# Pairing interaction in superconducting UCoGe tunable by magnetic field

K. Ishida<EMAIL_ADDRESS>S. Matsuzaki  M. Manago  T. Hattori
S. Kitagawa  Department of Physics, Graduate School of Science, Kyoto
University, Kyoto 606-8502, Japan  M. Hirata  T. Sasaki  Institute for
Materials Research, Tohoku University. Sendai 980-8577, Japan  D. Aoki
Institute for Materials Research, Tohoku University, Oarai, Ibaraki 311-1313,
Japan  Université Grenoble Alpes, CEA, IRIG, PHELIQS, F-38000 Grenoble, France

###### Abstract

The mechanism of unconventional superconductivity, such as high-temperature-
cuprate, Fe-based, and heavy-fermion superconductors, has been studied as a
central issue in condensed-matter physics. Spin fluctuations, instead of
phonons, are considered to be responsible for the formation of Cooper pairs,
and many efforts have been made to confirm this mechanism experimentally.
Although a qualitative consensus seems to have been obtained, experimental
confirmation has not yet been achieved. This is owing to a lack of the
quantitative comparison between theory and experiments. Here, we show a semi-
quantitative comparison between the superconducting-transition temperature
($T_{\rm SC}$) and spin fluctuations derived from the nuclear magnetic
resonance (NMR) experiment on the ferromagnetic (FM) superconductor UCoGe, in
which the FM fluctuations and superconductivity are tunable by external
fields. The enhancement and abrupt suppression of $T_{\rm SC}$ by applied
fields, as well as the pressure variation of $T_{\rm SC}$ around the FM
criticality are well understood by the change in the FM fluctuations on the
basis of the single-band spin-triplet theoretical formalism. The present
comparisons strongly support the theoretical formalism of spin-fluctuation-
mediated superconductivity, particularly in UCoGe.

###### pacs:

74

##  I Introduction

U-based ferromagnetic (FM) superconductors[1, 2] and the recently discovered
UTe2[3] similar to them have attracted much attention and have been studied
intensively. This is because superconductivity becomes stronger under external
fields[4], which is not explained by spin-singlet pairing, and spin-triplet
superconductivity is expected. The identification of the spin-triplet
superconducting (SC) state and unveiling of the SC pairing mechanism of
unconventional superconductivity, which have been considered to be other than
the electron-phonon interaction, are central issues in condensed-matter
physics. From these viewpoints, URhGe[5] and UCoGe[6] are located in special
situations, not only in ferromagnetic (FM) superconductors, but also in all
unconventional superconductors. This is because the superconductivity is
reentrant[7] or is enhanced under a high magnetic field parallel to the $b$
axis ($H\parallel b$)[8], and is extremely sensitive to the applied magnetic
field. When the high field is tilted to only a couple of degrees to the $c$
axis, the reentrant or enhancement of superconductivity disappears abruptly[7,
9]. This strongly suggests that the SC pairing interaction would be field
dependent; therefore, this feature is very useful for studying the SC pairing
interaction, because it is precisely controllable.

We studied the FM superconductor UCoGe, discovered by Huy et al. in 2007[6],
using nuclear magnetic resonance (NMR) and nuclear quadrupole resonance (NQR)
measurements from a microscopic point of view[10, 11]. In UCoGe, FM transition
occurs at the Curie temperature $T_{\rm Curie}\simeq 3$ K with a small ordered
moment $m_{0}\simeq 0.05\mu_{B}$ pointing to the $c$ axis, and
superconductivity sets in at $T_{\rm SC}\simeq 0.6$ K. From the detailed angle
dependence of the nuclear spin-lattice relaxation rate ($1/T_{1}$) and AC
magnetic-susceptibility measurements below 3 T, we found that magnetic fields
along the $c$ axis ($H\parallel c$) strongly suppress the FM fluctuations with
the Ising anisotropy along the $c$ axis and that superconductivity is observed
in the limited magnetic-field region where the Ising-type FM fluctuations are
active[12]. These experimental results suggest that the Ising-type FM
fluctuations tuned by $H\parallel c$ induce spin-triplet superconductivity. In
addition, we measured $1/T_{1}$ down to 2 K under the field along the $a$ and
$b$ axes up to $\sim 11$ T, and reported that $T_{\rm Curie}$ is suppressed
and $1/T_{1}$ at 2 K is enhanced in $H\parallel b$, although $T_{\rm Curie}$
and $1/T_{1}$ are unchanged in $H\parallel a$[13]. Therefore, we suggest that
the enhancement of the superconductivity observed in $H\parallel b$ originates
from the FM critical fluctuations induced by $H\parallel b$.

In the sister superconductor URhGe, the reentrant superconductivity (RSC) was
observed at approximately 13 T in $H\parallel b$, where the magnetization
along the $b$ axis shows a superlinear increase (metamagnetic transition)[7].
Tokunaga et al. investigated the magnetic fluctuations around the RSC region
from the measurement of $1/T_{1}$ and nuclear spin-spin relaxation rate
$1/T_{2}$, and showed that FM critical fluctuations develop in the same
limited field region where RSC is observed. In particular, by comparing the
divergence in $1/T_{1}$ and $1/T_{2}$, they revealed that the longitudinal
fluctuations along the applied $H$, which are detectable by $1/T_{2}$
measurements, are much more dominant than the transverse fluctuations
detectable by $1/T_{1}$ and become critical in the RSC field region in
URhGe[14]. Therefore, it is crucial to measure $1/T_{2}$ in UCoGe to identify
the dominant magnetic fluctuations in the $H$ region, where the
superconductivity is enhanced by $H\parallel b$. In addition, it is also
important to determine the direction of the critical FM fluctuations in the
high magnetic field region around and above the FM criticality.

In this paper, we report the $1/T_{1}$ and $1/T_{2}$ results obtained from the
59Co NMR measurements in the magnetic field $H$ up to 23 T and with the
temperature $T$ down to 1.5 K, as well as the upper critical field $H_{\rm
c2}$ of the same single-crystal sample with the AC susceptibility measurements
under a magnetic field exactly parallel to the $a$ or $b$ axis. We found that
$1/T_{1}$ and $1/T_{2}$ measured at 1.5 K show a clear peak at approximately
12.5 $T$, where $T_{\rm Curie}$ is nearly suppressed to zero in $H\parallel
b$, whereas $1/T_{1}$, $1/T_{2}$, and $T_{\rm Curie}$ are unchanged in
$H\parallel a$. From the analyses of the field dependence of $1/T_{1}$ and
$1/T_{2}$, it is concluded that the Ising-type FM fluctuations along the $c$
axis become critical at around 12.5 T, in contrast to the case of URhGe. By
using the field dependence of the critical behavior of $1/T_{1}T$, we compare
the field dependence of the difference of $T_{\rm SC}$ along the $a$ and $b$
axes with a difference of $1/T_{1}(H)T$ at 1.5 K on the basis of the
theoretical model. It was discovered that the characteristic feature of the
$H_{\rm c2}$ behaviors in UCoGe can be semi-quantitatively explained by the
field dependence of $1/T_{1}T$, following the theoretical formalism with
approximation. A similar clear relation between $T_{\rm SC}$ and $1/T_{1}T$
variation was observed in $H\parallel c$ as well as in the pressure-induced FM
critical region. These results provide strong evidence that the
superconductivity in UCoGe is induced by critical FM fluctuations.

##  II Experimental Methods

###  II.1 Samples

The present single-crystal sample was prepared by the Czochralski crystal
pulling method in a tetra-arc furnace under high-purity argon, and was cut
into cube-like shapes with dimensions $a\times b\times c=1.20\times 1.14\times
1.20$ mm3. The electrical-resistivity measurement along the $b$ axis revealed
a residual resistivity ratio RRR $\sim 98$ and $T_{\rm Curie}=2.7$ K. The
onset, midpoint, and zero-resistivity temperatures of the SC transition were
0.87 K, 0.74 K, and 0.64 K, respectively. Clear specific-heat jumps at $T_{\rm
Curie}$ and $T_{\rm SC}$ and a large RRR attest to the high quality of the
sample.

###  II.2 Measurement of AC susceptibility for the determination of $T_{\rm
SC}(H)$

We measured the AC susceptibility to determine $T_{\rm SC}(H)$ under a
magnetic field ($H$). As reported previously[9], $T_{\rm SC}(H)$ is extremely
sensitive to the orientation of a single crystal against the applied $H$. To
align the crystal accurately and to measure the AC magnetic susceptibility
down to 100 mK, the NMR coil with the crystal was mounted on a piezoelectric
rotator (ANRv51/ULT/RES+) with an angular resolution of $\sim 0.1^{\circ}$,
and all were immersed into the 3He-4He mixture to reduce the radio-frequency
heating. As we can rotate over one axis only, we rotated the $ca(bc)$ plane
for the $a(b)$ axis alignment to minimize the $c$-axis component.

