# Relationships between Superconductivity and Nematicity in FeSe1-xTex
($x=0-0.5$) Films Studied by Complex Conductivity Measurements

Hodaka Kurokawa  Department of Basic Science, The University of Tokyo, 3-8-1
Komaba, Meguro-ku, Tokyo, Japan<EMAIL_ADDRESS>Sota Nakamura
Department of Basic Science, The University of Tokyo, 3-8-1 Komaba, Meguro-ku,
Tokyo, Japan  Jiahui Zhao  Department of Basic Science, The University of
Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, Japan  Naoki Shikama  Department of
Basic Science, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, Japan
Yuki Sakishita  Department of Basic Science, The University of Tokyo, 3-8-1
Komaba, Meguro-ku, Tokyo, Japan  Yue Sun  Department of Physics, Aoyama Gakuin
University, 5-10-1 Fuchinobe, Chuou-ku, Sagamihara, Japan.  Fuyuki Nabeshima
Department of Basic Science, The University of Tokyo, 3-8-1 Komaba, Meguro-ku,
Tokyo, Japan  Yoshinori Imai  Department of Physics, Graduate School of
Science, Tohoku University, 6-3 Aramaki-Aoba, Aoba-ku, Sendai, Miyagi, Japan
Haruhisa Kitano  Department of Physics, Aoyama Gakuin University, 5-10-1
Fuchinobe, Chuou-ku, Sagamihara, Japan.  Atsutaka Maeda  Department of Basic
Science, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, Japan

###### Abstract

We measured the complex conductivity, $\sigma$, of FeSe1-xTex ($x=0-0.5$)
films in the superconducting state which show a drastic increase of the
superconducting transition temperature, $T_{\textrm{c}}$, when the nematic
order disappears. Since the magnetic penetration depth, $\lambda$ $(>$ 400
nm), of Fe(Se,Te) is longer than the typical thickness of the film ($\sim$100
nm), we combined the coplanar waveguide resonator and cavity perturbation
techniques to evaluate both the real and imaginary parts of $\sigma$. Films
with a nematic order showed a qualitatively different temperature dependence
in penetration depth and quasiparticle scattering time when compared with
those without nematic order, suggesting that nematic order influences the
superconducting gap structure. Conversely, the proportionality between
superfluid density, $n_{\textrm{s}}$ ($\propto\lambda^{-2}$), and
$T_{\textrm{c}}$ was observed irrespective of the presence or absence of
nematic order. This result indicates that the amount of superfluid has a
stronger impact on the $T_{\textrm{c}}$ of Fe(Se,Te) than the presence or
absence of nematic order. Combining these results with band dispersions
calculated using density functional theory, we propose that the change of the
Fermi surface associated with nematicity is the primary factor influencing the
change of $T_{\textrm{c}}$ and the superconducting gap structure in Fe(Se,Te).

##  I Introduction

Iron chalcogenide superconductor, FeSe, has been intensively studied by virtue
of its various intriguing properties:Liu  _et al._ (2015); Böhmer and Kreisel
(2018); Kreisel  _et al._ (2020) the potential for high-transition-temperature
superconductivity, the absence of magnetic order under ambient pressures, and
their ability to exhibit exotic electronic states as a result of their
extremely small Fermi surface. The superconducting transition temperature,
$T_{\textrm{c}}$, can be enhanced above 40 K from 9 K by intercalation,Hsu
_et al._ (2008); Burrard-Lucas  _et al._ (2013) carrier doping using an
electron double layer transistor,Shiogai  _et al._ (2016); Shikama  _et al._
(2020) and synthesis of a monolayer film.Wang  _et al._ (2012); Huang and
Hoffman (2017) The nematic phase without magnetic order in FeSe is ideal for
studying the origin of nematicity and the relationship between nematicity and
superconductivity.Fernandes  _et al._ (2014); Glasbrenner  _et al._ (2015)
Furthermore, the small Fermi surface ($\epsilon_{\textrm{F}}<10$ meV) can
easily be tuned by hydrostatic pressure,Margadonna  _et al._ (2009) chemical
pressure via isovalent substitution,Fang  _et al._ (2008); Mizuguchi  _et al._
(2009); Imai  _et al._ (2015) and the in-plane lattice strain.Nabeshima  _et
al._ (2018a) Since changes in the Fermi surface influence the superconducting,
nematic, and magnetic phases, various techniques have been applied to
investigate the electronic phase diagram and exotic superconductivity of FeSe.

Among the above-mentioned techniques to control the electronic state, chemical
isovalent substitution is advantageous since experiments can be performed
under ambient pressures. The S-substitution shrinks the lattice of FeSe,
resulting in positive chemical pressure. With increasing S content, the
nematic transition temperature, $T_{\textrm{n}}$, decreases, and
$T_{\textrm{c}}$ slightly increases and decreases.Watson  _et al._ (2015)
Although no significant changes in $T_{\textrm{c}}$ occur when the nematic
order disappears, some abrupt changes in the superconducting gap have been
observed in a measurement of thermal properties and the scanning tunneling
microscopy/spectroscopy.Sato  _et al._ (2018); Hanaguri  _et al._ (2018)
Hence, the nematic order or its fluctuation may exert some influence on the
superconducting state. Conversely, few systematic investigations of Te-
substituted FeSe, which are subject to negative chemical pressures, have been
conducted relative to those concerning Fe(Se,S) since the systematic synthesis
of bulk Fe(Se,Te) had, until recently, been hindered by the phase separation
region.Terao  _et al._ (2019) Since the superconducting gap structure of FeSe
is distinctly different from that of FeSe1-xTex ($x>0.5$),Hanaguri  _et al._
(2010); Kasahara  _et al._ (2014); Sprau  _et al._ (2017) it is necessary to
understand how the superconducting gap evolves with increasing Te content.

Before the systematic synthesis of bulk Fe(Se,Te), we succeeded in growing
single-crystalline thin films of FeSe1-xTex in the whole composition
($x=0-0.9$) using a pulsed laser deposition technique.Imai  _et al._ (2015,
2017) Although the $T_{\textrm{n}}$ of the Fe(Se,Te) films decreased after Te
substitution, $T_{\textrm{c}}$ was largely enhanced after the disappearance of
nematic order.Imai  _et al._ (2015) This enhancement of $T_{\textrm{c}}$ is
contrary to the Fe(Se,S) films and bulk Fe(Se,Te),Nabeshima  _et al._ (2018b);
Terao  _et al._ (2019) indicating that the effect of nematicity on
$T_{\textrm{c}}$ is complicated in these materials. Although a positive
correlation between $T_{\textrm{c}}$ and the carrier density or the density of
states (DOS) has been reported through magneto-transport and the angle-
resolved photoemission spectroscopy (ARPES) in the normal state,Nabeshima  _et
al._ (2020); Nakayama  _et al._ (2021) the superconducting properties of these
films and their relation to nematicity are yet to be fully understood. To
elucidate the effects of Te substitution and nematicity on superconductivity,
we investigated both the response of the superfluid and dynamics of
quasiparticles below $T_{\textrm{c}}$ in Fe(Se,Te) films.

In this paper, we report systematic measurements of the complex conductivity,
$\sigma$, of FeSe1-xTex ($x=0-0.5$) films below $T_{\textrm{c}}$. Since the
magnetic penetration depth, $\lambda$, is several times as long as the typical
thickness of the film ($\sim 100$ nm), measurement techniques applicable to
bulk crystals cannot be used. Hence, to evaluate both the real and imaginary
parts of $\sigma$, we combined the coplanar waveguide resonator and cavity
perturbation techniques. The quasiparticle scattering time, $\tau$, was
calculated from the real part of $\sigma$ and was found to increase at low
temperatures, as observed in bulk FeSe and FeSe0.4Te0.6.Takahashi  _et al._
(2011); Okada  _et al._ Moreover, the $\lambda$ and $1/\tau$ of films with
nematic order showed a quantitatively distinct behavior from films without
nematic order, suggesting that changes in the superconducting gap structure
are associated with nematic order. Conversely, the proportionality between
superfluid density, $n_{\textrm{s}}(\propto\lambda^{-2})$, and
$T_{\textrm{c}}$ was observed irrespective of the presence or absence of
nematic order. Additionally, using density functional theory (DFT)
calculations, we confirmed a change of Fermi surface associated with nematic
order, which is considered to influence the superconducting gap structure.
Moreover, DOS decreased in the nematic phase, which may have caused a decrease
in superfluid density. From these results, we suggest that the change of the
band structure in the nematic phase primarily influences the superconducting
gap structure and $T_{\textrm{c}}$ rather than the nematic fluctuation
developing near the nematic quantum critical point.Mukasa  _et al._ (2021)

##  II Experiments

###  II.1 Sample

All films were grown on CaF2 substrates ($\sim 5\times 5\times 0.5$ mm3) via
the pulsed laser deposition method using a KrF laser. The details of film
growth have been described elsewhere.Imai  _et al._ (2010a, b) The thicknesses
of grown films were measured using a stylus profiler. DC electrical
resistivity was measured using a standard four-probe method equipped with a
physical property measurement system (Quantum Design, PPMS).

