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# FeZnSb2: A possible NiAs-type hexagonal superconductor

Phinifolo Cambalame1,2
1 Centre for Physics of the University of Coimbra
2 Department of Physics  Eduardo Mondlane University
<EMAIL_ADDRESS>

###### Abstract

We present a first-principles investigation of electronic structure, lattice
dynamics, and electron-phonon coupling of NiAs-type structure FeZnSb2 and the
isostructural parent compound FeSb within the framework of density functional
theory. The calculation on partial disordered system FeZnSb2 was performed in
a fixed configuration. This hypothetical ordered structure is predicted to be
superconducting.

###### keywords:

NiAs-type structure, superconductivity, disordered partial systems

††terms: NiAs-type structure, partial disordered systems, superconductivity in
Iron-based compounds

##  1 Introduction

The recent discovery of superconductivity in the NiAs-type structure high-
entropy compound M1−xPtxSb (M = equimolar Ru, Rh, Pd and Ir) [8] triggered our
interest in the exploration of isostructural systems. Concurrently, the
discovery of Fe-based pnictide and chalcogenide superconductors with high
superconducting transition temperature ($T_{c}$) [10] has led to enormous
excitement and opening of new perspective to develop a promising new platform
to realize high-$T_{c}$ superconductors. Among these, FeZnSb2 stands out - an
unexplored Fe-based system that has a NiAs-type structure with disordered
atomic positions of Fe and Zn (2a Wyckoff position) while Sb atoms occupy the
2c position according to [2] who first reported this system, derived from FeSb
compound. To our knowledge, superconductivity has not been reported in an Fe-
based compound with hexagonal structure, though the simple binary system FeSe,
can crystallize in the superconducting phase $\beta$-FeSe (tetragonal) and
$\delta$-FeSe (hexagonal) non-superconducting phase [25, 16, 13]. The
occurrence of superconductivity within disordered NiAs-type structures [8] or
enhancement of $T_{c}$ through disorder in other crystal systems [12] and the
potential for ferroelectricity within distorted NiAs-type structures [24]
render this material a confluence of features of great interest for both
theoretical investigations and experimental studies. Disordered systems and
partial ordered systems can also exhibit magnetic frustrated phases [17],
spin-glass [23] due to complex exchange interactions between the magnetic
atoms introduced by the atomic disorder [20]. Partial ordered system generally
have at least one ordered atomic position and remaining position randomly
occupied. Fixed atomic configurations are relatively straightforward to handle
and can yield data for comparison with experiments. Experimental parameters
such as phonon vibrations and electronic structure can be calculated to infer
the degree of disorder present in this system. Some systems with 50 $\%$
chemical scattering centers have already been observed to display
superconductivity almost equal to pure elements [14].

In our current investigation, we present an ab initio prognosis regarding the
superconductivity and stability of FeZnSb2, under the assumption of fixed
atomic positions for Zn and Fe. Our study entails the structural optimization
of these compounds, juxtaposing the electronic structure and lattice dynamics
of FeZnSb2 with its parent compound, FeSb.

##  2 Computational methods

The DFT calculations were carried out on QUANTUM-ESPRESSO version 7.2 [5, 4,
6] using SSSP PBE Precision v1.2.0 pseudopotentials [18], except when
mentioned otherwise. The crystal visualization was carried on Xcrysden
software [11]. The electronic band structure calculation were performed on
relaxed structure, sampling the Brillouin zone (BZ) with 16 $\times$ 16
$\times$ 12 k-points in the irreducible Brillouin zone (IBZ) with a 0.02 Ry
smearing for BZ integration. The kinetic energy cutoff for wavefunctions was
chosen to be 60 Ry and 420 Ry for charge density and potential for electronic
structure and lattice dynamics calculations. As we are interested in
superconductivity in the framework of DFT, other correlations effects beyond
the DFT are not taken into account. Subsequently, to calculate the phonon
dispersion curves and electron–phonon coupling we have employed density
functional perturbation theory (DFPT), also within QUANTUM-ESPRESSO, using 20
$\times$ 20 $\times$ 16 electronic k-point grids and 5 $\times$ 5 $\times$ 4
phononic q-point grids.

