A FRESH APPROACH TO CLASSICAL EISENSTEIN SERIES AND THE NEWER HILBERT–EISENSTEIN SERIES

# A Fresh Approach to Classical Eisenstein Series and the Newer Hilbert–eisenstein Series

PAUL L. BUTZER Lehrstuhl A für Mathematik, RWTH Aachen, D–52056 Aachen, Germany
TIBOR K. POGÁNY John von Neumann Faculty of Informatics, Óbuda University, 1034–Budapest, Hungary
Faculty of Maritime Studies, University of Rijeka, HR–51000 Rijeka, Croatia
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###### Abstract

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.

Bernoulli numbers; Conjugate Bernoulli numbers; Butzer–Flocke–Hauss complete Omega function; Digamma function; Dirichlet Eta function; Eisenstein series; Exponential generating functions; Hilbert transform; Hilbert–Eisenstein series; Riemann Zeta function
\catchline{history}

Dedicated to the memory of Godfrey Harold Hardy, a Discoverer and Mentor of Srinivasa Aiyangar Ramanujan

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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

 πcot(πz)=1z+∑k∈N(1z+k+1z−k),

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 [16] 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 [37], [38] and Iwaniec [25])

 εr(z):=∑k∈Z1(z+k)r, (1)

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

 ε1(z)=∑ek∈Z1z+k:=limN→∞∑|k|≤N1z+k=1z+∑′k∈Z(1z+k−1k).

One observes that (Euler), and by differentiation

 ε2(z)=π2sin2(πz),ε3(z)=π3cot(πz)sin2(πz); (2)

this results in the intriguing relation [34, p. 299]

 ε3(z)=ε1(z)⋅ε2(z).

There immediately arises the question: ”do there exist further such that is valid?” Our answer is the following result.

{theorem}

The unique solution in of the equation

 εr+2(z)=εr+1(z)⋅εr(z),(z∈C∖Z) (3)

is .

{proof}

Obviously has to be odd. Indeed, setting in (1), we can write in terms of the Dirichlet’s Lambda–function

 λ(r)=∑k∈N01(2k+1)r,(r>1)

in the form

 εr(12)=2r(1+(−1)r)λ(r).

But by this result the initial equation (3) makes sense only for odd, since for even , the relation (3) becomes a contradiction

 22ℓ+3λ(2ℓ+2)=0⋅22ℓ+1λ(2ℓ)=0.

On the other hand, bearing in mind the essential differentiability property [37, pp. 6–13], [34, p. 299], namely

 εr(z)=(−1)r−1Γ(r)ε(r−1)1(z),(r∈N), (4)

and accordingly

 ε′′r(z)=r(r+1)εr+2(z),

we deduce from (3) the nonlinear second order ODE:

 y′′+(r+1)y′y=0,(y=εr(z)). (5)

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

 y=√2c1r+1tanh⎡⎣√c1(r+1)2(z+c2)⎤⎦,

where stand for integration constants. The function is –periodic, so

 √c1(r+1)2=iπ,

accordingly

 εr(z)=−2πr+1tanπ(z+c2). (6)

Now, we have

 ε′r(z) =−2π2r+1⋅1cos2π(z+c2) =−2π2r+1[tan2π(z+c2)+1] =−r+12ε2r(z)−2π2r+1,

which coincides exactly for with the Riccati–type differential identity [34, p. 268, Eq. (1)]

 ε′1=−ε21−π2.

Also, we observe that (6) becomes the Eisenstein series for .

The cotangent form of and the examples (2) are best expressed and extended when one recalls the beautiful reflection formula for the more-practical digamma–function , namely [1, p. 259, Eq. 6.3.7]

 ε1(z)=πcot(πz)=ψ(1−z)−ψ(z), (7)

for which the –th derivative [1, p. 260, Eq. 6.4.10], the so-called polygamma function reads

 (8)

Special attention is given to the case , that is

 ψ(z):=ψ0(z)=ψ(0)(z)=∑k∈N(1k−1z+k−1)−γ, (9)

where signifies the Euler–Mascheroni constant.