Fig. 1: (a) Schematic of the coplanar resonator. (b) The fabricated coplanar
resonator (FeSe0.8Te0.2). (c) Schematic of the cavity resonator.

###  II.2 Measurements and calculations

To measure $\lambda$ of FeSe1-xTex films, we fabricated the $\sim 5\times 5$
mm2 film into the coplanar waveguide resonator (Fig. 1 [a]) by Ar ion milling
and focused ion beam (FIB). Ar ion milling was used to fabricate the whole
structure, after which the 50 $\mu$m gap between the resonator and the
microwave input/output port was etched using FIB. The width of the resonator,
$w$, the gap between the resonator and the ground, $s$, and the length of the
resonator, $l$, were designed to be 120 $\mu$m, 30 $\mu$m, and 6.2-9.9 mm,
respectively. Figure 1 (b) shows the fabricated resonator on the FeSe0.8Te0.2
film. The resonator was mounted onto a printed circuit board, which was
connected to the resonator by Al wirebonding. This was cooled down to 2 K
using PPMS. Transmitted power was measured using a network analyzer (Keysight,
N5222A).

$\lambda$ was calculated from the resonance frequency, $f_{\textrm{c}}$, as
follows. For the half-wavelength coplanar resonator:

$f_{\textrm{c}}=\frac{1}{2l\sqrt{LC}},$ (1)

where $L$ is the inductance per unit length and $C$ is the capacitance per
unit length. Using an electromagnetic simulation software (WIPL-D), we
confirmed that the coupling between the resonator and the input port had
negligible effects on $f_{\textrm{c}}$. For a superconductor:

$L=L_{\textrm{m}}+L_{\textrm{k}},$ (2)

where $L_{\textrm{m}}$ is the magnetic inductance and $L_{\textrm{k}}$ is the
kinetic inductance corresponding to the response of the superfluid Watanabe
_et al._ (1994). $L_{\textrm{k}}$ is a quadratic function of $\lambda$ as
follows:

$L_{\textrm{k}}=\frac{\mu_{0}g(s,w,d)}{dw}\lambda^{2},$ (3)

where $\mu_{0}$ is vacuum permeability, $g(s,w,d)$ is a geometrical factor,
and $d$ is the thickness of the film Watanabe  _et al._ (1994); Clem (2013).
From eq. (1)-(3), $\lambda$ is expressed as follows:

$\lambda=\sqrt{\frac{dw}{g\mu_{0}}\left(\frac{1}{4l^{2}Cf_{\textrm{c}}^{2}}-L_{\textrm{m}}\right)}.$
(4)

All parameters on the right-hand side of eq. (4) can be determined from the
shape of the resonator ($d$, $w$, $s$) and measurements of $f_{\textrm{c}}$
and $C$. The length was measured using an optical microscope (Keyence,
VHS-6000), and $C$ was measured using an impedance analyzer (Hewlett-Packard,
4192A) in the frequency range, 10-1000 kHz. The measured values of $C$ were in
good agreement with the calculated values of $C$ assuming that the relative
permittivity of CaF2 is 6.5. Jacob  _et al._ (2003)

The dynamics of quasiparticles in the Fe(Se,Te) films were measured using the
cavity perturbation technique. For a thin film ($d<\lambda$), the cavity
perturbation formula for the analysis of a bulk crystal cannot be applied. In
such cases, the measured quantity is the effective impedance,
$Z_{\textrm{eff}}(Z_{\textrm{s}},d)$, where $Z_{\textrm{s}}$ is surface
impedance. Formulae of $Z_{\textrm{eff}}$ corresponding to various situations
have been derived, and are found to depend on configurations of both the
electromagnetic field and the sample.Klein  _et al._ (1990); Drabeck  _et al._
(1990); Peligrad  _et al._ (1998); Barannik  _et al._ (2014)

A flake of FeSe1-xTex film was cut from the coplanar resonator following the
measurement of $\lambda$. This flake ($\sim$ 0.5 $\times$ 0.5 mm2) was mounted
onto a sapphire rod at the center of the cavity resonator (Fig. 1 [c]). The
TE011 mode (44 GHz) of the resonator was used with a configuration in which
the magnetic field of the TE011 mode was parallel to the film well below
$T_{\textrm{c}}$ such that $Z_{\textrm{eff}}$ can be expressed as follows:

$Z_{\textrm{eff}}=-\frac{i}{2}Z_{\textrm{s}}\textrm{cot}\left(\frac{\omega\mu_{0}d}{2Z_{\textrm{s}}}\right),$
(5)

where $\omega$ is angular frequency Barannik  _et al._ (2014). Experimentally,
the effective surface resistance, $R_{\textrm{eff}}$, is determined by the
following equation:

$R_{\textrm{eff}}=G\left(\frac{1}{2Q_{\textrm{sample}}}-\frac{1}{2Q_{\textrm{blank}}}\right),$
(6)

where $G$ is the geometric factor, $Q_{\textrm{sample}}$ is the quality factor
of the cavity containing the sample, $Q_{\textrm{blank}}$ is the quality
factor of the cavity without the sample. Here, we have confirmed that the
effect of the CaF2 substrate was negligible by measurement of the substrate
alone. Also, the effective surface reactance, $X_{\textrm{eff}}$ is as
follows:

$X_{\textrm{eff}}(T)=G\left(\frac{f_{\textrm{c,sample}}(T_{0})-f_{\textrm{c,sample}}(T)}{f_{\textrm{c,sample}}(T_{0})}-\right.\\\
\left.\frac{f_{\textrm{c,blank}}(T_{0})-f_{\textrm{c,blank}}(T)}{f_{\textrm{c,blank}}(T_{0})}\right)+X_{\textrm{eff}}(T_{0}),$
(7)

where $f_{\textrm{c,sample}}$ is the resonance frequency with the sample,
$f_{\textrm{c,blank}}$ is the resonance frequency without the sample, and
$T_{0}$ is the minimum temperature during the measurement, typically 2.1 K.

To obtain $Z_{\textrm{s}}$ by solving eq. (5)-(7), we determined $G$ and
$X_{\textrm{eff}}(T_{0})$ as follows. At low temperatures where
$\sigma_{1}<<\sigma_{2}$,

$X_{\textrm{eff}}(T)=\frac{1}{2}\mu_{0}\omega\lambda\textrm{coth}\left(\frac{d}{2\lambda}\right)$
(8)

from eq. (5) and $X_{\textrm{s}}\approx\mu_{0}\omega\lambda$. Thus,
$X_{\textrm{eff}}(T)$ can be calculated by substituting the value of
$\lambda(T)$ measured by the coplanar resonator into eq. (8). Here,
$X_{\textrm{eff}}(T_{0})$ was obtained using eq. (8) and $\lambda(T_{0})$
measured by the coplanar resonator. Conversely, $G$ was determined by curve
fitting to satisfy $X_{\textrm{eff}}^{\textrm{coplanar}}(T)\simeq
X_{\textrm{eff}}^{\textrm{cavity}}(T)$ in the temperature range,
$0.2-0.5T_{\textrm{c}}$. After determining $G$ and $X_{\textrm{eff}}(T_{0})$,
we numerically solved eq. (5) and obtained $Z_{\textrm{s}}$. It should be
noted that eq. (5) is not applicable near $T_{\textrm{c}}$ due to the drastic
change of the electromagnetic field distribution around the film.Barannik  _et
al._ (2014) Here, we determined the upper temperature limit for an applicable
range of eq. (5) to be below 0.75 $T_{\textrm{c}}$ from a measurement of a
conventional superconductor, NbN film.