###  II.3 NMR measurements

As for the NMR measurement, the conventional spin-echo method was used. The
59Co-NMR measurement down to 1.5 K was carried out up to 23 T. We used a 25 T
cryogen-free SC magnet at the Institute for Materials Research at Tohoku
University for NMR measurements above 15 T. $1/T_{1}$ was measured by applying
the saturation $\pi/2$ pulse at time $t$ prior to the $\pi/2-\tau-\pi$ spin-
echo sequence and by monitoring the recovery $R(t)$ of the spin-echo intensity
as a function of $t$, and was evaluated by fitting $R(t)$ to the theoretical
function[15]. $1/T_{2}$ was measured by monitoring the decay of the spin-echo
intensity $I(\tau)$ as a function of the interval time $\tau$ between the
$\pi/2$ and $\pi$ pulses (Hahn method), and was evaluated by fitting $I(\tau)$
to the exponential function. Satisfactory fitting by a single component of
$T_{1}$($T_{2}$) was obtained in $R(t)$ ($I(\tau)$) in the entire $T$ and $H$
measurement region, indicating that magnetic fluctuations are homogeneous
across the entire region of the sample. $1/T_{1}$ and $1/T_{2}$ are measured
at the central transition peak of the 59 Co NMR spectrum in $H\parallel$ $a$
and $b$. The orientation of the external magnetic fields along the $a(b)$ axis
in the normal state was carefully controlled in situ using a laboratory-built
single-axis rotator by rotating the sample in the $ca(bc)$ plane. As reported
previously[13], $1/T_{1}$ just below $T_{\rm Curie}$, which is governed by the
FM critical fluctuation, is strongly suppressed when $H$ has a small $c$-axis
component; therefore, we measured the angle dependence of $1/T_{1}$ in the
$bc$ plane at 2.1 K and determined the $b$ axis from the peak of $1/T_{1}$, as
shown in Fig. 1.

Figure 1: Angle dependence of $1/T_{1}$ of 59Co in the $bc$ plane measured at
2.1 K for the determination of the $b$ axis. The peak angle was determined as
the $b$-axis.

##  III EXPERIMENTAL RESULTS

###  III.1 AC susceptibility measurements in magnetic fields

Figure 2: The AC susceptibility measured at various fields in (a) $H\parallel
a$ and (b) $H\parallel b$. The inset of (a) shows the angle dependence of the
decrease in $\chi_{ac}$ below $T_{\rm SC}$ measured at 3 T and 0.12 K. The
angle with the largest decrease was determined as the $a$-axis. The inset of
(b) shows the angle dependence of $T_{\rm SC}$ measured at 12.5 T. The angle
with the highest $T_{\rm SC}$ was determined as the $b$ axis.  Figure 3: Upper
critical field $H_{\rm c2}$ along each crystal axis. $H_{\rm c2}$ was
determined using the AC susceptibility measurements. The inset shows the zoom
of $H_{c2}$ along the $c$ axis.

Figures 2(a) and (b) show the results of the AC magnetic-susceptibility
measurements at various fields in (a) $H\parallel a$ and (b) $H\parallel b$.
For the measurements, the $a$ axis was determined from the angle dependence of
the decrease in $\chi_{\rm AC}$ below $T_{\rm SC}$ measured at 3 T, and the
$b$ axis was determined from the angle dependence of $T_{\rm SC}$ measured at
12.5 T, as shown in the inset of each figure. Here, $T_{\rm SC}$ in each field
was determined from the onset of the decrease in $\chi_{\rm AC}$ (Meissner
signal) from the constant value. The experimental errors are determined from
the noise level of $\chi_{\rm AC}$. Figure 3 shows the upper SC critical
fields ($H_{\rm c2}$) in the present single-crystal sample of UCoGe determined
with the above AC susceptibility measurement. The zero points in the $H$ axis
with the experimental error of 0.09 K in Fig. 3 mean that the Meissner signal
was not observed down to 0.09 K in the $T$-scan measurement of $\chi_{\rm
AC}$. In addition, $H_{\rm c2}$ value at $T\sim 0.09$ K in the inset was
determined from the $H$-scan measurement of $\chi_{\rm AC}$.

The $\mu_{0}H_{\rm c2}$ shows a huge anisotropy: $\mu_{0}H_{\rm c2}$ along the
$b$ seems to be beyond 20 T, but $\mu_{0}H_{\rm c2}$ along the $c$ is only 0.6
T, although the transport properties in UCoGe is rather three dimensional as
reported previously[12]. The remarkable enhancement of $H_{\rm c2}$
characterized by the S-shaped $H_{\rm c2}$ behavior[9, 16] was not observed in
the present sample. However, the different behavior in $H_{\rm c2}$ might be
due to sample dependence and/or difference of the experimental methods. The
comparison of $H_{\rm c2}$ values and the magnitude of the Meissner signals
between $H\parallel a$ and $H\parallel b$ indicates that the superconductivity
in the present sample also becomes robust in $H\parallel b$ as reported in the
previous measurement[9, 16].

###  III.2 $1/T_{1}$ and $1/T_{2}$ in magnetic fields

Figure 4: Magnetic field ($H$) dependence of $1/T_{1}T$ (a) and $1/T_{2}T$ (b)
measured in $H\parallel a$ and $H\parallel b$. (c) The $H$ dependence of
$G_{b}(0)$ and $G_{c}(\omega_{0})$ derived from $1/T_{1}T$ and $1/T_{2}T$ (see
text).

Figures 4 (a) and (b) show the $H$ dependence of $1/T_{1}T$ and $1/T_{2}T$ at
1.5 K in $H\parallel a$ and $H\parallel b$ measured up to 23 T. As reported
previously[13], $1/T_{1}T$ shows anisotropic behavior between the $a$ and $b$
axes. It was clarified that both $1/T_{1}T$ and $1/T_{2}T$ in $H\parallel b$
show a maximum at around 12.5 T in a similar manner, although both are $H$
independent in $H\parallel a$. When $1/T_{2}$ is determined by the electronic
contribution, $1/T_{1}T$ and $1/T_{2}T$ measured in $H\parallel x$ are
expressed as[17],

$\displaystyle\left(\frac{1}{T_{1}T}\right)_{x}$ $\displaystyle=$
$\displaystyle\frac{\gamma_{n}^{2}k_{\rm
B}}{(\gamma_{e}\hbar)^{2}}\sum_{{\bm{q}}}\left[|A^{y}_{\rm
hf}|^{2}\frac{\chi^{\prime\prime}_{y}({\bm{q}},\omega_{0})}{\omega_{0}}+|A^{z}_{\rm
hf}|^{2}\frac{\chi^{\prime\prime}_{z}({\bm{q}},\omega_{0})}{\omega_{0}}\right]$
$\displaystyle=$ $\displaystyle G_{\perp}(\omega_{0}),$

and

$\displaystyle\left(\frac{1}{T_{2}T}\right)_{x}=\frac{\gamma_{n}^{2}k_{\rm
B}}{(\gamma_{e}\hbar)^{2}}\lim_{\omega\rightarrow
0}\sum_{{\bm{q}}}\left[|A^{x}_{\rm
hf}|^{2}\frac{\chi^{\prime\prime}_{x}({\bm{q}},\omega)}{\omega}\right]$
$\displaystyle+\left[I(I+1)-m(m+1)-1/2\right]\left(\frac{1}{T_{1}T}\right)_{x}$
$\displaystyle=$ $\displaystyle G_{\parallel}(0)+\alpha
G_{\perp}(\omega_{0}).$

Here, $\chi^{\prime\prime}_{i}({\bm{q}},\omega)$ ($i=x,y,$ and $z$) is the
dynamical spin susceptibility components along the $i$ axis, and $\omega_{0}$
is the NMR frequency. $G_{k}(\omega)=\int_{-\infty}^{\infty}\langle
h_{k}(t)h_{k}(0)\rangle\exp{(i\omega t)}dt$ ($k=\parallel$ and $\perp$) is the
spectral density of the fluctuating hyperfine field $h_{k}(t)$. It should be
noted that $1/T_{1}$ in $H\parallel x$ is determined by magnetic fluctuations
perpendicular to $H$, but $1/T_{2}T$ in $H\parallel x$ can detect nearly
static magnetic fluctuations parallel to $H$ in addition to an extra
contribution from $1/T_{1}T$ in $H$. The coefficient of the second term
$\alpha$ in the $1/T_{2}T$ is 15.5, for the central transition
($m_{z}=1/2\leftrightarrow 1/2$) of the Co NMR spectrum ($I$ =7/2). When
$H\parallel b$, $G_{\parallel}(0)=G_{b}(0)$ and $G_{\perp}(\omega_{0})\sim
G_{c}(\omega_{0})$, because
$\chi^{\prime\prime}_{c}\gg\chi^{\prime\prime}_{a}$ is shown from the axial
dependence of $1/T_{1}$ at low $T$[18]. Using these formulae, the $H$
dependence of $G_{c}(\omega_{0})$ and $G_{b}(0)$ normalized with each low-$H$
value is shown in Fig. 4 (c). The $H$ dependence of $G_{c}(\omega_{0})$
exhibits a clear maximum around FM criticality, although $G_{b}(0)$ shows a
small hump. It is obvious that the critical fluctuations originate from
$G_{c}(\omega_{0})$; the Ising FM magnetic fluctuations along the $c$ axis,
inherent to UCoGe, become critical for the application of $H\parallel b$. This
situation is different from the FM criticality in URhGe induced by
metamagnetism in $H\parallel b$, where the magnetic fluctuations along the $H$
direction become critical. The difference is related to the anisotropy and the
character of the magnetization in two compounds; the Ising anisotropy along
the $c$ axis is much stronger in UCoGe. This is shown from the absence of the
metamagnetic transition up to 40 T in UCoGe, and the metamagnetic transition
was observed at 12 T in URhGe[19]. The different magnetic properties between
UCoGe and URhGe becomes more transparent from the recent study on UCo1-xRhxGe,
although the 5$f$ electrons are in the itinerant regime in both compounds[20].