##  3 Results

###  3.1 Mechanical and Lattice Parameters

As mentioned earlier, FeZnSb2 is a partially disordered system with NiAs-type
crystal structure, while the Sb atoms occupy exclusively the As- crystal site
in NiAs, the Ni (2a) site is now occupied by Zn and Fe. The translational
asymmetric unit cell representation gives two equivalent atomic site, that
were atribute randomly to Zn and Fe atoms, resulting in the fixed or ordered
configuration as depicted in Fig. 1, with the iron unit cell occupying the
corner of the unit cell , and the zinc atoms in the intermediate edge position
between the Fe atoms. The inverse occupation (Zn and Fe interchanged) gives
the same Fermi surface and phonon dispersions and electron-phonon coupling
strength, as could be expected. In FeSb, we have a similar picture, with the
obvious difference being that Zn is now replaced by Fe atoms.

(a)

(b)

Figure 1: FeZnSb2. (a) View to xz-plane and (b) xy-plane view. Table 1: Equilibrium crystal lattice parameters, a and b, Bulk modulus B and total energy Emin. | a | b | B (GPa) | Emin (Ry) |
---|---|---|---|---|---
FeSb | 3.99 | 4.97 | 140.4 | -1028.4026 |
Exp. [22] | 4.1 | 5.14 | – | — |
FeZnSb2 | 4.19 | 5.17 | 82.4 | -1160.70637 |
Exp. [2] | 4.38 | 5.73 | – | – |

The optimization of the crystal structure from the experimental lattice
parameters[22, 2] provides good agreement between experimental and relaxed
structure for FeSb but not in the FeZnSb2 case. There’s 10$\%$ disagreement of
the c lattice parameter, while for both FeSb lattice parameters this
underestimation falls around $3\%$ (see Table LABEL:tab1). To understand this
uncommon underestimation of cell volume (and the c lattice constant, in
particularly) we performed a series of calculations with FeZnSb2 supercells of
2 $\times$ 2 $\times$ 2 dimensions [21], where the theoretical sites of Zn and
Fe were gradually interchanged, the results are plotted at Fig. 2. Our
calculations suggest that the increment of disorder in system, will shrink the
unit cell volume. This result rules out disorder in the Zn and Fe positions as
the primary cause for the abnormal underestimation of the lattice constant,
thereby suggesting the potential involvement of other interactions, such as
magnetism, in this discrepancy. We advance that the diffusion of Fe (or Zn) to
the double tetrahedral interstitial positions within the primitive unit cell
[19] constitutes the primary factor. This phenomenon has been previously
documented in FeSb, NiSb, and related structures [7]. Afterwards, we have
performed magnetic calculations with different magnetic states and have come
to the conclusion that the ground state of FeSb is ferromagnetic(FM), while
for FeZnSb2, the FM state and collinear antiferromagnetic(AFM) state are
separated by a mere 2 mRy (calculation on 2 $\times$ 1 $\times$ 1 supercell).
While the FM state predominates as the lowest-energy configuration, this small
difference suggest a more complex ground state and more detailed calculations
is needed to elucidate further.

Figure 2: Variation of ordered FeZnSb2 with the Zn/Fe iter-site substitution

###  3.2 Electronic Structure

The calculated electronic band structure are shown in Fig. 3. There are
relatively more bands crossing the Fermi level in FeSb (Fig. 3(b)) than in
FeZnSb2 (Fig. 3(a)). There are multiple flat bands above the Fermi level along
the $\Gamma$-M-K-$\Gamma$ in FeSb, and in FeZnSb2 two degenerate flat bands
along the L-H-K high-symmetry path are observed. The projected band structure
calculations for FeZnSb2 suggest a splitting into two sets of doubly
degenerate states: one of the $d_{xz}$ and $d_{yz}$ character and another of
the $d_{{x^{2}}-{y^{2}}}$ and $d_{xy}$ character, along with a non-degenerate
state derived from the $d_{z^{2}}$ orbital. Unlike the FeZnSb2 degeneracy
picture, where we have a complete degeneracy along the displayed high-symmetry
path (Fig. 3(a)), the set of $d_{xz}$ and $d_{yz}$ in FeSb have their
degeneracy lifted along the A-L-H path, the set of $d_{{x^{2}}-{y^{2}}}$ and
$d_{xy}$ are partially degenerate only close to the Fermi level and
significantly along M-K-$\Gamma$-A and lifted elsewhere. In FeZnSb2, there’s a
significant hybridization between Sb ($p_{x}$ and $p_{y}$) with Fe
($d_{{x^{2}}-{y^{2}}}$). Hybridization is marginally in FeSb.