Besides the measurement of $\sigma$, we performed DFT calculation using
FPLO-18. The exchange functional was a generalized gradient approximation
(GGA+U). The $k$-mesh was $12\times 12\times 6$. For the calculation of the
nematic phase, we applied the technique proposed in Ref. Long  _et al._
(2020), using lattice constants for FeSe as follows: $a=3.76976$ Å,
$c=5.52122$ Åand $z_{\textrm{Se}}=0.2688$.Glasbrenner  _et al._ (2015)

##  III Results and Discussion

###  III.1 Measurements of complex conductivity

Fig. 2: Temperature dependence of dc resistivity of the FeSe1-xTex ($x=0-0.5$)
films. The inset shows a schematic of the phase diagram of FeSe1-xTex
films.Imai  _et al._ (2015)

Figure 2 shows the temperature dependence of DC resistivity in FeSe1-xTex
($x=0-0.5$) films. $T_{\textrm{c}}$ increased from $x=0$ to $x=0.2$,
consistent with previous reports.Imai  _et al._ (2015) Subsequently,
$T_{\textrm{c}}$ gradually decreased with increasing Te content, and the
resistivity of $T_{\textrm{c,onset}}$ increased from $\sim$ 100 $\mu\Omega$ cm
at $x=0$ to $\sim$ 400 $\mu\Omega$ cm at $x=0.5$, which was again a typical
value for these films.Imai  _et al._ (2017)

Fig. 3: (a) Temperature dependence of $\lambda$ of FeSe0.8Te0.2 film. The red
line corresponds a fitted curve with the equation,
$\lambda(T)=\lambda_{0}+A(T/T_{\textrm{c}})^{n}$. (b) $n$ and (c)
$A/\lambda_{0}$ as a function of Te content. Fig. 4: (a) $T_{\textrm{c}}$ and
$\lambda$ at 0 K in FeSe1-xTex ($x=0-0.5$) films. (b) $T_{\textrm{c}}$ as a
function of $\lambda_{0}^{-2}$ in FeSe1-xTex ($x=0-0.5$) films. Results of
bulk FeSe and bulk FeSe1-xTex ($x>0.5$) are also shown.Khasanov  _et al._
(2009); Kasahara  _et al._ (2014); Okada  _et al._ ; Takahashi  _et al._
(2011); Kim  _et al._ (2010); Biswas  _et al._ (2010) The blue line and yellow
dashed lines correspond to the data of hole-doped cuprates and electron-doped
cuprates, respectively.Luetkens  _et al._ (2008)

Using the temperature dependence of $f_{\textrm{c}}(T)$ of the coplanar
resonator, we calculated $\lambda(T)$ using eq. (4). We plotted the result of
FeSe0.8Te0.2 film as a representative (Fig. 3 [a]).The obtained $\lambda(T)$
was extrapolated to 0 K assuming that
$\lambda(T)=\lambda_{0}+A(T/T_{\textrm{c}})^{n}$, where $\lambda_{0}$ is the
penetration depth at 0 K, and $A$ and $n$ are constants. Here, we performed
curve fitting in the temperature range from 2 K to 0.3 $T_{\textrm{c}}$.
Figure 3 (b) shows $n$ of FeSe1-xTex films. For films in nematic phase ($x$=0,
0.1), $n$ was almost 1, whereas $n$ was 2$-$3 in non-nematic phase. The
$T$-linear behavior indicates that the superconducting gap has nodes or gap
minima in nematic phase. Kasahara  _et al._ (2014); Okada  _et al._ .
Conversely, $n>2$ in non-nematic phase probably denote the existence of
nodeless gaps which are subjected to pair-breaking effects.Prozorov and Kogan
(2011) On the other hand, no systematic changes in the dimensionless
coefficient, $A/\lambda_{0}$, were observed as shown in Fig.3 (c).

Figure 4 (a) shows Te content versus $T_{\textrm{c,zero}}$ and $\lambda_{0}$.
The negative correlation between $T_{\textrm{c}}$ and $\lambda_{0}$ seems to
exist irrespective of the presence or absence of nematic order. Subsequently,
we plotted $T_{\textrm{c}}$ as a function of $\lambda_{0}^{-2}$ (Fig. 4 [b]),
which is the so-called Uemura plot. $T_{\textrm{c}}$ exhibited an obvious
positive correlation with $\lambda_{0}^{-2}$, corresponding to
$n_{\textrm{s}}$. The observed proportionality between $T_{\textrm{c}}$ and
$n_{\textrm{s}}$ is consistent with the correlation between $T_{\textrm{c}}$
and the carrier density of the Fe (Se,Te) and Fe(Se,S) films in their normal
state.Nabeshima  _et al._ (2020) These results indicate that either
$n_{\textrm{s}}$ or carrier density plays a crucial role in determining the
$T_{\textrm{c}}$ of Fe(Se,Te) films irrespective of the presence or absence of
nematic order or its fluctuation.

Compared with bulk samples, whereas the trend between $T_{\textrm{c}}$ and
$\lambda_{0}^{-2}$ in the films was similar to that of bulk FeSe1-xTex
($x>0.5$),Takahashi  _et al._ (2011); Kim  _et al._ (2010) a discrepancy with
bulk FeSe was observed.Kasahara  _et al._ (2014); Okada  _et al._ Namely, the
slope of the data of FeSe1-xTex ($x=0-0.5$) and bulk FeSe1-xTex ($x>0.5$) was
larger than that of bulk FeSe. Such differences in the Uemura plot have
already been reported in cuprate superconductors, in which the data of hole-
doped cuprates show the larger slopes than that of electron-doped cuprates
(dotted lines in Fig. 4 [b]). Interestingly, whereas
$n_{\textrm{h}}=1.1-1.4n_{\textrm{e}}$ in the bulk FeSe,Ovchenkov  _et al._
(2019) $n_{\textrm{h}}=1.0-2.8n_{\textrm{e}}$ in the FeSe1-xTex films which
showed the steeper slope,Sawada  _et al._ (2016); Yoshikawa  _et al._ (2019);
Nabeshima  _et al._ (2020) where $n_{\textrm{h}}$ is the hole density and
$n_{\textrm{e}}$ is the electron density. The carrier density of the bulk as-
grown FeSe1-xTex ($x>0.5$) is also estimated to be
$n_{\textrm{h}}>n_{\textrm{e}}$ from measurements of the Hall coefficient.Liu
_et al._ (2009); Sun  _et al._ (2014) This correspondence between Fe(Se,Te)
and the cuprates suggests the possibility that hole-doping increases the slope
of the Uemura plot, even in multi-band superconductors.

Fig. 5: (a) Temperature dependence of $\sigma_{1}$ and (b) the inverse of
$\tau$ in the FeSe1-xTex ($x=0-0.5$) films. The result of bulk crystals are
also shown Takahashi  _et al._ (2011); Okada  _et al._ . (c) The exponent $n$
in the equation, $1/\tau=aT^{n}+b$, determined from curve fitting. The maximum
temperature for the fitting was varied from 0.25$T/T_{\textrm{c}}$ to
0.5$T/T_{\textrm{c}}$. Orange circles and blue squares correspond to $1/\tau$
of the bulk FeSe Okada  _et al._ ; Li  _et al._ (2016). Pink triangles denote
$1/\tau$ of the bulk FeSe0.4Te0.6 Takahashi  _et al._ (2011).