Figure 5: (a) Temperature dependence of $1/T_{1}T$ of 59Co measured at various
magnetic fields from $\mu_{0}H$ = 1.5 to 23 T in $H\parallel b$. The
phenomenological fitting of the experimental $1/T_{1}T$ is shown by the dotted
curves with $\theta$ = 2.5, 0, and -5 K. (b) $H$ dependence of $T_{\rm Curie}$
determined by the peak or maximum of $1/T_{1}T$ measured in $H\parallel a$ and
$H\parallel b$ in the present sample. The characteristic temperature
$\theta(H)$ corresponding to the magnetic anomaly in the present sample is
also plotted. The $T_{\rm Curie}$ determined by the bending of the electric
resistance is also plotted for comparison.

Next, we measured the temperature ($T$) dependence of $1/T_{1}T$ in various
magnetic fields along the $b$ axis to investigate the critical behavior of the
Ising FM fluctuations along the $c$ axis. Figure 5(a) shows the $T$ dependence
of $1/T_{1}T$ in various magnetic fields along the $b$ axis. $1/T_{1}T$ shows
a clear peak at 1.5 T and 3 T and a broad maximum at 10 and 12 T at $T_{\rm
Curie}$. In the $H$ region greater than 18.5 T, $1/T_{1}T$ becomes almost
constant below 3 K, and the constant values of $1/T_{1}T$ decrease with
increasing $H$. It is noteworthy that the $T$ dependence of $1/T_{1}T$ in the
entire $H$ region is well fitted by the following equation:

$\left(\frac{1}{T_{1}(H)T}\right)_{H\parallel b}=\frac{200}{T-\theta(H)}+15$

only by changing $\theta(H)$. Here, $\theta(H)$ is a parameter of how close
the system is to the FM criticality. The evaluated $\theta(H)$ from the $T$
dependence of $1/T_{1}(H)T$ is plotted in Fig. 5 (b), along with the Curie
temperature $T_{\rm Curie}$ determined by the peak of $1/T_{1}T$. It is noted
that the evaluated $\theta(H)$ systematically changes from positive to
negative values and becomes zero at approximately 14 T, where $T_{\rm Curie}$
is assumed to be zero. This shows the presence of the FM critical point at
approximately 14 T, and that the $1/T_{1}(H)T$ can trace the FM criticality
systematically. In fact, the $H$ dependence of $\theta$ can be well fitted
with the $H$ quadratic behavior, which is expected in the mean-field
theory[21]. In the self-consistent renormalization theory describing weak
ferromagnetism, $1/T_{1}T$ is proportional to the static susceptibility $\chi$
in a three-dimensional system[22]. Therefore, it is experimentally indicated
that $1/T_{1}T$ in $H\parallel b$ is a good measure of magnetic susceptibility
$\chi_{c}$ related to the FM criticality.

##  IV DISCUSSION

In this section, we present a semi-quantitative discussion of the relationship
between FM critical fluctuations and superconductivity. As mentioned above, we
have shown the relationship between the Ising-type FM fluctuations and the
superconductivity in UCoGe based on the 59Co NMR measurements. From a
theoretical perspective, a theoretical models of spin-triplet
superconductivity observed in U-based ferromagnets were discussed by Mineev
[23, 24, 25], and Tada and Fujimoto[12, 26, 27], Hattori and Tsunetsugu[28],
independently. Mineev discussed the triplet pairing triggered by the exchange
of magnetic fluctuations with a phenomenological fluctuation spectrum[23, 24,
25]. Tada and Fujimoto showed that the experimentally observed anomalous
$H_{c2}$ behaviors against $T$ and the angle in the $bc$ plane are well
reproduced by the theoretical model with appropriately taking into account the
critical FM fluctuations observed in experiments[12, 26, 27]. Hattori and
Tsunetsugu discussed the RSC in URhGe by strong attractive interactions
generated by soft magnons in the Ising systems with transverse fields[28, 29].
These models were developed to understand the unusual response of
superconductivity against the applied $H$ observed in U-based FM
superconductors.

When an orbital suppression of superconductivity is neglected and the largest
critical temperature corresponds to the pairing of quasiparticles with spin
up-up and spin down-down, Mineev expressed BCS-type $T_{\rm SC}$ formula in a
weak coupling interaction as [24, 25],

$T_{\rm SC}(H)=\varepsilon\exp{\left(-\frac{1}{g(H)}\right)},$

where $g(H)$ is the pairing interaction for the single-band model, in which
only spin up-up pairing is formed, and is considered to be a function of $H$,
and $\varepsilon$ is the energy cutoff of the pairing interaction. In
$H\parallel a(b)$, $g$ is expressed as $g\sim[\langle
N_{0}(\mbox{\boldmath$k$})\chi_{c}^{u}(H^{a(b)})\rangle\cos^{2}{\varphi}+\langle
N_{0}(\mbox{\boldmath$k$})\chi_{a(b)}^{u}(H^{a(b)})\rangle\sin^{2}{\varphi}]I^{2}$,
where $I$ is the coupling constant of electrons with spin fluctuations,
$\tan{\varphi}=H^{a(b)}/h$, and $h$ is the exchange field acting on the
electron spins. The angular brackets denote the average over the Fermi
surface, and $N_{0}(\mbox{\boldmath$k$})$ is the angular dependent density of
the electronic states on the Fermi surface. $\chi_{c}^{u}$ and
$\chi_{a(b)}^{u}$ are odd in the momentum part of the static susceptibilities
along the $c$ and $a$ ($b$) axes, respectively. In UCoGe near $T_{\rm SC}$,
the Ising anisotropy in the static susceptibility is so large that
$\chi_{a(b)}^{u}$ is negligibly smaller than $\chi_{c}^{u}$, and $\varphi\ll
1$ such that $\cos{\varphi}\approx 1$, since the exchange field $h$ is
considered to be $\sim 50$ T, much larger than $H^{a(b)}$, from the
metamagnetic behavior in $M/H$ in $H\parallel b$[19]. Thus, we approximately
obtain

$T_{\rm
SC}(H^{a(b)})\sim\varepsilon\exp{\left(-\frac{\alpha_{1}}{\chi_{c}(H^{a(b)})}\right)}.$

Here, $\alpha_{1}$ is a constant, and we assume that $\chi_{c}$ is
proportional to $1/T_{1}T$ from the $T$ and $H$ dependence of $1/T_{1}T$, as
shown above. If we take the difference between $T_{\rm SC}(H^{b})$ and $T_{\rm
SC}(H^{a})$, the orbital suppression can be canceled out in the lower $H$
region, and the superconductivity enhanced by the FM criticality can be mainly
extracted, although the cancelation of the orbital suppression is insufficient
in the higher $H$ region. Therefore, $T_{\rm SC}(H^{b})-T_{\rm SC}(H^{a})$
normalized by $T_{\rm SC}(0)$, which is defined as $\delta T_{\rm SC}(H^{b})$,
is extracted and can be expressed in the lowest order as

$\displaystyle\frac{T_{\rm SC}(H^{b})-T_{\rm SC}(H^{a})}{T_{\rm SC}(0)}$
$\displaystyle\sim$
$\displaystyle\frac{\exp{\left(-\frac{\alpha_{1}}{\chi_{c}(H^{b})}\right)}-\exp{\left(-\frac{\alpha_{1}}{\chi_{c}(H^{a})}\right)}}{\exp{\left(-\frac{\alpha_{1}}{\chi_{c}(0)}\right)}}$
$\displaystyle\propto\alpha_{1}\left[\frac{1}{\chi_{c}(0)}-\frac{1}{\chi_{c}(H^{b})}\right]$
$\displaystyle\propto$
$\displaystyle\frac{T_{1}(0)T-T_{1}(H^{b})T}{T_{1}(0)T},$

by taking $\chi_{c}(H^{a})\sim\chi_{c}(0)$ on the basis of $H^{a}$ independent
$1/T_{1}T$, and using the experimental relation of
$\chi_{c}(H^{b})\propto(T_{1}(H^{b})T)^{-1}$. Figure 6 shows the $H$
dependence of the $\delta T_{\rm SC}(H^{b})\equiv[T_{\rm SC}(H^{b})-T_{\rm
SC}(H^{a})]/T_{\rm SC}(0)$ in UCoGe reported so far for the left axis[8, 16]
and that of the normalized $T_{1}T$ difference, $\delta
T_{1}(H^{b})T\equiv[T_{1}(0)T-T_{1}(H^{b})T]/T_{1}(0)T$ at 1.5 K measured on
the present UCoGe [Fig. 4(a)] for the right axis. It is interesting that
$\delta T_{\rm SC}(H^{b})$ and $\delta T_{1}(H^{b})T$ show almost the same $H$
dependence, as suggested by the theoretical model[24, 25].