A first new, but simple result in this respect reads, noting (4),

{proposition}

For all we have

 εr(z)=1Γ(r)(ψr−1(1−z)+(−1)rψr−1(z))

As to the proof, it follows immediately from (7) and (4).

Our first more important result is a new integral representation of .

{theorem}

There holds the integral representation

 εr(z)=(z−[R(z)])−r+1Γ(r)∫∞0tr−1et−1(e−(z−[R(z)])t+(−1)re(z−[R(z)])t)dt,

for all and for all . Here stands for the integer part of  .

{proof}

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

we conclude

 εr(z) =∑k∈Z1(z+k)r=1zr+∑k∈N(1(z+k)r+(−1)r(k−z)r) =1zr+1Γ(r)∫∞0tr−1(∑k∈Ne−kt)(e−zt+(−1)rezt)dt =z−r+1Γ(r)∫∞0tr−11−e−t(e−(z+1)t+(−1)re−(1−z)t)dt.

The integral converges for , when , as the integrand’s behavior is controlled near to the origin and at infinity. The rest is clear.

{corollary}

For all and , we have

 εr(z)=1(z−[R(z)])r+2Γ(r)∫∞0tr−1et−1{cosh(z−[R(z)])t−sinh(z−[R(z)])t}dt,{revenrodd.

## 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

 ∑n∈N0Bn(x)znn!=zezxex−1(z∈C,|z|<2π,x∈R). (10)

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

 B∼n(x):=H1[Bn(⋅)](x),(n∈N).

Here is the (periodic) Hilbert transform of the –periodic function defined by

 H1[φ](x)=PV∫12−12φ(x−u)cot(πu)du,

so that

 B∼n(x)=PV∫12−12Bn(x−u)cot(πu)du,

with for all , since . Written as a Fourier series, they then are to be [10]

 B∼2n+1(x)=−2(2n+1)!∑k∈Nsin(2πkx−(2n+1)π2)(2πk)2n+1,(n∈N0). (11)

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

 B∼2m+1(12)=(2−2m−1)⋅B∼2m+1(1),B∼2m(12)=0.

Some values of the conjugate Bernoulli numbers are (see [10, p. 69])

 (12)

and

 B∼1(x)=−1πlog(2sin(πx)).

Of basic importance is also the exponential generating function of , given for by

 ∑k∈N0B∼k(12)zkk!=−zez2ez−1Ω(z)=−z2sinhz2Ω(z), (13)

first established by M. Hauss [23, p. 91–95], [24] (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 [12] in the form

It is the Hilbert transform at zero of the –periodic function , defined by the periodic extension of the exponential function , , thus

 H1[e−zx](0)=PV∫12−12ezucot(πu)du=Ω(z).

As to the Omega function, we further need its basic partial fraction development for , namely

 Ω(2πz)=1π(e−πz−eπz)∑k∈N(−1)kkz2+k2=−isinh(πz)π∑ek∈Z(−1)ksgnkz+ik, (14)

the proof of which depends upon a new Hilbert–Poisson formula, introduced by Hauss; see [23, pp. 97–103] or [24].

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)])

 ∑k∈Nζ(2k+1)z2k=−12[ψ(1+z)+ψ(1−z)]+γ,(|z|<1) (15)

or, replacing , then

 ∑k∈N(−1)k−1ζ(2k+1)z2k=12[ψ(1+iz)+ψ(1−iz)]+γ,(|z|<1). (16)

A second formula in this respect reads [1, 6.3.17], [22, Eq. (54.3.5)]

 (17)

## 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 [23].

{definition}

The Hilbert–Eisenstein series are defined for and by

 hr(z):=∑k∈Z(−1)ksgn(k)(z+ik)r=∑k∈N(−1)k(1(z+ik)r−1(z−ik)r),

and, for recalling (14), by

 h1(z):=∑ek∈Z(−1)ksgn(k)z+ik=iπΩ(2πz)sinhπz=iΩ(2πz)∑k∈Z(−1)kz+ik,

with , noting .