Next, we evaluated the results of measurements of the dynamics of
quasiparticles using the cavity perturbation technique. From the measurement
of the $Q^{-1}(T)$ and $f_{\textrm{c}}(T)$ in each films placed in the cavity
resonator, $R_{\textrm{s}}$ and $X_{\textrm{s}}$ were calculated using eq. (5)
below $0.75\ T_{\textrm{c}}$.The real part of the complex conductivity,
$\sigma_{1}$, was calculated using
$\sigma_{1}=2\omega\mu_{0}R_{\textrm{s}}X_{\textrm{s}}/(R_{\textrm{s}}^{2}+X_{\textrm{s}}^{2})^{2}$.
When calculating $\sigma_{1}$, we subtracted residual resistance from
$R_{\textrm{s}}$, which was estimated from the linear extrapolation of
$R_{\textrm{s}}$ to 0 K. Figure 5 (a) shows the value of $\sigma_{1}$
corresponding to each film. With decreasing temperature, $\sigma_{1}$
decreased in all tested films. The decrease of $\sigma_{1}$ at low
temperatures is consistent with the measurement of bulk FeSe and
FeSe0.4Te0.6.Li  _et al._ (2016); Takahashi  _et al._ (2011)

Assuming the two-fluid model and Drude-like single-carrier normal fluid, the
quasiparticle scattering time, $\tau$, can be expressed as follows:

$\omega\tau=\frac{\tilde{\sigma_{1}}}{1-\tilde{\sigma_{2}}},$ (9)

where
$\tilde{\sigma}=\tilde{\sigma_{1}}+i\tilde{\sigma_{2}}=\mu_{0}\omega\lambda_{0}^{2}(\sigma_{1}+i\sigma_{2})$,
which gives the dimensionless conductivity.Takahashi  _et al._ (2011) Here, we
should be careful this single-carrier treatment because Fe(Se,Te) is a multi-
band superconductor. Since FeSe has highly anisotropic gaps in both hole and
electron pockets,Sprau  _et al._ (2017) whereas FeSe1-xTex ($x>0.5$) has
nodeless gaps in both pockets ,Hanaguri  _et al._ (2010) the corresponding
values $\tau$ of the electron pockets are expected to show similar temperature
dependence to that of the hole pocket in FeSe1-xTex ($x=0-0.9$). Hence, in eq.
(9), we assumed that the temperature dependence of $\tau$ in both pockets
could be captured using a single $\tau$ as a first approximation.

In all films, $1/\tau$ was observed to decrease at low temperatures (Fig. 5
[b]), indicating rapid suppression of the inelastic scattering of the
electron, which was already established in bulk FeSe and FeSe0.4Te0.6.Okada
_et al._ ; Takahashi  _et al._ (2011) Furthermore, the slope of $1/\tau$ seems
to be different among these films as shown in Fig. 5 (b). To obtain further
insights, we performed curve fitting with $1/\tau=aT^{n}+b$, where $a$, $b$,
and $n$ are positive constants. Figure 5 (c) shows $n$ of each film as a
function of the maximum temperature used for the curve fitting,
$T_{\textrm{max}}^{\textrm{fit}}$. The exponent, $n$, showed differing
behavior among these films when $T_{\textrm{max}}^{\textrm{fit}}$ was
decreased. Although $n$ remained constant around 1 in $x=$0, 0.1 films below
$T_{\textrm{max}}^{\textrm{fit}}=0.5T_{\textrm{c}}$, $n$ tended to increase
with decreasing $T_{\textrm{max}}^{\textrm{fit}}$ in the other films,
resulting in $n\geq 2$ at low $T_{\textrm{max}}^{\textrm{fit}}$. Additionally,
in bulk FeSe,Okada  _et al._ ; Li  _et al._ (2016) which is in the nematic
phase same as $x=$0 0.1 films, $n$ was almost equal to 1. In bulk
FeSe0.4Te0.6,Takahashi  _et al._ (2011) which does not show nematic order, $n$
increased to over 2 with decreasing $T_{\textrm{max}}^{\textrm{fit}}$. As was
pointed out by Li $et\ al.$,Li  _et al._ (2016) the $T$-linear behavior
($n=1$) in $1/\tau$ may be the consequence of gap structure with line nodes or
deep gap minima.Li  _et al._ (2016); Hirschfeld  _et al._ (1993); Özcan  _et
al._ (2006) Conversely, the $n>2$ behavior observed for $x\geq 0.2$ films
could denote the nodeless superconducting gap, since an exponential decrease
in $1/\tau$ is expected in nodeless superconductor.Quinlan  _et al._ (1994);
Hashimoto  _et al._ (2009) Hence, considering the variation in $n$ in samples
with different Te contents, we consider that the superconducting gap structure
changes from those with line nodes or deep minima in nematic phase to those
that are nodeless outside the nematic phase. The result is consistent with
other measurement techniques claiming that bulk FeSe has line nodes or the
deep minima, Sprau  _et al._ (2017); Sun  _et al._ (2017) whereas bulk
FeSe1-xTex ($x>0.5$) shows nodeless superconducting gaps.Hanaguri  _et al._
(2010)

###  III.2 DFT calculations

Fig. 6: Fermi surface of FeSe in the (a) non-nematic phase and (b) nematic
phase obtained by DFT calculations. Band dispersion of FeSe in the (c) non-
nematic phase and (d) nematic phase. (e) Ground state energy of FeSe as a
function of $U$ in the non-nematic and nematic phases. (f) Averaged density of
states per unit cell of FeSe as a function of $U$ in the non-nematic and
nematic phases.

To interpret our experimental results with respect to electronic band
structure, we calculated the band dispersion of FeSe in both non-nematic and
nematic phases. Figure 6 (a) and (b) show the Fermi surface of FeSe in the
non-nematic and nematic phases calculated using $U=3$ eV. Figs. 6 (c) and (d)
show band dispersion along the points depicted in Figs. 6 (a) and (b). In the
nematic phase, the Fermi surface shows two-fold symmetry and the disappearance
of one of the electron pockets at the $M$ point (Fig. 6 [b]). Although the
size of Fermi surface was larger than that of experimentally observed one as
it is widely accepted,Liu  _et al._ (2015); Coldea and Watson (2018) the band
dispersion agrees with the previous calculation and qualitatively captures the
experimentally observed band structure.Long  _et al._ (2020); Yi  _et al._
(2019)

The disappearance of the electron pocket in the nematic phase has been
confirmed by several photoemission experiments.Rhodes  _et al._ (2018); Yi
_et al._ (2019); Rhodes  _et al._ (2020) It has been previously reported that,
when considering the absence of one electron pocket, the strongly anisotropic
superconducting gap of FeSe can be reproduced via calculation of the
superconducting gap equation.Rhodes  _et al._ (2018) Based on these results,
the observed nodal gap in FeSe1-xTex films with nematic order is probably
caused by the disappearance or shrinkage of the electron pocket due to the
nematic transition.

In Fig. 6 (e), the ground state energies of the non-nematic and nematic phases
are shown. The nematic phase became energetically favorable compared with the
non-nematic phase when $U$ increased to values greater than 3 eV. Furthermore,
we compared the averaged DOS near the Fermi surface, which determines the
carrier density. Figure 6 (f) shows DOS averaged over
$\epsilon_{\textrm{F}}\pm 10$ meV in the non-nematic and nematic phases, where
$\epsilon_{\textrm{F}}$ is the Fermi level. The averaged DOS in the nematic
phase is considerably lower than that in the non-nematic phase because of the
disappearance of one electron pocket. The difference in averaged DOS between
the non-nematic and nematic phases became larger with increasing $U$. The
decrease of DOS near the Fermi surface should result in an observed reduction
of normal-carrier density in the nematic phase.Nabeshima  _et al._ (2020)
Superfluid density is also considered to be reduced in the nematic phase,
resulting in a decrease of $T_{\textrm{c}}$ in the FeSe1-xTex films in nematic
phase. Of note, nematic order disappears at different Te content in Fe(Se,Te)
film on CaF2 substrate, LaAlO3 substrate and bulk crystal.Imai  _et al._
(2017); Terao  _et al._ (2019) This difference is possibly due to existence of
compression strain in Fe(Se,Te) films, which results in the difference in
electronic phase diagram of Fe(Se,Te) film and bulk crystal.

Fig. 7: Schematic phase diagram of FeSe1-xTex ($x=0-0.5$) films, summarizing
the experimental and calculated results.