Figure 6: The $H$ dependence of the difference of $T_{\rm SC}$ between
$H\parallel b$ and $H\parallel a$ normalized by zero-field $T_{\rm SC}(0)$,
$[T_{\rm SC}(H^{b})-T_{\rm SC}(H^{a})]/T_{\rm SC}(0)\equiv\delta T_{\rm
SC}(H^{b})$ on the present sample and the $H$ dependence of the normalized
difference of $T_{\rm SC}$ evaluated from the previous reports[9, 16] are also
plotted for the left axis, and the vertical scale of the results reported by
Wu et al. is reduced to half for the comparison. The $H$ dependence of
$[T_{1}(H\sim 0)T-T_{1}(H^{b})T]/T_{1}(H\sim 0)T\equiv\delta T_{1}(H^{b})T$
measured at 1.5 K on the present sample is plotted on the right axis.

We apply this theoretical model to other two cases observed in UCoGe. One is
the steep decrease of $H_{\rm c2}$ in $H\parallel c$. It is obvious that such
a steep decrease cannot be explained only by the orbital suppression effect,
which was experimentally demonstrated by Wu et al. [16], and we have shown the
relationship between the suppression of the Ising fluctuations and the steep
decrease of $T_{\rm SC}$ in $H\parallel c$, as pointed out previously[12].
Following the above theoretical analysis, $T_{\rm SC}(H^{c})$ in $H\parallel
c$ is approximately expressed as[24, 25]:

$\displaystyle T_{\rm SC}(H^{c})$ $\displaystyle=$
$\displaystyle\varepsilon\exp{\left(-\frac{1}{\langle
N_{0}(\mbox{\boldmath$k$})\chi_{c}^{u}(H^{c})\rangle
I^{2}}\right)}\sim\varepsilon\exp{\left(-\frac{\alpha_{2}}{\chi_{c}(H^{c})}\right)}.$

Here, we assume that UCoGe is still close to the FM criticality, where the
magnetic coherence length $\xi_{m}$ is much larger than the lattice constant
($\xi k_{\rm F}\gg 1$). Using this relation and
$\chi_{c}(H^{c})\propto(T_{1}(H^{c})T)^{-1}$, the normalized decreasing rate
$T_{\rm SC}$ against $H^{c}$, defined as $\delta T_{\rm SC}(H^{c})$ is
expressed as

$\displaystyle\delta T_{\rm SC}(H^{c})$ $\displaystyle\equiv$
$\displaystyle\frac{T_{\rm SC}(H^{c})-T_{\rm SC}(0)}{T_{\rm SC}(0)}$
$\displaystyle\propto$
$\displaystyle\frac{T_{1}(0)T-T_{1}(H^{c})T}{T_{1}(0)T}\equiv\delta
T_{1}(H^{c})T.$

Figure 7 shows the $H^{c}$ dependence of $\delta T_{\rm SC}(H^{c})$ in UCoGe
reported so far for the left axis[12, 16] and that of $\delta T_{1}(H^{c})T$
at 1.5 K reported previously for the right axis[12]. The $H^{c}$ dependence of
$T_{1}T$ was measured in $H\parallel b$ by adding a small $H^{c}$. This
corresponds to the $T_{1}T$ measurements by the rotating $H$ from the $b$ to
$c$ axis, and the details are reported in [12].

Figure 7: The $H$ dependence of $[T_{\rm SC}(H^{c})-T_{\rm SC}(0)]/T_{\rm
SC}(0)\equiv\delta T_{\rm SC}(H^{c})$ on the present sample and its $H$
dependence evaluated from previous reports are also plotted for the left
axis[12, 16]. The $H$ dependence of $[T_{1}(H\sim
0)T-T_{1}(H^{c})T]/T_{1}(H\sim 0)T\equiv\delta T_{1}(H^{c})T$ measured at 1.5
K reported by our group is plotted on the right axis[12]. The $H^{c}$
dependence of $1/T_{1}T$ is evaluated from the angle dependence of $1/T_{1}T$
in the $bc$ plane.

The other case is the pressure ($P$) induced FM criticality. We studied the
relationship between $P$-induced FM criticality and superconductivity by 59Co-
NQR measurements in a zero magnetic field[30, 31]. The superconductivity is
also enhanced by the $P$-induced FM critical region, which has been reported
previously[32, 33, 34]. This suggests that the pairing interaction can be
tuned by pressure, as in the case of $H\parallel b$. We assume that $T_{\rm
SC}(P)$ in $H=0$ can be approximated as:

$T_{\rm
SC}(P)\sim\varepsilon\exp{\left(-\frac{\alpha_{3}}{\chi_{c}(P)}\right)}.$

as in the case of $H\parallel b$. Following the above analysis, the normalized
$T_{\rm SC}$ variation by $P$, defined as $\delta T_{\rm SC}(P)$ can be
expressed as

$\displaystyle\delta T_{\rm SC}(P)$ $\displaystyle\equiv$
$\displaystyle\frac{T_{\rm SC}(P)-T_{\rm SC}(0)}{T_{\rm SC}(0)}$
$\displaystyle\propto$
$\displaystyle\frac{T_{1}(0)T-T_{1}(P)T}{T_{1}(0)T}\equiv\delta T_{1}(P)T.$

In Fig. 8, the pressure dependence of $\delta T_{\rm SC}(P)$ evaluated from
our UCoGe results is plotted on the left axis, and that of $\delta T_{1}(P)T$
measured at 1 K is plotted on the right axis[30, 31]. It should be noted that
the $1/T_{1}T$ measured by NQR is determined by the $c$-axis fluctuations,
because the principal axis of the electric quadrupole interaction is almost
parallel to the $a$ axis. As the FM and paramagnetic (PM) signals coexist at
$P$ = 0.3 GPa, we take an average for $T_{1}(0.3)T$. It is noteworthy that the
good relationship between the variation in $\delta T_{\rm SC}(P)$ and that in
$\delta T_{1}(P)T$ holds beyond the FM criticality. This indicates that the FM
fluctuations are intrinsically important, and that the presence of the ordered
moments is not very sensitive to the superconductivity in UCoGe.

Figure 8: The pressure ($P$) dependence of $[T_{\rm SC}(P)-T_{\rm
SC}(0)]/T_{\rm SC}(0)\equiv\delta T_{\rm SC}(P)$ in the zero field is plotted
on the left axis[30, 31]. The $P$ dependence of
$[T_{1}(0)T-T_{1}(P)T]/T_{1}(0)T\equiv\delta T_{1}(P)T$ measured at 1 K on the
same sample of the $P$ dependence of $T_{\rm SC}$ is plotted on the right
axis[30, 31]. The inset shows the $P$ dependence of $1/T_{1}T$ measured at 1
K. PM (FM) means $1/T_{1}T$ in the paramagnetic (ferromagnetic) state.

It is quite interesting that a good relation holds between $\delta T_{\rm SC}$
and $\delta T_{1}T$ in all three cases, where FM fluctuations are tuned by
different external perturbations. Since $1/T_{1}T$ probes the low-energy spin
fluctuations, it was experimentally shown that this SC mechanism becomes
dominant in UCoGe near the magnetic instability, where the low-magnetic
fluctuations are enhanced. The SC mechanism by the electron-electron
interactions, other than the electron-phonon interactions, was first pointed
out by Kohn and Luttinger[35]. They pointed out that $T_{\rm SC}$ by their
mechanism was so low in metals ($T_{\rm SC}$ was estimated as only $\sim 1$ mK
in $s$-electron metals) but pointed out the possibility of “high” $T_{\rm SC}$
in metals where a flattening of the Fermi surface or an abnormally high
density of states could be realized. This means that their mechanism becomes
more effective in metallic compounds near the magnetic instability, where
magnetic susceptibility and density of states become enhanced. It is
reasonable to consider that the magnetic compounds, in which critical
behaviors in $1/T_{1}$ are observed, show superconductivity also from their
theoretical point of view.

