula (27) we have to connect the Hilbert–Eisenstein series , which is to be understood in the sense of Eisenstein summation, with the normally convergent HE series (for the latter see Remmert [34, p. 290]), which is termwise integrable (see [33, p. 42]).
From Definition 3.1 and the series form (9) of the digamma function, using the straightforward representation of the alternating series, say
we have for all ,
actually, we employ here the trigamma function , which normally converges in .
Term–wise integration then implies
Integrating (4) directly on too, we obtain
which completes the proof of the desired closed form representation (27).
where and , applied to (4), we obtain
which completes the proof of the theorem.
The restriction of (4) to