Figure 7 summarizes the above results and discussion. The deformation of the
Fermi surface in the nematic phase induces strongly anisotropic
superconducting gaps with nodes or gap minima. We are not able to determine
whether the change in the gap structure takes place exactly at the nematic end
point composition or not, since it is rather difficult to change Te content so
finely to cover the nematic end point almost continuously. However, it can be
mentioned safely that there is the gross correspondence between the
presence/absence of the nematicity and the gap structure. In nematic phase,
superfluid density is suppressed because of the reduction of DOS near the
Fermi surface, resulting in a decrease of $T_{\textrm{c}}$. Although an origin
of the disappearance of nematic order by Te substitution is difficult to infer
from our results alone, a recent ARPES study on our films revealed an upward
shift of the $d_{xy}$ orbital with increasing Te content because of the change
of the chalcogen height.Nakayama  _et al._ (2021) The approach of the $d_{xy}$
orbital to Fermi energy reduces the relative contribution of the $d_{xz/yz}$
orbitals which is important for nematicity, resulting in instability of
nematic order. Also, we should comment on the difference in the behaviors of
the $T_{\textrm{c}}$ and other quantities such as DOS between Fe(Se,Te) films
and Fe(Se,S) films, which do not show a drastic increase of $T_{\textrm{c}}$
when nematic order disappears.Nabeshima  _et al._ (2018b) In Fe(Se,S) films,
our recent $\mu$-SR study showed the appearance of the short range magnetic
order at high S contents,Nabeshima  _et al._ (2018b, ) which is absent in Te
substituted films. This may be one of the possible origins to explain the
contrasting behavior of $T_{\textrm{c}}$ in Fe(Se,Te) and Fe(Se,S) films.
Another important feature for S-substituted system is the weakening of the
electronic correlation with increasing S content,Coldea  _et al._ (2019);
Kreisel  _et al._ (2020) which is again, in contrast to Te substituted
films.Nakajima  _et al._ In summary, phase diagrams of the FeSe1-xTex
($x=0-0.5$) films can be explained by considering their band structure as a
primary factor. This indicates that band deformation in the nematic phase
rather than the existence of the nematic order itself or possible nematic
fluctuations developing near the quantum critical point is the predominant
factor on superconductivity. Topological nature established in highly Te
substituted materials might be another important factor,Zhang  _et al._ (2018)
which is the subject of future works.

##  IV Conclusion

To conclude, we measured the complex conductivity of FeSe1-xTex ($x=0-0.5$)
films below $T_{\textrm{c}}$ combining coplanar waveguide resonator and cavity
perturbation techniques. In the presence of nematic order, the temperature
dependences of superfluid density and quasiparticle scattering time were
qualitatively distinct from those of films without nematic order. This
difference indicates that nematic order strongly influences the formation of
nodes or gap minima in its superconducting gap structure. Conversely, the
proportionality between $T_{\textrm{c}}$ and $\lambda_{0}^{-2}$ was observed
irrespective of the presence or absence of nematic order, suggesting that the
amount of superfluid exerts a more direct influence on the $T_{\textrm{c}}$ of
Fe(Se,Te) than the nematic order itself. Combining those results with the band
dispersion calculated based on DFT, we propose that the change of the Fermi
surface in the nematic phase is the main factor for changes of
$T_{\textrm{c}}$ and the corresponding superconducting gap structure in
Fe(Se,Te).

## Acknowledgments

This work was supported by JSPS KAKENHI Grant Numbers JP20H05164, JP19K14661.