It is also noted that the ratio between the left and right axes is the same as
that in Fig. 6 (b) and (c) ($\sim 3$), and the ratio in Fig. 6 (a) is $\sim
1.4$, indicating that the proportional constants are nearly the same in these
three cases. The good relation between $T_{\rm SC}$ and $T_{1}T$ variations
strongly suggests that $T_{\rm SC}$ is determined by the FM fluctuations and
is approximately expressed as $T_{\rm
SC}\sim\varepsilon\exp{\left(-1/g\right)}$, where $g$ is intimately related to
the anisotropic Ising spin susceptibility along the $c$ axis $\chi_{c}$ in
UCoGe.

It is expected that the character of the spin susceptibility is highly
dependent on SC compounds, even in U-based FM superconductors. Although all
U-based FM superconductors are considered as itinerant $f$-electron metals[36,
37, 38, 39, 40], the itinerant-localized components are different within them.
Since both the FM and SC transitions occur far below the Kondo coherent
temperature and the ordered moment in the FM state is small (0.05 $\mu_{B}$),
the itinerant character is dominant in UCoGe. In contrast, the FM ordered
moments in URhGe and UGe2 are nearly one order of magnitude larger than those
in UCoGe, thus the localized character is stronger in URhGe, and UGe2. This is
the reason why a clear metamagnetic transition was observed in these two
compounds[2]. It is important to check whether similar semi-quantitative
analyses hold for URhGe and UGe2.

Finally, we comment on the advantage of the study of FM superconductors. As
the possibility of the spin-fluctuation-mediated superconductivity was pointed
out[41, 42, 43, 44], a lot of studies have been done to clarify the
relationship between the spin fluctuation and superconductivity, and in most
of cases the relationship between antiferromagnetic (AFM) fluctuations and
superconductivity has been investigated mainly on Ce-based heavy-ferimon (HF),
cuprates, and Fe-based superconductors[45, 46, 47]. Although the
superconductivity in these systems can be enhanced by critical fluctuations
around the AFM critical points, the spin fluctuations cannot be tuned by $H$
because the AFM fluctuations are not coupled with $H$. This point is crucially
different from the present case of the FM superconductors. We point out that
FM superconductors are a key system for understanding the mechanism of spin-
fluctuation-mediated superconductivity. It is also important to note that the
pairing interactions originate from the spin-spin exchange process between the
nearly localized 5$f$ electrons through the conduction electrons[24]. This
process is the same as the Ruderman-Kittel-Kasuya-Yosida interaction[48, 49],
which also plays an important role in the realization of the HF state. We
point out that this is the reason why spin-fluctuation-mediated
superconductivity has often been observed in HF compounds. It is also noted
that the pairing interaction is cut off by the band energy of itinerant quasi-
particles in the theoretical model[24, 25]. We consider this to be the reason
why $T_{\rm SC}$ of most HF superconductors has a linear relationship with
their Fermi energy ($T_{\rm F}$), which is known as the Uemura plot[50], the
empirical relation holding in various unconventional superconductors.

##  V CONCLUSION

We measured $1/T_{1}$ and $1/T_{2}$ of 59Co in $H\parallel a$ and $H\parallel
b$ up to 23 T, as well as $H_{\rm c2}$ in each crystalline axis under the
precise alignment of the single-crystal UCoGe. Although the $1/T_{1}T$ and
$1/T_{2}T$ at 1.5 K are almost $H$ independent in $H\parallel a$, it was found
that the FM criticality in both $1/T_{1}T$ and $1/T_{2}T$ was observed in
$H\parallel b$. From the comparison between $1/T_{1}T$ and $1/T_{2}T$, it can
be concluded that the FM criticality arises from the Ising spin susceptibility
along the $c$ axis $\chi_{c}$, and that $1/T_{1}T$ is a good measure of $H$
and $T$ dependence of $\chi_{c}$. Based on the developed weak-coupling
theory[24, 25] and using the phenomenological relation
$(T_{1}T)^{-1}\propto\chi_{c}$, we examined the relationship between $T_{\rm
SC}$ and FM fluctuations derived from $1/T_{1}T$ when the FM fluctuations are
enhanced by the suppression of $T_{\rm Curie}$ in $H\parallel b$, and are
suppressed by $H\parallel c$. It was revealed that the variations of $T_{\rm
SC}$ and FM fluctuations are well understood by this theoretical model at a
semi-quantitative level. A similar good relationship was also confirmed when
the FM criticality was induced by pressure $P$. The present semi-quantitative
discussion concludes that superconductivity in UCoGe is induced by FM
fluctuations tunable by $H$ and $P$.

###### Acknowledgements.

One of the authors (K. I. ) wish to acknowledge V. P. Mineev for the
theoretical inputs and fruitful discussions, and K. Deguchi, T. Yamamura, and
N. K. Sato for the long-term collaboration on UCoGe. The authors thank Y.
Ihara, Y. Tokunaga, S. Yonezawa, Y. Maeno, H. Harima, H. Ikeda, S. Fujimoto,
Y. Tada, A. de Visser, J-P. Brison, G. Knebel, and J. Flouquet for fruitful
discussions. This work was supported by the Kyoto University LTM Center,
Tohoku University (Project No. 202012-HMKPB-0052, 202012-IPKAC-0008), Grants-
in-Aid for Scientific Research (Grant No. JP15H05745, JP17K14339, JP19K03726,
JP16KK0106, JP19K14657, JP19H04696, JP20H00130, and JP20KK0061, and a Grant-
in-Aid for JSPS Research Fellows (Grant No. JP20J11939) from JSPS. We would
like to thank Editage (www.editage.com) for English language editing.

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Fibonacci and Lucas Riordan Arrays and
Construction of Pseudo-Involutions

Candice Marshall and Asamoah Nkwanta

Morgan State University, Baltimore, MD 21251

###### Abstract

Riordan arrays, denoted by pairs of generating functions $(g(z),f(z))$, are
infinite lower-triangular matrices that are used as combinatorial tools. In
this paper, we present Riordan and stochastic Riordan arrays that have
connections to the Fibonacci and modified Lucas numbers. Then, we present some
pseudo-involutions in the Riordan group that are based on constructions
starting with a certain generating function $g(z)$. We also present a theorem
that shows how to construct pseudo-involutions in the Riordan group starting
with a certain generating function $f(z)$ whose additive inverse has
compositional order $2$. The theorem is then used to construct more pseudo-
involutions in the Riordan group where some arrays have connections to the
Fibonacci and modified Lucas numbers. A MATLAB algorithm for constructing the
pseudo-involutions is also given.

##  1 Introduction

In this paper, we present examples of Riordan and stochastic Riordan arrays
that have connections to the Fibonacci A000045 [16] and modified Lucas numbers
A000204 [16]. In particular, we present some pseudo-involutions in the Riordan
group; the first array that is a pseudo-involution includes the Fibonacci
numbers in the first column, the second has the modified Lucas numbers in the
first column, and the third contains the convolved Fibonacci numbers A001629
[16] in the first column. See OEIS [16] for more information on the convolved
Fibonacci numbers. The constructions start with a certain generating function
$g(z)$. We also present a theorem for constructing Riordan group pseudo-
involutions starting with a certain generating function $f(z)$ whose additive
inverse has compositional order $2$. The theorem is then used to construct
more pseudo-involutions in the Riordan group. Other constructions of pseudo-
involutions in the Riordan group can be found in [2, 7, 8].

##  2 Preliminaries

We begin with presenting some preliminary information. For those readers
familiar with Riordan arrays, you may skip this section and start with Section
3.

The Fibonacci numbers are given by the following recurrence relation,
$\mathcal{F}_{0}=\mathcal{F}_{1}=1$ and
$\mathcal{F}_{n}=\mathcal{F}_{n-1}+\mathcal{F}_{n-2}$ [12]. The first few
Fibonacci numbers A000045 [16] are 1, 1, 2, 3, 5, 8, 13, … The $n^{th}$
Fibonacci number is given by the following expression

$\mathcal{F}_{n}=\tfrac{\left(\tfrac{1+\sqrt{5}}{2}\right)^{n+1}-\left(\tfrac{1-\sqrt{5}}{2}\right)^{n+1}}{\sqrt{5}}.$

The generating function of the Fibonacci numbers is

$\mathcal{F}(z)=\tfrac{1}{1-z-z^{2}}=\sum_{n\geq 0}\mathcal{F}_{n}z^{n}.$

In this paper, we also use the Lucas numbers, sequence A000032 [16]. However,
we use a modified version, denoted by $\mathcal{L}_{n}$, because the Lucas
numbers begin with a $2$. We prefer that the leading (constant) term be $1$.
The corresponding generating function of the modified Lucas numbers A000204
[16] is

$\mathcal{L}(z)=\tfrac{1+z^{2}}{1-z-z^{2}}=\sum_{n\geq
0}\mathcal{L}_{n}z^{n}.$

We now introduce the definition of Riordan arrays. The arrays are known as
Riordan matrices.