(a)

(b)

Figure 3: Electronic structure of (a) FeZnSb2 and (b) FeSb.

(a)

(b)

Figure 4: Total and partial density of state (DOS) (a) for FeSb and (b)
FeZnSb2.

The density of states at the Fermi level of these two materials exhibits a
metallic behaviour (see Fig. 4). The contribution of Sb to the density of
states is higher in FeZnSb2 than in FeSb. This can be explained by the
stronger covalent binding between Sb and Fe in the former compound. The
contribution of Zn to the total DOS of FeZnSb is marginal. In both FeSb and
FeZnSb2 the highest DOS is observed along the L-H-K near the Fermi Level.

####  3.2.1 Fermi surfaces

At the $\Gamma$ point, we observe the primary topological difference in the
Fermi surface (FS) see Fig. 6 and 6. In FeSb, we identify two anisotropic and
co-centered 2-D hole pocket, whereas in FeZnSb2 a cylinder-like structure
encloses a complex dumbbell-like formation. In these two systems, we observe
along the boundaries a typical metallic open FS extending over almost the
entire BZ.

(a)

(b)

Figure 5: Fermi surface of FeZnSb2. (a) view to xz-plane, (b) xy-plane view.

(c)

(d)

Figure 6: Fermi surface FeSb. (a) view to xz-plane, (b) xy-plane view.

###  3.3 Phonons and electron-phonon coupling

The calculated phonon dispersion curves are shown in Fig.7. The absence of any
imaginary or negative vibrational modes suggests that both FeZnSb2 and FeSb
are dynamically stable systems within a hexagonal structure of NiAs-type. The
origin of phonon branches is similar; the highest frequency branches result
from the contribution of Fe and Zn in FeZnSb2 and Fe in FeSb. The three (two)
atomic species are involved in the acoustic vibration modes of FeZnSb2 (FeSb).
This pattern of vibration explains the depression in the phonon DOS when
transiting to high frequencies. Raman and infrared (IR) were calculated at
$\Gamma$. The calculation on the FeZnSb2 (FeSb) relaxed structure assigns the
crystallographic symmetry to D3d (C2v), which decomposes into the vibrational
modes in ($\Gamma$) to the following irreducible representations:

$\Gamma=3A_{2u}+3E_{u}+E_{g}+A_{1g}$ (1)

(a)

(b)

Figure 7: Phonon dispersion curves along high symmetry curves and the density of states (DOS) (a) for FeZnSb2 and (b) FeSb . Table 2: The calculated frequencies of the Raman (R) and infrared (I) active modes of FeSb and FeZnSb2. The modes are classified by the irreducible representations (irreps) according to which they transform. FeSb |  |  | FeZnSb2 |  |
---|---|---|---|---|---
$\omega$ cm-1 | (irrep) |  |  $\omega$ cm-1 | (irrep) |
8.1 | ($B_{1}$) | I+R | 11.3 | ($A_{2u}$) | I
152.1 | ($A_{1},B_{2}$) | I+R | 17.7 | ($E_{u}$) | I
164.8 | ($A_{2}$) | R | 88.2 | ($A_{2u}$) | I
170.5 | ($B_{1}$) | I+R | 108.1 | ($E_{u}$) | I
205.9 | ($B_{2}$) | I+R | 125.3 | ($E_{g}$) | R
213.3 | ($A_{2}$) | R | 143.1 | ($A_{1g}$) | R
213.7 | ($B_{1}$) | I+R | 209.3 | ($E_{u}$) | I
236.8 | ($A_{1}$) | I+R | 212.7 | ($A_{2u}$) | I

The calculated Raman and infrared active modes can be validated through
experimentation, enabling inference about the disorder of the system and its
crystallographic symmetry, particularly in the case of FeZnSb2. The $E_{g}$
Raman mode exhibits degeneracy. Both Raman modes are attributed exclusively to
Sb vibrations in FeZnSb2, whereas in FeSb, one Raman mode (164.8 cm-1)
originates from Sb and 213.13 cm-1 from Fe. The presence of flat bands in this
material may suggest low thermal conductivity and potential thermoelectric
capacity [3].