## References

  * (1)
  * Liu  _et al._ (2015) X. Liu, L. Zhao, S. He, J. He, D. Liu, D. Mou, B. Shen, Y. Hu, J. Huang, and X. J. Zhou, Journal of Physics: Condensed Matter 27, 183201 (2015).
  * Böhmer and Kreisel (2018) A. E. Böhmer and A. Kreisel, Journal of Physics: Condensed Matter 30, 023001 (2018).
  * Kreisel  _et al._ (2020) A. Kreisel, P. J. Hirschfeld, and B. M. Andersen, Symmetry 12, 1402 (2020).
  * Hsu  _et al._ (2008) F.-C. Hsu, J.-Y. Luo, K.-W. Yeh, T.-K. Chen, T.-W. Huang, P. M. Wu, Y.-C. Lee, Y.-L. Huang, Y.-Y. Chu, D.-C. Yan, and M.-K. Wu, Proceedings of the National Academy of Sciences 105,14262 (2008).
  * Burrard-Lucas  _et al._ (2013) M. Burrard-Lucas, D. G. Free, S. J. Sedlmaier, J. D. Wright, S. J. Cassidy, Y. Hara, A. J. Corkett, T. Lancaster, P. J. Baker, S. J. Blundell, and S. J. Clarke, Nature Materials 12, 15 (2013).
  * Shiogai  _et al._ (2016) J. Shiogai, Y. Ito, T. Mitsuhashi, T. Nojima, and A. Tsukazaki, Nature Physics 12, 42 (2016).
  * Shikama  _et al._ (2020) N. Shikama, Y. Sakishita, F. Nabeshima, Y. Katayama, K. Ueno, and A. Maeda, Applied Physics Express 13, 083006 (2020).
  * Wang  _et al._ (2012) Q. Y. Wang, Z. Li, W. H. Zhang, Z. C. Zhang, J. S. Zhang, W. Li, H. Ding, Y. B. Ou, P. Deng, K. Chang, J. Wen, C. L. Song, K. He, J. F. Jia, S. H. Ji, Y. Y. Wang, L. L. Wang, X. Chen, X. C. Ma, and Q. K. Xue, Chinese Physics Letters 29, 037402 (2012).
  * Huang and Hoffman (2017) D. Huang and J. E. Hoffman, Annual Review of Condensed Matter Physics 8, 311 (2017).
  * Fernandes  _et al._ (2014) R. M. Fernandes, A. V. Chubukov, and J. Schmalian, Nature Physics 10, 97 (2014).
  * Glasbrenner  _et al._ (2015) J. K. Glasbrenner, I. I. Mazin, H. O. Jeschke, P. J. Hirschfeld, R. M. Fernandes, and R. Valentí, Nature Physics 11, 953 (2015).
  * Margadonna  _et al._ (2009) S. Margadonna, Y. Takabayashi, Y. Ohishi, Y. Mizuguchi, Y. Takano, T. Kagayama, T. Nakagawa, M. Takata, and K. Prassides, Physical Review B 80, 064506 (2009).
  * Fang  _et al._ (2008) M. H. Fang, H. M. Pham, B. Qian, T. J. Liu, E. K. Vehstedt, Y. Liu, L. Spinu, and Z. Q. Mao, Physical Review B 78, 224503 (2008).
  * Mizuguchi  _et al._ (2009) Y. Mizuguchi, F. Tomioka, S. Tsuda, T. Yamaguchi, and Y. Takano, Journal of the Physical Society of Japan 78, 074712 (2009).
  * Imai  _et al._ (2015) Y. Imai, Y. Sawada, F. Nabeshima, and A. Maeda, Proceedings of the National Academy of Sciences 112, 1937 (2015).
  * Nabeshima  _et al._ (2018a) F. Nabeshima, M. Kawai, T. Ishikawa, N. Shikama, and A. Maeda, Japanese Journal of Applied Physics 57, 120314 (2018a).
  * Watson  _et al._ (2015) M. D. Watson, T. K. Kim, A. A. Haghighirad, S. F. Blake, N. R. Davies, M. Hoesch, T. Wolf, and A. I. Coldea, Physical Review B 92, 121108 (2015).
  * Sato  _et al._ (2018) Y. Sato, S. Kasahara, T. Taniguchi, X. Xing, Y. Kasahara, Y. Tokiwa, Y. Yamakawa, H. Kontani, T. Shibauchi, and Y. Matsuda, Proceedings of the National Academy of Sciences 115, 1227 (2018).
  * Hanaguri  _et al._ (2018) T. Hanaguri, K. Iwaya, Y. Kohsaka, T. Machida, T. Watashige, S. Kasahara, T. Shibauchi, and Y. Matsuda, Science Advances 4, eaar6419 (2018).
  * Terao  _et al._ (2019) K. Terao, T. Kashiwagi, T. Shizu, R. A. Klemm, and K. Kadowaki, Physical Review B 100, 224516 (2019).
  * Hanaguri  _et al._ (2010) T. Hanaguri, S. Niitaka, K. Kuroki, and H. Takagi, Science 328, 474 (2010).
  * Kasahara  _et al._ (2014) S. Kasahara, T. Watashige, T. Hanaguri, Y. Kohsaka, T. Yamashita, Y. Shimoyama, Y. Mizukami, R. Endo, H. Ikeda, K. Aoyama, T. Terashima, S. Uji, T. Wolf, H. Von Löhneysen, T. Shibauchi, and Y. Matsuda, Proceedings of the National Academy of Sciences 111, 16309 (2014).
  * Sprau  _et al._ (2017) P. O. Sprau, A. Kostin, A. Kreisel, A. E. Böhmer, V. Taufour, P. C. Canfield, S. Mukherjee, P. J. Hirschfeld, B. M. Andersen, and J. C. Davis, Science 357, 75 (2017).
  * Imai  _et al._ (2017) Y. Imai, Y. Sawada, F. Nabeshima, D. Asami, M. Kawai, and A. Maeda, Scientific Reports 7, 46653 (2017).
  * Nabeshima  _et al._ (2018b) F. Nabeshima, T. Ishikawa, K.-i. Oyanagi, M. Kawai, and A. Maeda, Journal of the Physical Society of Japan 87, 073704 (2018b).
  * Nabeshima  _et al._ (2020) F. Nabeshima, T. Ishikawa, N. Shikama, and A. Maeda, Physical Review B 101, 184517 (2020).
  * Nakayama  _et al._ (2021) K. Nakayama, R. Tsubono, G. N. Phan, F. Nabeshima, N. Shikama, T. Ishikawa, Y. Sakishita, S. Ideta, K. Tanaka, A. Maeda, T. Takahashi, and T. Sato, Physical Review Research 3, L012007 (2021).
  * Takahashi  _et al._ (2011) H. Takahashi, Y. Imai, S. Komiya, I. Tsukada, and A. Maeda, Physical Review B 84, 132503 (2011).
  * (30) T. Okada, Y. Imai, T. Urata, Y. Tanabe, K. Tanigaki, and A. Maeda, arXiv:1801.00262 .
  * Mukasa  _et al._ (2021) K. Mukasa, K. Matsuura, M. Qiu, M. Saito, Y. Sugimura, K. Ishida, M. Otani, Y. Onishi, Y. Mizukami, K. Hashimoto, J. Gouchi, R. Kumai, Y. Uwatoko, and T. Shibauchi, Nature Communications 12, 381 (2021).
  * Imai  _et al._ (2010a) Y. Imai, T. Akiike, M. Hanawa, I. Tsukada, A. Ichinose, A. Maeda, T. Hikage, T. Kawaguchi, and H. Ikuta, Applied Physics Express 3, 043102 (2010a).
  * Imai  _et al._ (2010b) Y. Imai, R. Tanaka, T. Akiike, M. Hanawa, I. Tsukada, and A. Maeda, Japanese Journal of Applied Physics 49, 023101 (2010b).
  * Watanabe  _et al._ (1994) K. Watanabe, K. Yoshida, T. Aoki, and S. Kohjiro, Japanese Journal of Applied Physics 33, 5708 (1994).
  * Clem (2013) J. R. Clem, Journal of Applied Physics 113, 013910 (2013).
  * Jacob  _et al._ (2003) M. V. Jacob, J. Mazierska, D. Ledenyov, and J. Krupka, Journal of the European Ceramic Society 23, 2617 (2003).
  * Klein  _et al._ (1990) N. Klein, H. Chaloupka, G. Müller, S. Orbach, H. Piel, B. Roas, L. Schultz, U. Klein, and M. Peiniger, Journal of Applied Physics 67, 6940 (1990).
  * Drabeck  _et al._ (1990) L. Drabeck, K. Holczer, G. Grüner, J.-J. Chang, D. J. Scalapino, A. Inam, X. D. Wu, L. Nazar, and T. Venkatesan, Physical Review B 42, 10020 (1990).
  * Peligrad  _et al._ (1998) D.-N. Peligrad, B. Nebendahl, C. Kessler, M. Mehring, A. Dulčić, M. Požek, and D. Paar, Physical Review B 58, 11652 (1998).
  * Barannik  _et al._ (2014) A. A. Barannik, N. T. Cherpak, M. S. Kharchenko, Y. Wu, S. Luo, Y. He, and A. Porch, Low Temperature Physics 40, 492 (2014).
  * Long  _et al._ (2020) X. Long, S. Zhang, F. Wang, and Z. Liu, npj Quantum Materials 5, 50 (2020).
  * Khasanov  _et al._ (2009) R. Khasanov, M. Bendele, A. Amato, K. Conder, H. Keller, H. H. Klauss, H. Luetkens, and E. Pomjakushina, Physical Review Letters 104, 087004 (2009).
  * Kim  _et al._ (2010) H. Kim, C. Martin, R. T. Gordon, M. A. Tanatar, J. Hu, B. Qian, Z. Q. Mao, R. Hu, C. Petrovic, N. Salovich, R. Giannetta, and R. Prozorov, Physical Review B 81, 180503 (2010).
  * Biswas  _et al._ (2010) P. K. Biswas, G. Balakrishnan, D. M. Paul, C. V. Tomy, M. R. Lees, and A. D. Hillier, Physical Review B 81, 092510 (2010).
  * Luetkens  _et al._ (2008) H. Luetkens, H.-H. Klauss, R. Khasanov, A. Amato, R. Klingeler, I. Hellmann, N. Leps, A. Kondrat, C. Hess, A. Köhler, G. Behr, J. Werner, and B. Büchner, Physical Review Letters 101, 097009 (2008).
  * Prozorov and Kogan (2011) R. Prozorov and V. G. Kogan, Reports on Progress in Physics 74, 124505 (2011).
  * Ovchenkov  _et al._ (2019) Y. A. Ovchenkov, D. A. Chareev, V. A. Kulbachinskii, V. G. Kytin, S. V. Mishkov, D. E. Presnov, O. S. Volkova, and A. N. Vasiliev, Superconductor Science and Technology 32, 065005 (2019).
  * Sawada  _et al._ (2016) Y. Sawada, F. Nabeshima, Y. Imai, and A. Maeda, Journal of the Physical Society of Japan 85, 073703 (2016).
  * Yoshikawa  _et al._ (2019) N. Yoshikawa, M. Takayama, N. Shikama, T. Ishikawa, F. Nabeshima, A. Maeda, and R. Shimano, Physical Review B 100, 035110 (2019).
  * Liu  _et al._ (2009) T. J. Liu, X. Ke, B. Qian, J. Hu, D. Fobes, E. K. Vehstedt, H. Pham, J. H. Yang, M. H. Fang, L. Spinu, P. Schiffer, Y. Liu, and Z. Q. Mao, Physical Review B 80, 174509 (2009).
  * Sun  _et al._ (2014) Y. Sun, Y. Tsuchiya, T. Taen, T. Yamada, S. Pyon, A. Sugimoto, T. Ekino, Z. Shi, and T. Tamegai, Scientific Reports 4, 4585 (2014).
  * Li  _et al._ (2016) M. Li, N. R. Lee-Hone, S. Chi, R. Liang, W. N. Hardy, D. A. Bonn, E. Girt, and D. M. Broun, New Journal of Physics 18, 082001 (2016).
  * Hirschfeld  _et al._ (1993) P. J. Hirschfeld, W. O. Putikka, and D. J. Scalapino, Physical Review Letters 71, 3705 (1993).
  * Özcan  _et al._ (2006) S. Özcan, P. J. Turner, J. R. Waldram, R. J. Drost, P. H. Kes, and D. M. Broun, Physical Review B 73, 064506 (2006).
  * Quinlan  _et al._ (1994) S. M. Quinlan, D. J. Scalapino, and N. Bulut, Physical Review B 49, 1470 (1994).
  * Hashimoto  _et al._ (2009) K. Hashimoto, T. Shibauchi, S. Kasahara, K. Ikada, S. Tonegawa, T. Kato, R. Okazaki, C. J. Van Der Beek, M. Konczykowski, H. Takeya, K. Hirata, T. Terashima, and Y. Matsuda, Physical Review Letters 102, 207001 (2009).
  * Sun  _et al._ (2017) Y. Sun, A. Park, S. Pyon, T. Tamegai, and H. Kitamura, Physical Review B 96, 140505 (2017).
  * Coldea and Watson (2018) A. I. Coldea and M. D. Watson, Annual Review of Condensed Matter Physics 9, 125 (2018).
  * Yi  _et al._ (2019) M. Yi, H. Pfau, Y. Zhang, Y. He, H. Wu, T. Chen, Z. R. Ye, M. Hashimoto, R. Yu, Q. Si, D.-H. Lee, P. Dai, Z.-X. Shen, D. H. Lu, and R. J. Birgeneau, Physical Review X 9, 041049 (2019).
  * Rhodes  _et al._ (2018) L. C. Rhodes, M. D. Watson, A. A. Haghighirad, D. V. Evtushinsky, M. Eschrig, and T. K. Kim, Physical Review B 98, 180503 (2018).
  * Rhodes  _et al._ (2020) L. C. Rhodes, M. D. Watson, A. A. Haghighirad, D. V. Evtushinsky, and T. K. Kim, Physical Review B 101, 235128 (2020).
  * (62) F. Nabeshima, Y. Kawai, N. Shikama, Y. Sakishita, A. Suter, T. Prokscha, S. E. Park, S. Komiya, A. Ichinose, T. Adachi, and A. Maeda, arXiv:2103.07056 .
  * Coldea  _et al._ (2019) A. I. Coldea, S. F. Blake, S. Kasahara, A. A. Haghighirad, M. D. Watson, W. Knafo, E. S. Choi, A. McCollam, P. Reiss, T. Yamashita, M. Bruma, S. C. Speller, Y. Matsuda, T. Wolf, T. Shibauchi, and A. J. Schofield, npj Quantum Materials 4, 2 (2019).
  * (64) M. Nakajima, K. Yanase, M. Kawai, D. Asami, T. Ishikawa, F. Nabeshima, Y. Imai, A. Maeda, and S. Tajima, arXiv:2103.13659.
  * Zhang  _et al._ (2018) P. Zhang, K. Yaji, T. Hashimoto, Y. Ota, T. Kondo, K. Okazaki, Z. Wang, J. Wen, G. D. Gu, H. Ding, and S. Shin, Science 360, 182 (2018).