In this respect recall that

The basic properties of for and , are

 h′r(z) =−rhr+1(z) h(s)r(z) =(−1)s(r)shr+s(z), (18)

as well as their difference and symmetry property [23, Eq. (6.5.72)], [10, Eq. (9.7)]

 hr(z)+hr(z+i)=z−r−(z+i)−r;hr(−z)=(−1)r+1hr(z),(z∈C∖iZ).

Above

 (ρ)σ:=Γ(ρ+σ)Γ(ρ)={1,(σ=0;ρ∈C∖{0})ρ(ρ+1)⋯(ρ+σ−1)(σ∈N;ρ∈C);

stands for the Pochhammer symbol (or shifted, rising factorial). Note that it being understood conventionally that .

{proposition}

For , one has

 ∑ek∈Z(−1)ksgn(k)z+ik=2i∑n∈N0(−1)nη(2n+1)z2n. (19)

Moreover

 B∼2n+1(12) =(−1)n+1(2n+1)!2−2nπ−2n−1η(2n+1) (20) =(−1)n(2n+1)!(4−2n−2−2n)π−2n−1ζ(2n+1), (21)

where stands for the Dirichlet’s Eta function.

{proof}

We have

 limN→∞ ∑|k|≤N(−1)ksgn(k)z+ik =limN→∞N∑k=1(−1)k−1{1z−ik−1z+ik} =2i∑k∈N(−1)k−1kz2+k2=2i∑k∈N(−1)k−1k∑n∈N0(−1)n(zk)2n =2ilimN→∞N∑k=1(−1)k−1k+2i∑n∈N(−1)n{∑k∈N(−1)k−1k2n+1}z2n =2i∑n∈N0(−1)nη(2n+1)z2n,

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)

 12z∑n∈N0B∼n(12)(2πz)nn!=πΩ(2πz)e−πz−eπz.

Comparing coefficients with (19), gives and results in (20). As to (21), it follows from (20) by noting that , .

Alternatively, (21) follows from (11), by setting , which yields

 B∼2n+1(12)=2(2n+1)!∑k∈Nsin((k+12)π)(2πk)2n+1=(−1)n2(2n+1)!(2π)2n+1η(2n+1).

This finishes the proof of proposition.

Now, the generating function of can be expressed in terms of the digamma function. In fact,

{theorem}

For with there holds

 ∑k∈N0B∼k(12)zkk! (22)
{proof}

Substituting formula (21) for into the series below, and observing (15), we have

 ∑k∈N0B∼k(12)zkk! =−log2π⋅z+∑k∈NB∼k(12)zkk! =−log2π⋅z+4∑k∈N(−1)k(z4π)2k+1ζ(2k+1) −2∑k∈N(−1)k(z2π)2k+1ζ(2k+1) =−log2π⋅z+4{z8π[−2γ−ψ(1−iz4π)−ψ(1+iz4π)]}

This establishes the representation in (3) for complex . That for real follows from (17).

Observe that the proof of Theorem 3.3 has the same outward appearance as that of Theorem 7.3. in [10, p. 74], but it uses the correct formula (12), provided with two proofs in Proposition 3.2.

Now, the can also be represented in terms of the classical digamma function.

{theorem}

There holds

First proof. According to (19) of Proposition 3.2 and Definition 3.1, we have, noting , ,

 h1(z) =2i∑k∈N0(−1)kη(2k+1)z2k =2i∑k∈N0(−1)k(1−2−2k)ζ(2k+1)z2k =2i∑k∈N0((iz)2k−(iz2)2k)ζ(2k+1) =2i{limh→0+((iz)2h−(iz2)2h)ζ(2h+1) +∑k∈N((iz)2k−(iz2)2k)ζ(2k+1)}.

We now express the sum via the linear combination of digamma functions, recalling (16), that means

 h1(z) =2i{limh→0+((iz)2h−(iz2)2h)ζ(2h+1) +12[ψ(1+iz2)+ψ(1−iz2)]−12[ψ(1+iz)+ψ(1−iz)]}.