####  3.3.1 Electron-phonon coupling and superconductivity

The frequency dependence of the Eliashberg spectral function
$\alpha^{2}F(\omega)$ and the average $\lambda$ value for FeSb and FeZnSb2 are
presented in Table LABEL:tab3 and Fig. 8. We have also calculated the relevant
parameters for the description of the superconducting properties within the
Eliashberg formalism, namely the logarithmically averaged phonon frequency
$\omega_{log}$ and the superconducting transition temperature $T_{c}$,
calculated by the McMillan [15] formula modified by Allen-Dynes [1] Eq. 2,
presented in the table LABEL:tab3.

$T_{c}=\frac{\omega_{log}}{1.2}exp\left(-\frac{1.04(1+\lambda)}{\lambda-\mu^{*}(1+0.62\lambda)}\right)$
(2)

Table 3: The calculated DOS at EF, N(0), the average electron–phonon coupling constant $\lambda$, the logarithmically averaged $\omega_{log}$ phonon frequencies and Tc calculated from modified McMillan formula eqn. 2 | N(0) | $\lambda$ | $\omega_{log}[K]$ | T${}_{c}[K]$ ($\mu=0.1$) |
---|---|---|---|---|---
FeSb | 39.471550 | 0.33 | 191.886 | 0.217 |
FeZnSb2 | 21.530679 | 0.59 | 139.067 | 3 |

(a)

(b)

Figure 8:  Electron-phonon spectral function $\alpha^{2}F(\omega)$ and
cumulative electron-phonon coupling strength $\lambda$ for (a) FeSb and (b)
FeZnSb2

To validate the accuracy of our findings, we conducted a fresh computation
employing the projector augmented-wave (PAW) pseudopotentials [9] version
2.0.1, yielding closely aligned results. Specifically, we observed an average
Tc of 3.7 K and an electron-phonon coupling strength of 0.66.

###  3.4 Discussion and conclusions

In this study, we demonstrate the stability of synthesized compounds FeSb and
FeZnSb2 with the NiAs structure. The simulated disorder in FeZnSb2 reveals
that the lattice parameter will shrink with disorder, and the substituion of
Fe by Zn increases the relevance of Sb in the DOS next to the Fermi level. If
the phonon mediated pairing mechanism for superconductivity is assumed for
these systems, superconducting transition temperature is expected to be
significant in the unexplored system FeZnSb2. While disorder within the system
may negatively impact superconductivity, the potential occurrence of
superconductivity in this Fe-based compound offers a novel platform for
material manipulation and comparison with unconventional superconductors,
particularly Fe-based layered compounds. The findings of this theoretical
study serve as strong motivation for the synthesis and experimental
exploration of this compound, as well as for employing more sophisticated
methodologies in theoretical calculations. This includes the introduction of
supercells and disorder to better align with the realistic characteristics of
the FeZnSb2 system.

###### Acknowledgements.

This work was produced with the support of CFISUC computing resources, cluster
Adamastor.

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# A Family of New Formulas for the Euler-Mascheroni Constant

Noah Ripke

###### Abstract

We introduce and prove several new formulas for the Euler-Mascheroni Constant.
This is done through the introduction of the defined E-Harmonic function,
whose properties, in this paper, lead to two novel formulas, alongside a
family of formulas. While the paper does introduce many new approximations, it
does not exhaust the possibilities of the E-Harmonic function but provides a
strong first dive into its natural conclusions. We hope that the diversity of
new formulas may provide stepping stones to a proof (or disproof) of the
irrationality of the Euler-Mascheroni constant.

##  1 Introduction

The Euler-Mascheroni constant is the limiting difference between the harmonic
series and the natural logarithm [Eul40] proving a fascinating connection
between the discrete and continuous.

$\lim_{n\rightarrow\infty}\sum_{i=1}^{n}{\frac{1}{i}}-\ln{n}=\gamma$

But the constant’s claim to fame is that a proof (or disproof) of its
irrationality has eluded Mathematicians for centuries. Several formulas have
been discovered in the hope of finding a path towards irrationality, and
faster convergence. This paper will introduce two new concise and effective
formulas that may be used towards a theory of irrationality, and the methods
used to discover these formulas can be replicated to create an infinite family
of formulas (Most of which are slightly more convoluted). The star of this
paper is the E-Harmonic function $E(n)$ whose interesting properties will
hopefully pave a path towards many new discoveries involving the Euler-
Mascheroni constant.