######  Definition 1.

[14] An infinite matrix $L=(l_{n,k})_{n,k\in\mathbb{N}^{\ast}}$ with entries
in $\mathbb{C}$ is called a Riordan matrix if the $k^{th}$ column satisfies

$\sum_{n\geq 0}l_{n,k}\ z^{n}=g(z)f(z)^{k}$

where $g(z)=g_{0}+g_{1}z+g_{2}z^{2}+\cdots$ and
$f(z)=f_{1}z+f_{2}z^{2}+\cdots$ belong to the ring of formal power series
$\mathbb{C}[[z]],$ and $f_{1}\neq 0$ and $g_{0}\neq 0.$

A Riordan matrix, denoted by $L$, is usually written in pair form as
$\left(g(z),f(z)\right)$ or $(g,f)$.

######  Example 2.

A typical example of a Riordan matrix is the Pascal matrix. In this case,
$g(z)=\frac{1}{1-z}$ and $f(z)=\frac{z}{1-z}$. In pair form notation,
$P=\left(\frac{1}{1-z},\frac{z}{1-z}\right)$ where the entries of the Pascal
Riordan matrix are Pascal’s triangle A007318 [16] written in infinite lower-
triangular form.

The following theorem is called the Fundamental Theorem of the Riordan Group.
It leads to the next theorem by applying the fundamental theorem to an
arbitrary Riordan matrix $N$, one column of $N$ at a time.

######  Theorem 3.

[14] If $L=(l_{n,k})_{n,k\in\mathbb{N}^{\ast}}=\left(g(z),f(z)\right)$ is a
Riordan matrix and $h(z)$ is the generating function of the sequence
associated with the entries of the column vector
$h=(h_{k})_{k\in\mathbb{N}^{\ast}},$ then the product of $L$ and $h(z)$,
defined by

$L\otimes h(z)=g(z)h(f(z)),$

is the generating function of the sequence associated with the entries of the
column vector
$\left(\sum_{k=0}^{n}l_{n,k}h_{k}\right)_{n\in\mathbb{N}^{\ast}}.$

Let us denote by $L*N$, or by simple juxtaposition $LN$, the row-by column
product of two Riordan matrices.

######  Theorem 4.

[14] If

$L=(l_{n,k})_{n,k\in\mathbb{N}^{\ast}}=\left(g(z),f(z)\right)$

and

$N=(\nu_{n,k})_{n,k\in\mathbb{N}^{\ast}}=\left(h(z),l(z)\right)$

are Riordan matrices, then

$\displaystyle L\ast
N=\left(\sum_{j=0}^{n}l_{n,j}\nu_{j,k}\right)_{n,k\in\mathbb{N}^{\ast}}$
$\displaystyle=\left(g(z),f(z)\right)\ast\left(h(z),l(z)\right)$
$\displaystyle=\left(g(z)h(f(z)),l(f(z))\right),$

and the set $(\mathcal{R},\ast)$ of all Riordan matrices is a group under the
operation of matrix multiplication.

A Riordan matrix is also known as a proper Riordan array. A Riordan array
which is not proper does not belong to the Riordan group. Given that
$L=(g(z),f(z))$ is a Riordan matrix, then the identity element is $(1,z)$ and
the inverse of $(g(z),f(z))$ is
$\left(\tfrac{1}{g(\bar{f}(z))},\bar{f}(z)\right)$, where $\bar{f}(z)$ is the
compositional inverse of $f(z)$. Some well known subgroups of the Riordan
group mentioned in this paper are given in [14]. A special property of Riordan
matrices is that every element (except those in the first row and first
column) can be expressed as a linear combination of the elements in the
preceding row starting from the preceding column [13]. Also, every element in
the first column other than the first element, can be expressed as a linear
combination of all the elements of the preceding row [10]. These properties
are stated in the following theorem.

######  Theorem 5.

[10, 13] Let $L=(l_{n,k})$ be an infinite triangular matrix. Then $L$ is a
Riordan matrix if and only if there exists two sequences
$A={a_{0},a_{1},a_{2},...}$ and $Z={z_{0},z_{1},z_{2},...}$ with $a_{0}\neq
0,z_{0}\neq 0$ such that

$l_{n+1,k+1}=\sum_{j=0}^{\infty}a_{j}l_{n,k+j}\hskip 7.22743pt(k,n=0,1...),$

and

$l_{n+1,0}=\sum_{j=0}^{\infty}z_{j}l_{n,j}\hskip 7.22743pt(n=0,1,...).$

These sequences are called the A and Z sequences. We can find the A and Z
sequences of a Riordan matrix by calculating its production matrix. The
production matrix is the product of the inverse of the Riordan matrix and the
Riordan matrix with the first row removed [1].

We now introduce a few definitions and propositions about pseudo-involutions
that will be useful in the next section.

######  Definition 6.

A stochastic array is an array whose row sums equal one.

######  Definition 7.

[6] A stochastic Riordan array is an array which can be written as a pair of
generating functions $(g(z),f(z))$ with row sums equal to one. A stochastic
Riordan matrix is a proper stochastic Riordan array. It is invertible and
therefore belongs to the Riordan group.

######  Definition 8.

[3] An element $L$ of the Riordan group is called a pseudo involution or is
said to have pseudo-order $2$ if $LM$ or equivalently $ML$ has order $2$ where
$M=(1,-z)$. The entries of $M$ consist of alternating $1$ and $-1$ on the main
diagonal and $0$’s everywhere else.

One interesting property of pseudo-involutions is that one can calculate its
inverse very easily by placing negative signs on alternate column and row
entries. This is given by the following proposition.

######  Proposition 9.

[4, 5] If $A$ is a Riordan matrix that is a pseudo-involution, then
$A^{-1}=MAM.$

######  Example 10.

Consider the Pascal Riordan matrix given in Example 2. Since P is a pseudo-
involution [3], then the first few entries of $P^{-1}$ A130595 [16] are shown
below:

$\begin{bmatrix}\phantom{-}1\\\ -1&\phantom{-}1\\\
\phantom{-}1&-2&\phantom{-}1\\\ -1&\phantom{-}3&-3&\phantom{-}1\\\
\phantom{-}1&-4&\phantom{-}6&-4&\phantom{-}1\\\
-1&\phantom{-}5&-10&\phantom{-}10&-5&\phantom{-}1\\\
\phantom{-}1&-6&\phantom{-}15&-20&\phantom{-}15&-6&\phantom{-}1&\end{bmatrix}$.

##  3 Pseudo-involutions and Subgroups of the Riordan Group

######  Proposition 11.

A Riordan matrix $(g(z),F(z))$ is an involution if and only if

$g(z)\cdot g(F(z))=1$

and

$F(F(z))=z.$

###### Proof.

$\displaystyle(g(z),F(z))\hskip 1.4457pt\text{is an involution}$
$\displaystyle\iff(g(z),F(z))*(g(z),F(z))=(1,z)$ $\displaystyle\iff(g(z)\cdot
g(F(z)),F(F(z)))=(1,z).$

∎

Letting $F(z)=-f(z)$, we get the following corollary.

######  Corollary 12.

A Riordan matrix $(g(z),-f(z))$ is a pseudo-involution if and only if

$g(z)\cdot g(-f(z))=1$

and

$-f(-f(z))=z.$

######  Proposition 13.

A Riordan matrix denoted by $A$ is a pseudo-involution if and only if $A^{-1}$
is a pseudo-involution.

We conclude this section by presenting some simple facts about certain
subgroups of the Riordan group and pseudo-involutions.

######  Proposition 14.

An element of the Appell subgroup, $(g(z),z)$, is a pseudo-involution if and
only if $g(z)\cdot g(-z)=1$.

###### Proof.

$\displaystyle\big{(}(g(z),z)*(1,-z)\big{)}^{2}=(1,z)$
$\displaystyle\iff(g(z),-z)^{2}=(1,z)$ $\displaystyle\iff(g(z)\cdot
g(-z),z)=(1,z)$ $\displaystyle\iff g(z)\cdot g(-z)=1.$

∎

######  Example 15.

Let $g(z)=\tfrac{1+kz}{1-kz}$, where $k$ is any real number. Then,
$g(-z)=\tfrac{1-kz}{1+kz}$ and so
$\tfrac{1+kz}{1-kz}\cdot\tfrac{1-kz}{1+kz}=1.$ Hence,
$\left(\tfrac{1+kz}{1-kz},z\right)$ is a pseudo-involution in the Appell
subgroup for any real number $k$.