* Wing . (2014) Wing2014Wing, S., Johnson, JR., Chaston, CC., Echim, M., Escoubet, CP., Lavraud, B., Lemon, C., Nykyri, K., Otto, A., Raeder, J. Wang, CP.  2014\.  Review of Solar Wind Entry into and Transport Within the Plasma Sheet Review of solar wind entry into and transport within the plasma sheet.  Space Science Reviews1841-433-86.  0.1007/s11214-014-0108-9U2014
  * Winslow . (2015) Winslow2015Winslow, RM., Lugaz, N., Philpott, LC., Schwadron, NA., Farrugia, CJ., Anderson, BJ. Smith, CW.  2015\.  Interplanetary coronal mass ejections from MESSENGER orbital observations at Mercury Interplanetary coronal mass ejections from MESSENGER orbital observations at Mercury.  Journal of Geophysical Research: Space Physics12086101-6118.  10.1002/2015JA021200
  * Xu . (2020) Xu2020aXu, Z., Hartinger, MD., Oliveira, DM., Coyle, S., Clauer, CR., Weimer, D. Edwards, T.  2020\.  Inter-hemispheric asymmetries in the ground magnetic response to interplanetary shocks: The role of shock impact angle Inter-hemispheric asymmetries in the ground magnetic response to interplanetary shocks: The role of shock impact angle.  Space Weather183e2019SW002427.  10.1029/2019SW002427
  * Yagova . (2021) Yagova2021Yagova, NV., Pilipenko, VA., Sakharov, YA. Selivanov, VN.  2021\.  Spatial scale of geomagnetic Pc5/Pi3 pulsations as a factor of their efficiency in generation of geomagnetically induced currents Spatial scale of geomagnetic Pc5/Pi3 pulsations as a factor of their efficiency in generation of geomagnetically induced currents.  Earth, Planets and Space7388.  10.1186/s40623-021-01407-2
  * Zhou Tsurutani (2001) Zhou2001Zhou, X. Tsurutani, BT.  2001\.  Interplanetary shock triggering of nightside geomagnetic activity: Substorms, pseudobreakups, and quiescent events Interplanetary shock triggering of nightside geomagnetic activity: Substorms, pseudobreakups, and quiescent events.  Journal of Geophysical Research106A918957-18967.  10.1029/2000JA003028
  * Zong . (2012) Zong2012Zong, QG., Wang, YF., Zhang, H., Fu, SY., Zhang, H., Wang, CR., Yuan, CJ. Vogiatzis, I.  2012\.  Fast acceleration of inner magnetospheric hydrogen and oxygen ions by shock induced ULF waves Fast acceleration of inner magnetospheric hydrogen and oxygen ions by shock induced ULF waves.  Journal of Geophysical Research117A11.  10.1029/2012JA018024
  * Zong . (2021) Zong2021Zong, QG., Yue, C. Fu, SY.  2021\.  Shock Induced Strong Substorms and Super Substorms: Preconditions and Associated Oxygen Ion Dynamics Shock Induced Strong Substorms and Super Substorms: Preconditions and Associated Oxygen Ion Dynamics.  Space Science Reviews21733.  10.1007/s11214-021-00806-x


# Type I Chebyshev polynomials

U. Fidalgo  Department of Mathematics and Statistics, Case Western Reserve
University, Cleveland, Ohio 43403<EMAIL_ADDRESS>
###### Abstract.

We consider a sequence of type I star Chebyshev multiple orthogonal
polynomials. We find explicit formula for its components in terms of an
algebraic equation, and the strong asymptotic behaviors. We also study the
behavior of their roots’ distributions.

###### Key words and phrases:

Multiple orthogonal polynomials, Asymptotic behavior

###### 2000 Mathematics Subject Classification:

Primary 54C40, 14E20; Secondary 46E25, 20C20

##  1\. Introduction

This work concerns about type I Chebyshev polynomials. They were introduced in
[4], where we study a connection between bulk queuing theory and multi-
orthogonality of polynomials. Let $c$ and $m$ be two positive number with $m$
being integer. The sequence of type I Chebyshev polynomials corresponding to
$c\in\mathbb{R}_{+}\setminus\\{0\\}$ and
$m\in\mathbb{N}=\mathbb{Z}_{+}\setminus\\{0\\}$ is conformed by polynomial
vectors

(1.1) $\displaystyle\left\\{{\bf
t}_{r}=\left(t_{0,r},t_{1,r},\ldots,t_{m-1,r}\right)\right\\}_{r\in\mathbb{Z}_{+}}$

that satisfy the following recurrence relation

(1.2) $\begin{array}[]{l}{\bf t}_{-1}=\left(0,0,\ldots,0\right),\quad{\bf
t}_{0}=\left(1,0,\ldots,0\right),\ldots,{\bf
t}_{m-1}=\left(0,\ldots,0,1\right)\\\ \\\ c{\bf t}_{n}=x{\bf t}_{n-m}-{\bf
t}_{n-m-1},\quad n\in\mathbb{Z}_{+}\setminus\\{0,1,\ldots,m-1\\}.\end{array}$

The type I Chebyshev polynomials in (1.1) is certain sense dual to a very well
studied sequence of polynomials
$\displaystyle\left\\{T_{n}\right\\}_{n\in\mathbb{Z}_{+}}$ (see for instance
[2, 3]), that they are constructed with the recurrence relation

(1.3) $T_{n+1}=xT_{n}-cT_{n-m}(x),\quad n\in\mathbb{Z}_{+},$

and initial conditions

$T_{-m}=T_{-m+1}=\cdots=T_{-1}=0\quad\mbox{and}\quad T_{0}=1.$

Set two complex numbers $u$ and $v$. We denote by $[u,\,v]$ the compact
segment of straight line that connects $u$ with $v$. A real valued function
$r$ defined on a certain domain $D\subset\overline{\mathbb{C}}$ of the Riemann
sphere $\overline{\mathbb{C}}=\mathbb{C}\cup\\{\infty\\}$ is said to be a
weight if it does not change sign on $D$. This is
$r:D\subset\overline{\mathbb{C}}\rightarrow\mathbb{R}_{+}$ or
$r:D\subset\overline{\mathbb{C}}\rightarrow\mathbb{R}_{-}=(\mathbb{R}\setminus\mathbb{R}_{+})\cup\\{0\\}$.
Fixed $n\in\mathbb{Z}_{+}$ and $j\in\\{0,1,\ldots,m-1\\}$ the polynomial
$T_{n}$ satisfies the multi-orthogonality relations

(1.4) $0=\int x^{\nu}T_{n}(x)\,d\sigma_{j}(x),\quad\nu=0,1,\ldots,d_{j},$

where $n-j-1=d_{j}m+r$ $d\in\mathbb{Z}_{+}$ and $r\in\\{0,1,\ldots,m-1\\}$,
with $\sigma_{j}$, $j=0,1,\ldots,m-1$ being measures which have Radon Nikodym
derivatives

$\frac{d\sigma_{j}}{dx}=\rho_{j}(x)\rho(x),\quad j=0,1,\ldots,m-1,$

denoting $\rho_{0}\equiv 1$. The expression $\rho_{j}(x)\rho(x)$ is a function
defined on the starlike set

(1.5)
$S_{0}=\bigcup_{k=0}^{m}\left[0,\frac{m+1}{m}\sqrt[m+1]{mc}\exp\frac{2\pi
ik}{m+1}\right],$

where for each $k\in\\{0,1,\ldots,m\\}$ and each point

$x_{k}=t\exp\frac{2\pi ik}{m+1}\in S_{0},\quad 0\leq
t\leq\frac{m+1}{m}\sqrt[m+1]{mc},$

the following symmetry holds

$\rho_{j}\left(x_{k}\right)\rho\left(x_{k}\right)=\exp\left(-\frac{2\pi
ij}{m+1}\right)\rho_{j}\left(t\right)\rho\left(t\right),$

and $\displaystyle\rho_{j}\left(t\right)\rho\left(t\right)$ is a weight.