##  2 The E-Harmonic Function

The limiting connection between the Harmonic Series and the natural logarithm
leads us to the intuitive analysis of what I call the E-Harmonic function:

$E(n)=e^{H_{n}}$

Note that $H_{n}$ is simply $\sum_{i=1}^{n}{\frac{1}{i}}$. The best way to
understand the power of the E-Harmonic Function is to see its graph:

$1$$2$$3$$4$$5$$2$$4$$6$$8$$n$$E(n)$

The linear-looking aspect of this graph should be somewhat expected due to the
Euler-Mascheroni Constant, but the rate at which the E-Harmonic function
approaches linear is what makes the formulas derived from it converge so
quickly. When first looking at this graph, the next intuitive step is to ask
what the limiting difference between each successive term is: and it happens
to be $e^{\gamma}$!

######  Theorem 2.1 (E-Harmonic Linearization Theorem).

An integral theorem to all E-Harmonic derived formulas for the Euler-
Mascheroni constant

$\lim_{n\rightarrow\infty}E(n)=\lim_{n\rightarrow\infty}e^{\gamma}n$

###### Proof.

A simple proof comes from the original definition for the Euler-Mascheroni
constant:

$\lim_{n\rightarrow\infty}H_{n}-\ln{n}=\gamma$

By raising each side to the power of $e$, we receive this true statement:

$\lim_{n\rightarrow\infty}\frac{E(n)}{n}=e^{\gamma}$

Trivial application of limit theory will obtain the desired result. ∎

The power of the E-Harmonic Linearization Theorem is our ability to now
substitute the limiting behavior of $E(n)$ with a simplistic function
involving $\gamma$. This will prove to be endlessly fruitful in the following
sections.

##  3 Two Novel Formulas

The following two sub-sections will show the two main results of this paper.
These formulas, derived from $E(n)$, are both interesting in their own right
and are generally the most simplistic of all the other formulas described in
this paper (And thus will be discussed separately).

###  3.1 $E(n+1)-E(n)$

In this subsection, we will prove and discuss the following formula:

$\large\lim_{n\rightarrow\infty}H_{n}+\ln{(e^{\frac{1}{n+1}}-1)}=\gamma$ (1)

###### Proof.

Using Theorem 2.1, we start with the following true statement:

$\lim_{n\rightarrow\infty}E(n+1)-E(n)=e^{\gamma}$

By rewriting $E(n)$ as $e^{H_{n}}$, and factoring out a common $E(n)$:

$\lim_{n\rightarrow\infty}e^{H_{n}}(e^{\frac{1}{n+1}}-1)=e^{\gamma}$

Finally, when we take the natural log of both sides, we obtain the desired
result. ∎

This formula is interesting in a few aspects. For one, its similarity to the
original definition of the Euler-Mascheroni constant allows us to notice that
the functions $f(x)=e^{\frac{1}{x+1}}-1$, and $g(x)=\frac{1}{x}$ behave very
similarly as $x$ approaches infinity. This might be obvious from an early look
at the Taylor Series expansion for $f(x)$, but it leads us to an interesting
observation.

In a philosophical manner, it seems the original definition of the Euler-
Mascheroni constant is a simple byproduct (and approximation) of formula
3.1.1. While formula 3.1.1 is more complex than the original simplistic nature
of $\gamma$, it seems to be the ”father” of our primary definition. And in
fact, this new formula converges much faster! To use in actual computations,
one can create an $n$-degree Taylor Series for $e^{\frac{1}{x+1}}-1$, and
choose some large $n$; the larger the $n$, the faster the convergence. Of
course, this approximation would also involve a second Taylor Series for
$\ln{(x)}$ for small $x$ approaching 0.

Hopefully, this formula may also be used as a device towards a theory of
irrationality of $\gamma$. For one, this formula separates itself from other
equations in that it involves $e$, and not some value shooting off towards
infinity, but a function of $e$ that converges to 0.