######  Proposition 16.

An element of the Bell subgroup, $\left(\tfrac{f(z)}{z},f(z)\right)$, is a
pseudo-involution iff $-f(z)$ has compositional order $2$.

######  Proposition 17.

An element of the Associated subgroup, $(1,f(z))$, is a pseudo-involution iff
$-f(z)$ has compositional order $2$.

######  Proposition 18.

An element of the Derivative subgroup, $(f^{\prime}(z),f(z))$, is a pseudo-
involution iff $-f(z)$ has compositional order $2$.

######  Proposition 19.

An element of the Hitting time subgroup, $(\tfrac{f^{\prime}(z)}{f(z)},f(z))$,
is a pseudo-involution iff $-f(z)$ has compositional order $2$.

##  4 Lucas Riordan Array

###  4.1 Stochastic Lucas Array

In this section, we create a stochastic Lucas array based on the modified
Lucas numbers and then generate a stochastic Riordan matrix.

######  Lemma 20.

[5] If a Riordan array is stochastic, then $f(z)=-g(z)+zg(z)+1.$

Using this lemma, we can generate a stochastic array for any sequence of
counting numbers.

######  Example 21.

Using the generating function of the modified Lucas numbers as $g(z)$, we can
generate the stochastic Lucas array. Since $g(z)=\tfrac{1+z^{2}}{1-z-z^{2}}$,
by Lemma 20

$f(z)=\tfrac{-2z^{2}+z^{3}}{1-z-z^{2}}.$

Thus, the stochastic Lucas array is

$(g(z),f(z))=\left(\frac{1+z^{2}}{1-z-z^{2}},\frac{-2z^{2}+z^{3}}{1-z-z^{2}}\right)$

where the first few entries of the array are shown below:

$\begin{bmatrix}1\\\ 1&\phantom{-}0\\\ 3&-2&0\\\ 4&-3&0&\phantom{-}0\\\
7&-10&4&\phantom{-}0&0\\\ 11&-18&8&\phantom{-}0&0&0&\\\ 18&-38&29&-8&0&0&0\\\
29&-71&63&-20&0&0&0&0\\\ 47&-134&150&-78&16&0&0&0&0\\\
76&-245&317&-195&48&0&0&0&0&0\end{bmatrix}.$

This is an example of a vertically stretched Riordan array. After removing the
leading 1 from the generating function of the modified Lucas numbers by
subtracting 1 and dividing by $z$, and applying Lemma 20, we can generate a
stochastic Lucas matrix
$\left(\frac{1+2z}{1-z-z^{2}},\frac{-2z+z^{2}}{1-z-z^{2}}\right)$. The first
few entries are shown below:

$\begin{bmatrix}1\\\ 3&-2\\\ 4&-7&4\\\ 7&-14&16&-8\\\ 11&-31&41&-36&16\\\
18&-60&105&-110&80&-32&\\\ 29&-116&235&-315&280&-176&64\\\
47&-216&512&-790&880&-688&384&-128\\\
76&-397&1063&-1894&2425&-2344&1648&-832&256\\\
123&-718&2153&-4298&6303&-7002&6032&-3872&1792&-512\end{bmatrix}.$

The Z sequence of the stochastic Lucas matrix is infinite. The first few terms
are:

$3,\tfrac{5}{2},\tfrac{25}{8},\tfrac{25}{8},\tfrac{375}{128},\tfrac{375}{128},\tfrac{3125}{1024},\tfrac{3125}{1024},...$

The A sequence is also infinite. The first few terms are:

$-2,\tfrac{1}{2},-\tfrac{5}{8},0,\tfrac{25}{128},0,-\tfrac{125}{1024},0,...$

It is easy to confirm the above new array and matrix are stochastic.
See [5] for an example of a stochastic Fibonacci array.

##  5 Construction of a Lucas Pseudo-involution

The following theorem is used to construct a Lucas pseudo-involution.

######  Theorem 22.

[9] If $g(z)=g_{0}+g_{1}z+g_{2}z^{2}+...,$ with $g_{0}=1$ and $g_{1}\neq 0$,
then there exists a unique $f(z)$ such that $(g(z),f(z))$ is a pseudo-
involution. In fact,

$f(z)=-\bar{G}\big{(}\tfrac{-G(z)}{g(z)}\big{)}$

where $G(z)=g(z)-g_{0}$.

###  5.1 Lucas Pseudo-involution

The next array is another pseudo-involution in the Riordan group. The modified
Lucas numbers make up the first column of the array.

######  Example 23.

We use the generating function of the modified Lucas numbers and Theorem 22 to
generate another pseudo-involution. Let $g(z)=\tfrac{1+z^{2}}{1-z-z^{2}}$.
Then,

$G(z)=\tfrac{z+2z^{2}}{1-z-z^{2}}$

and

$\bar{G}(z)=\tfrac{-(1+z)+\sqrt{5z^{2}+10z+1}}{2(2+z)}$

Now,

$\tfrac{1}{g(z)}-1=\tfrac{-z-2z^{2}}{1+z^{2}}$

Hence,

$F(z)=\bar{G}(\tfrac{-z-2z^{2}}{1+z^{2}})=\tfrac{-1+z+z^{2}+\sqrt{z^{4}+10z^{3}-13z^{2}-10z+1}}{4-2z}.$

So,

$(g(z),f(z))=\Big{(}\tfrac{1+z^{2}}{1-z-z^{2}},\tfrac{1-z-z^{2}-\sqrt{z^{4}+10z^{3}-13z^{2}-10z+1}}{4-2z}\Big{)}$

is a pseudo-involution of the Riordan group with the modified Lucas numbers in
the first column. The first few entries of this Riordan matrix are shown
below:

$\begin{bmatrix}1\\\ 1&1\\\ 3&6&1\\\ 4&33&11&1\\\ 7&214&88&16&1\\\
11&1572&699&168&21&1\\\ 18&12686&5787&1584&273&26&1\\\
29&108583&50036&14652&2994&403&31&1\\\
47&967294&447998&136436&30792&5054&558&36&1\end{bmatrix}.$

This can be confirmed by direct calculations. The Lucas pseudo-involution also
has infinite A and Z sequences. Its Z sequence is

$1,2,-11,58,-384,2872,-23416,201608,...$

and its A sequence is

$1,5,0,45,-225,1980,-16200,142920,...$

###  5.2 Convolved Fibonacci Pseudo-involution

The generating function of the convolved Fibonacci numbers is
$\left(\tfrac{1}{1-z-z^{2}}\right)^{n}.$ Using this as $g(z)$, following
Proposition 24 below, we can construct a convolved Fibonacci pseudo-
involution.

######  Proposition 24.

If $(g(z),f(z))$ is a pseudo-involution, then $(g^{n}(z),f(z))$ is a pseudo-
involution.

###### Proof.

Let $F(z)=-f(z)$. Notice that the Riordan matrix $(g(z),f(z))$ is a pseudo-
involution $\iff(g(z),F(z))$ is an involution. This means that $g(z)\cdot
g(F(z))=1$ and $F(F(z))=z$. To show that $(g^{n}(z),F(z))$ is an involution,
we need to show that $g^{n}(z)\cdot g^{n}(F(z))=1$ since we already have that
$F(F(z))=z.$ Now,

$g^{n}(z)\cdot g^{n}(F(z))=(g(z)\cdot g(F(z)))^{n}=1^{n}=1.$

∎

The above proposition confirms that
$\left((\tfrac{1}{1-z-z^{2}})^{n},\tfrac{1-z-z^{2}-\sqrt{5z^{4}+10z^{3}-z^{2}-6z+1}}{2-2z-2z^{2}}\right)$
is a pseudo-involution. Recall the case for $n=1$ is a pseudo-involution given
earlier. When $n=2$, we get

$(g^{2}(z),f(z))=\left(\tfrac{1}{(1-z-z^{2})^{2}},\tfrac{1-z-z^{2}-\sqrt{5z^{4}+10z^{3}-z^{2}-6z+1}}{2-2z-2z^{2}}\right).$

The first few entries of the Riordan matrix are shown below, where the
convolved Fibonacci numbers A001629 [16] are given in the leftmost column:

$\begin{bmatrix}1\\\ 2&1\\\ 5&5&1\\\ 10&20&8&1\\\ 20&75&44&11&1\\\
38&285&212&77&14&1\\\ 71&1138&976&448&119&17&1\\\
130&4820&4476&2390&810&170&20&1\\\
235&21545&20838&12266&4905&1325&230&23&1\end{bmatrix}.$

The Z-sequence of the convolved Fibonacci pseudo-involution is infinite and
starts with

$2,1,-5,20,-77,308,-1303,5805,...$

The A-sequence is also infinite and starts with

$1,3,0,5,-15,70,-310,1455,...$

Notice that this is the same A sequence as that of the Fibonacci pseudo-
involution given in [8].