On the other hand

 limh→0+((iz)2h−(iz2)2h)ζ(2h+1) =limh→0+((iz)2h−(iz2)2h)(12h+γ+o(h)) =log(iz)−log(iz2)=log2.

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

 Rψ(1+iz2π)=−γ+2∫∞0e−usin2(zu2π)sinh(u)du.

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

 ∑k∈N0B∼k(12)(2πz)kk!=−πzsinh(πz)Ω(2πz),(z∈C∖iZ), (23)

and, on the other hand

 h1(z)=iπsinh(πz)Ω(2πz). (24)

Thus, following the argument along the lines of the proof of Theorem 3.3,

 −zih1(z) =∑k∈N0B∼k(12)(2πz)kk! =−2zlog2+2∑k∈N(−1)k(2πz4π)2k+1ζ(2k+1) −∑k∈N(−1)k(2πz2π)2k+1ζ(2k+1) =−2zlog2+2⋅z4{2ψ(1)−ψ(1+iz2)−ψ(1−iz2)} −z2{2ψ(1)−ψ(1+iz)−ψ(1−iz)}.

Therefore

 h1(z)=2ilog2+i{ψ(1+iz2)+ψ(1−iz2)−ψ(1+iz)−ψ(1−iz)}.

This completes the proof of the first formula of Theorem 3.4. The second one follows immediately by the mirror symmetry formula .

It is important to mention that this representation of is not given in [10], but contained implicitly in a more complicated form in the proof of Proposition 6.4.1. in [24].

{corollary}

The Omega function has the representation

for .

The proof is immediate from Theorem 3.3 in view of

 Ω(z)=−iπsinh(z2)⋅h1(z2π),

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

 ζ(2m+1)=(−1)m22mπ2m−1B∼2m+1(0)(2m+1)!,(m∈N),

as well as the delicate formulae (16) and (17) given in tables by Hansen [22] and Abramowitz–Stegun [1]. 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

 Sr(x) =∑k∈N2k(k2+x2)r,(r>1) (25) ˜Sr(x) =∑k∈N(−1)k−12k(k2+x2)r,(r>0); (26)

see among others [15], [31], [32] 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.

{theorem}

For all there holds

 (27)

Moreover, for the same –domain we have for

 hr(z) =irΓ(r){2−r+1ψr−1(1−iz2)+(−2)−r+1ψr−1(1+iz2) −ψr−1(1−iz)−(−1)r−1ψr−1(1+iz)}. (28)
{proof}

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

 ∑k∈Z(−1)k−1ak=∑koddak−∑kevenak=∑k∈Zak−2∑k∈Za2k,

we have for all ,

 h2(z) =−∑k∈N0(1(z+ik)2−1(z−ik)2)+2∑k∈N0(1(z+2ik)2−1(z−2ik)2) =−∑k∈N0(1(k+iz)2−1(k−iz)2)+12∑k∈N0(1(k+iz2)2−1(k−iz2)2) =12[ψ1(1+iz2)−ψ1(1−iz2)]−ψ1(1+iz)+ψ1(1−iz); (29)

actually, we employ here the trigamma function , which normally converges in .

Term–wise integration then implies

 ∫z0h2(t)dt=h1(0)−h1(z)=2ilog2−h1(z).

Integrating (4) directly on too, we obtain

 ∫z0h2(t)dt=−i[ψ(1+iz2)+ψ(1−iz2)−ψ(1+iz)−ψ(1−iz)],

which completes the proof of the desired closed form representation (27).

In view of the differentiation property (3) of the HE series (see also [4, p. 796, Eq. (41)]),

 hr(z)=(−1)rΓ(r)h(r−2)2(z), (30)

where and , applied to (4), we obtain

 hr(z) =irΓ(r){2−r+1ψr−1(1−iz2)+(−2)−r+1ψr−1(1+ix2) −ψr−1(1−ix)−(−1)r−1ψr−1(1+ix)},

which completes the proof of the theorem.

The restriction of (4) to