###  3.2 $E(n+1)*E(n)$

In this subsection, we will prove and discuss the following formula:

$\large\lim_{n\rightarrow\infty}H_{n}+\frac{1}{2(n+1)}-\frac{1}{2}\ln{(n^{2}+n)}=\gamma$
(2)

###### Proof.

In a similar manner to the proof in 3.1, we start by rewriting $E(n+1)E(n)$
using Theorem 2.1:

$\lim_{n\rightarrow\infty}E(n+1)E(n)=\lim_{n\rightarrow\infty}e^{2\gamma}(n^{2}+n)$

Since $H_{n+1}=H_{n}+\frac{1}{n+1}$, we can take the natural log of both sides
to obtain this statement:

$\lim_{n\rightarrow\infty}2H_{n}+\frac{1}{n+1}-\ln{(n^{2}+n)}=2\gamma$

The equation is perfectly fine in this format, but by dividing by two, we can
reach the desired result. ∎

While this formula seems interesting, it actually converges just as fast as
the original definition for $\gamma$. This may be disappointing until an
observation is made. We seem to have a ”leftover” term of $\frac{1}{2(n+1)}$,
which converges to 0 of course as $n$ approaches infinity. It happens to be
that removing this term allows for significantly faster convergence.

$n$ | With ”Leftover” Term | Without ”Leftover” Term
---|---|---
1 | 0.90342640972 | 0.65342640972
10 | 0.624182616527 | 0.578728071072
100 | 0.582182661274 | 0.577232166225
1,000 | 0.577715331901 | 0.577215831402
10,000 | 0.577265661568 | 0.577215666568

This idea doesn’t come from nowhere. If you were to combine our original
formula for $\gamma$ and a first-degree approximation of the formula derived
in 3.1, you would achieve the same equation! This equation will be expanded
upon in the next section which aims to discuss these formulas in a more broad,
and generalized way.

##  4 A Family of Formulas

We generalize the formulas found in the previous sections and notice
interesting results that can be derived from them.

###  4.1 $E(n+k)-E(n)$

This is a clear continuation of the idea brought up in 3.1, and gives rise to
the following equation:

$\large\lim_{n\rightarrow\infty}H_{n}+\ln(e^{\sum_{i=1}^{k}\frac{1}{n+i}}-1)-\ln(k)=\gamma$
(3)

###### Proof.

Clearly, by applying Theorem 2.1:

$\lim_{n\rightarrow\infty}E(n+k)-E(n)=ke^{\gamma}$

Thus, by factoring the left side into:

$\lim_{n\rightarrow\infty}E(n)(e^{\frac{1}{n+1}+...+\frac{1}{n+k}}-1)=ke^{\gamma}$

We see that our desired formula arises easily by taking the natural log of
both sides. We will call this equation:

$L(n,k)=H_{n}+\ln(e^{\sum_{i=1}^{k}\frac{1}{n+i}}-1)-\ln(k)$

∎

Now, clearly from construction, as $n$ limits to $\infty$ we get $\gamma$, but
is the same true for constant $n$ and limiting $k$? We verify:

$\lim_{k\rightarrow\infty}L(n,k)=\gamma$

###### Proof.

By working backwards from the definition of $L(n,k)$, we let:

$\lim_{k\rightarrow\infty}H_{n}+\ln(e^{\sum_{i=1}^{k}\frac{1}{n+i}}-1)-\ln(k)=L$

Thus by raising both sides to the $e$:

$\lim_{k\rightarrow\infty}\frac{E(n+k)-E(n)}{k}=e^{L}$

Since $E(n+k)$ limits to infinity, we can apply Theorem 2.1 to see that:

$\lim_{k\rightarrow\infty}\frac{e^{\gamma}(n+k)-E(n)}{k}=e^{L}$

Then after basic re-arranging we find that:

$e^{L}=e^{\gamma}-\lim_{k\rightarrow\infty}\frac{e^{\gamma}n-E(n)}{k}$

Since our lasting limit clearly goes to 0 since the numerator is a constant,
we find that $L=\gamma$. ∎

Now from the proof, it’s evident that for larger $n$, our approximation can
become more accurate since $|e^{\gamma}n-E(n)|$ is decreasing. But one can
conveniently choose $n=0$ and out comes a completely new formula:

$\large\lim_{k\rightarrow\infty}\ln(E(k)-1)-\ln(k)=\gamma$ (4)

The formula is amazingly close to the original definition of $\gamma$, and
yet, can get away with being much more accurate.