The modified Lucas and convolved Fibonacci pseudo-involutions are new arrays
and none of their columns, other than the first, are sequences found in OEIS.
Their row sums, alternating row sums and diagonal sums are also not in OEIS.

###  5.3 MATLAB Algorithm for Constructing Pseudo-involutions Starting with
g(z)

The following MATLAB algorithm is used to find the corresponding $f(z)$ that
will make $(g(z),f(z))$ a pseudo-involution, after entering a bi-invertible
generating function $g(z)$. Note that a bi-invertible generating function
$g(z)$ satisfies $g_{0}\neq 0$ and $g_{1}\neq 0$.

syms z

g(z) = input(’Enter g(z): ’); $G(z)=g(z)-1\\\ \\\ G_{(}z)=finverse(G)$

R=-G(z)/g(z)

$F(z)=subs(G_{(}z),z,R)$

$simplify(F(z))\\\ \\\ f(z)=-subs(G_{(}z),z,R)\\\ \\\ simplify(f(z))$.

Construction of pseudo-involutions starting with bi-invertible generating
functions $g(z)$ involving the Motzkin A001006 [16] and Hex numbers A003215
[16] can be found in [8].

##  6 Constructing Pseudo-Involutions Starting with f(z)

Earlier we constructed pseudo-involutions starting with $g(z)$. We saw that if
we were given $g(z)$ such that $g_{0}=\pm 1$ and $g_{1}\neq 0$, then we can
find a unique $f(z)$ such that $(g(z),f(z))$ is a pseudo-involution. Now, we
will construct pseudo-involutions starting with $f(z)$. Note that this time we
are restricted because we need to start with an $f(z)$ such that
$-f(-f(z))=z$.

######  Theorem 25.

If $f(z)$ is a formal power series such that $-f(z)$ has compositional order
$2$, then $\\{g(z)\in F_{0}[[z]]\mid(g(z),f(z))$ is a pseudo-involution$\\}$
is an infinite subgroup of $F_{0}[[z]]$.

###### Proof.

By Propositions 16,17,18 and 19, $g(z)=f(z)/z,g(z)=1$, $g(z)=f^{\prime}(z)$
and,
$g(z)=\tfrac{zf^{\prime}(z)}{f(z)}$ all make $(g(z),f(z))$ a pseudo-
involution. These are just a few of the $g(z)$’s that make $(g(z),f(z))$ a
pseudo-involution.

Let $F(z)=-f(z)$. We first show closure under multiplication by showing that
if $(g_{1}(z),F(z))$ and $(g_{2}(z),F(z))$ are involutions, then
$(g_{1}(z)g_{2}(z),F(z))$ is also an involution. Let $(g_{1}(z),F(z))$ and
$(g_{2}(z),F(z))$ be involutions. Then,

$\displaystyle(g_{1}(z)g_{2}(z),F(z))(g_{1}(z)g_{2}(z),F(z))$
$\displaystyle=(g_{1}(z)g_{2}(z)\cdot g_{1}(F(z))\cdot g_{2}(F(z)),F(F(z)))$
$\displaystyle=(g_{1}(z)g_{1}(F(z))\cdot g_{2}(z)g_{2}(F(z)),F(F(z)))$
$\displaystyle=(1,z).$

Next, we show closure under taking of inverses by showing that
$(g_{1}^{-1}(z),F(z))$ is also an involution. Thus,

$\displaystyle(g_{1}^{-1}(z),F(z))(g_{1}^{-1}(z),F(z))$
$\displaystyle=(g_{1}^{-1}(z)\cdot g_{1}^{-1}(F(z)),F(F(z)))$
$\displaystyle=\left(\tfrac{1}{g_{1}(z)}\tfrac{1}{g_{1}(F(z))},F(F(z))\right)$
$\displaystyle=\left(\tfrac{1}{g_{1}(z)g_{1}(F(z))},F(F(z))\right)$
$\displaystyle=(1,z).$

∎

###  6.1 Construction of Fibonacci-type Pseudo-Involutions in the Riordan
Group

We will now use the $f(z)$ that made the generating function for the Fibonacci
numbers and the generating function of the convolved Fibonacci numbers into
pseudo-involutions. Recall below, $f(z)$ as given in Subsection 5.2.

$f(z)=\tfrac{1-z-z^{2}-\sqrt{5z^{4}+10z^{3}-z^{2}-6z+1}}{2-2z-2z^{2}}.$

We know that if we let $g(z)=1,g(z)=f(z)/z$ or $g(z)=f^{\prime}(z)$, then we
have constructed pseudo-involutions. We start by illustrating the first few
rows of $(1,f(z))$.

$\begin{bmatrix}1\\\ 0&1\\\ 0&3&1\\\ 0&9&6&1\\\ 0&32&27&9&1\\\
0&126&118&54&12&1\\\ 0&538&525&285&90&15&1\\\ 0&2429&2408&1440&560&135&18&1\\\
0&11412&11378&7203&3195&970&189&21&1\\\
0&55201&55146&36162&17488&6195&1542&252&24&1\end{bmatrix}$

Next, we construct a pseudo-involution in the Bell subgroup. Now,

$f(z)/z=\tfrac{1-z-z^{2}-\sqrt{5z^{4}+10z^{3}-z^{2}-6z+1}}{2z-2z^{2}-2z^{3}}.$

We illustrate the first few rows of $(f(z)/z,f(z))$.

$\begin{bmatrix}1\\\ 3&1\\\ 9&6&1\\\ 32&27&9&1\\\ 126&118&54&12&1\\\
538&525&285&90&15&1\\\ 2429&2408&1440&560&135&18&1\\\
11412&11378&7203&3195&970&189&21&1\\\
55201&55146&36162&17488&6195&1542&252&24&1\\\
272993&272904&183132&93926&37043&10926&2303&324&27&1\end{bmatrix}$

Now, if

$f(z)=\tfrac{1-z-z^{2}-\sqrt{5z^{4}+10z^{3}-z^{2}-6z+1}}{2-2z-2z^{2}},$

then,

$f^{\prime}(z)=\tfrac{-2z-1}{(z^{2}+z-1)\sqrt{5z^{4}+10z^{3}-z^{2}-6z+1}}.$

Next, we illustrate the first few rows of $(f^{\prime}(z),f(z))$, an element
of the Derivative subgroup.

$\begin{bmatrix}1\\\ 6&1\\\ 27&9&1\\\ 128&54&12&1\\\ 630&295&90&15&1\\\
3228&1575&570&135&18&1\\\ 17003&8428&3360&980&189&21&1\\\
91296&45512&19208&6390&1552&252&24&1\\\
496809&248157&108486&39348&11151&2313&324&27&1\\\
2729930&1364520&610440&234815&74086&18210&3290&405&30&1\end{bmatrix}$

Finally, we illustrate the first few rows of
$\left(\tfrac{zf^{\prime}(z)}{f(z)},f(z)\right)$, an element of the Hitting
Time subgroup.

$\begin{bmatrix}1\\\ 3&1\\\ 9&6&1\\\ 42&27&9&1\\\ 201&128&54&12&1\\\
1043&630&295&90&15&1\\\ 5544&3228&1575&570&135&18&1\\\
30012&17003&8428&3360&980&189&21&1\\\
164281&91296&45512&19208&6390&1552&252&24&1\\\
906693&496809&248157&108486&39348&11151&2313&324&27&1\end{bmatrix}$

Thus far, we have six Riordan pseudo-involutions with the same $f(z)$. There
are many more Riordan pseudo-involutions using this same $f(z)$ since we
showed earlier that the $g(z)$’s that make $(g(z),f(z))$ a pseudo-involution,
form a group under multiplication. We therefore know that we can multiply
$g(z)$’s to get others and we can take powers of these $(g(z))$’s.

##  7 Conclusion

In this paper, we presented examples of Riordan arrays that involve the
Fibonacci and modified Lucas numbers. There are also other papers on Riordan
arrays that involve the Fibonacci numbers [1, 11, 15].

Most of the Riordan matrices, as well as the Lucas stochastic array presented
in this paper are new and contain many new sequences of integers. The arrays
were observed while studying certain algebraic properties of the Riordan
group. Finding combinatorial interpretations of the arrays was not the focus
of this paper. Thus, the arrays are open for combinatorial interpretations.
Constructing a pseudo-involution starting with a modified Lucas generating
function $f(z)$ would also be of interest.

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Keywords: Fibonacci numbers, Lucas numbers, Riordan group, Riordan matrix,
Riordan array, pseudo-involution.

(Concerned with sequences A000045, A000204, A130595, A007318, A000032,
A003215, A001006 and A001629.)