###  4.2 $E(n)*E(n+1)*...*E(n+k)$

There are multiple natural continuations of the product formula, but this one
I find to be most interesting.

$\large\lim_{n\rightarrow\infty}H_{n+k}-\frac{1}{k+1}\sum_{i=1}^{k}{\frac{i}{n+i}}-\frac{1}{k+1}\ln(\frac{(n+k)!}{(n-1)!})=\gamma$
(5)

###### Proof.

We start out with our product and note that using Theorem 2.1:

$\lim_{n\rightarrow\infty}E(n)*...*E(n+k)=\lim_{n\rightarrow\infty}e^{(k+1)\gamma}n(n+1)...(n+k)$

Then by taking the natural log of both sides, we see that after grouping the
$(k+1)H_{n}$ and the leftover fractional terms into a sum, we get:

$\lim_{n\rightarrow\infty}H_{n}+\frac{1}{k+1}\sum_{i=1}^{k}{\frac{k-i+1}{n+i}}-\frac{1}{k+1}\ln(\prod_{i=0}^{k}(n+i))=\gamma$

Then finally, turning our product into a factorial, and re-arranging our sum,
we get the desired result. ∎

For fixed $k$, we can see that our approximation will approach $\gamma$, but
as was discussed in the previous subsection, does increasing $k$ also limit to
$\gamma$? Calling our new equation $P(n,k)$, we verify:

$\large\lim_{k\rightarrow\infty}P(n,k)=\gamma$

###### Proof.

We start with our limit, and let it equal $L$, from there we will prove that
$L=\gamma$

$\lim_{k\rightarrow\infty}H_{n+k}-\frac{1}{k+1}\sum_{i=1}^{k}{\frac{i}{n+i}}-\frac{1}{k+1}\ln(\frac{(n+k)!}{(n-1)!})=L$

Rewrite the equation as such:

$\lim_{k\rightarrow\infty}H_{n+k}-\frac{k}{k+1}+\frac{1}{k+1}\sum_{i=1}^{k}{\frac{n}{n+i}}-\frac{1}{k+1}\ln((n+k)!)+\frac{1}{k+1}\ln((n-1)!)=L$

Since $(n-1)!$ is constant, and our sum will clearly go to 0, we get:

$\lim_{k\rightarrow\infty}H_{n+k}-1-\frac{1}{k+1}\ln((n+k)!)=L$

Now, we use Stirling’s Approximation for $\ln(n!)\approx n\ln(n)-n$. This is
rigorous since $(n+k)$ approaches infinity, so as long as we maintain our
limit, our statement still holds.

$\lim_{k\rightarrow\infty}H_{n+k}-1-\frac{1}{k+1}((n+k)\ln(n+k)-(n+k))=L$

Since $\frac{n+k}{k+1}$ approaches 1, we get:

$\lim_{k\rightarrow\infty}H_{n+k}-\ln(n+k)=L$

But this is simply the limiting difference between the Harmonic Series and the
natural log, which is the original definition of $\gamma$! Thus $L=\gamma$. ∎

Now that we have this powerful fact, we can use $n=1$ as our convenient value,
and quickly derive this new formula:

$\large\lim_{k\rightarrow\infty}H_{k+1}+\frac{H_{k+1}}{k+1}-1-\frac{1}{k+1}\ln((k+1)!)=\gamma$

But in this case, it may be more easily understood to replace $k+1$ with
simply $k$ to get:

$\large\lim_{k\rightarrow\infty}H_{k}+\frac{H_{k}}{k}-1-\frac{1}{k}\ln(k!)=\gamma$
(6)

Now, one can apply Stirling’s approximation to our lasting factorial term, but
the resulting formula is one that is not too interesting, nor helpful for
approximations.

As can be seen, the formulas shown in this paper just scratch the surface of
the possibilities for new formulas. Abusing properties of linearity and
products are just natural first steps, but further research can be done on new
approaches to the E-Harmonic function that may provide even more insightful
results. Such approaches may include unique sums, further study of $L(n,k)$
and $P(n,k)$, etc.

## References

  * [Eul40] L. Euler  “De progressionibus harmonicis observationes”  In _Commentarii academiae scientiarum Petropolitanae_ 7, 1740, pp. 150–161