######  Theorem 1.1.

Consider the sequences $\left\\{{\bf t}_{r}\right\\}_{r\in\mathbb{Z}_{+}}$ and
$\displaystyle\left\\{T_{n}\right\\}_{n\in\mathbb{Z}_{+}}$ satisfying the
recurrence relations (1.2) and (1.3), respectively. Denote the function vector

$\displaystyle{\bf w}=\left(1,\rho_{1},\ldots,\rho_{m-1}\right)^{\top},$

and the linear form

$\mathcal{L}_{r}(x)={\bf t}_{r}(x){\bf
w}(x)=\sum_{j=0}^{m-1}t_{j,r}(x)\,\rho_{j}(x).$

Then

$\delta_{n,r}=\int
T_{n}(x)\,\mathcal{L}_{r}(x)\,d\rho(x),\quad(n,r)\in\mathbb{Z}_{+}^{2}.$

Note that the recurrence relations (1.2) and (1.3) have fixed coefficients,
when $m=1$ their elements

$\displaystyle\left\\{{\bf
t}_{r}=\left(t_{0,r},t_{1,r},\ldots,t_{m-1,r}\right)\right\\}_{r\in\mathbb{Z}_{+}}\quad\mbox{and}\quad\displaystyle\left\\{T_{n}\right\\}_{n\in\mathbb{Z}_{+}},$

are both reduced to the standard Chebyshev polynomials (see [1, Chapter 20]).
The terminology of type I and type II is currently used in the multi-
orthogonality of polynomials. The historical origin of these terms is not well
sharp. However their concepts were introduce by K. Mahler (see [6]) in the
context of simultaneous rational approximation during lectures delivered at
the University of Groningen in 1934-1935, when he was a refugee from Germany
in Netherland. Instead of using type I and type II, Mahler named them Latin
and German ones respectively, correlating these names with the fonts that they
used to denote their corresponding polynomials (Latin with Roman font and
German for Gothic font). The following Theorem 1.2 summarizes some algebraic
properties that the elements of the sequence $\\{{\bf
t}_{r}\\}_{r\in\mathbb{Z}_{+}}$ in (1.1) satisfy.

######  Theorem 1.2.

Consider the sequence of type I Chebyshev polynomials

$\mathcal{T}=\left\\{{\bf
t}_{r}=\left(t_{0,r},\ldots,t_{m-1,r}\right)\right\\}_{r\in\mathbb{Z}_{+}}$

as in (1.1) defined by the recurrence relation (1.2). For each
$r\in\mathbb{Z}_{+}$ we have that

(1.6) $t_{r}=t_{0,r}=t_{1,r+1}=t_{2,r+2}=\cdots=t_{m-1,r+m-1}.$

Set $r\in\mathbb{Z}_{+}$ with $r=dm+k$ where $d\in\mathbb{Z}_{+}$ and
$k\in\\{0,1,\ldots,m-1\\}$. Take $d-k=(m+1)\tau+\ell$ with
$\tau\in\mathbb{Z}_{+}\cup\\{-1\\}$ and $\ell\in\\{0,1,\ldots,m\\}$. There
exists a polynomial $h_{r}$ with degree $\tau$ and real coefficients such that
$t_{r}(z)=(-1)^{k}z^{\ell}h_{r}(z^{m+1})$. The polynomial elements of the
sequence $\displaystyle\left\\{h_{r}\right\\}_{r\in\mathbb{Z}_{+}}$ satisfy
the following initial conditions and recurrence relations, respectively

(1.7) $h_{0}=1,\quad h_{1}=\cdots=h_{m-1}=0$

and when $r\geq m$

(1.8) $c\,h_{r}(z)=\left\\{\begin{array}[]{l c
l}z\,h_{r-m}(z)+(-1)^{m-1}h_{r-m-1}(z)&\mbox{if}&\ell=0,\\\ &&\\\
h_{r-m}(z)+h_{r-m-1}(z)&\mbox{if}&\ell\in\\{1,\ldots,m\\}.\end{array}\right.$

We suspect that the roots of the polynomials $h_{r}$, $r\in\mathbb{Z}_{+}$ are
real and simple, which would imply that the $t_{j,r}$,
$(j,r)\in\\{0,\ldots,m-1\\}\times\mathbb{Z}_{+}$ have their simple zeros lying
in the starlike set corresponding to the $m+1$roots of real numbers. This
happens with the type II polynomials $T_{n}$, $n\in\mathbb{Z}_{+}$ (see in [2,
3, 4]). The way to prove such behavior for the $T_{n}$’s is using their
corresponding recurrence relations. Despite we have not been able to obtain an
analogous result for type I polynomials, we show that the roots of $t_{j,r}$
tend to accumulate near the mentioned starlike set as $r\to\infty$.

Let us denote $[0,\infty)\times\exp\left(i\theta\right),$
$\theta\in\mathbb{R}$ the ray that starts at the origin with slope
$\tan\theta$. With the expression
$[-\infty,\infty)\times\exp\left(i\theta\right)$ we refer the straight line
with slope $\tan\theta$ containing the origin. Let us declare the following
starlike sets

(1.9) $\begin{array}[]{l}\displaystyle
S_{0}=\bigcup_{k=0}^{m}\left[0,a\exp{\frac{2\pi ik}{m+1}}\right],\quad\mbox{we
introduced this set in (\ref{scero})}\\\ \\\ \displaystyle
S_{\infty}^{e}=\bigcup_{k=0}^{m}[-\infty,\,\infty)\times\exp\left(\frac{(2k+1)i\pi}{m+1}\right),\\\
\\\ \displaystyle
S_{\infty}^{o}=\bigcup_{k=0}^{m}[0,\infty)\times\exp\left(\frac{2ki\pi}{m+1}\right).\end{array}$

We denote

$\Omega_{0}=\mathbb{C}\setminus
S_{0},\,\,\Omega_{j}=\mathbb{C}\setminus\\{S_{\infty}^{o}\cup
S_{\infty}^{e}\\},\,\,1\leq
j<m\,\,\mbox{and}\,\,\Omega_{m}=\left\\{\begin{array}[]{l
l}\mathbb{C}\setminus S_{\infty}^{e}&\mbox{if }m\mbox{ even},\\\
\mathbb{C}\setminus S_{\infty}^{o}&\mbox{if }m\mbox{ odd},\end{array}\right.$

where

$a=\frac{m+1}{m}\left(\frac{c}{m}\right)^{m/m+1}.$

We consider the algebraic equation

(1.10) $c\,\lambda^{m+1}(z)-z\lambda(z)+1=0.$

Observe that

(1.11) $\omega=1/\lambda$

is a solution of the equation $\omega^{m+1}-z\omega^{m}+c=0$, which is very
well studied in [2, 3]. The following statement can be easily shown making the
change of variable (1.11) in [4, Proposition 2.1] (which summarizes some
results of [2]).

There is a multivalued solution $\lambda(z)$ of the equation (1.10) which is
an algebraic function of order $m+1$ with global branches of $\lambda(z)$ in
(1.10) satisfying for each $\displaystyle j\in\left\\{0,1,\ldots,m\right\\}$,
that $\lambda_{j}$ is a holomorphic function on $\Omega_{j}$ (Notation:
$\lambda_{j}\in\mathcal{H}(\Omega_{j})$) and

(1.12) $|\lambda_{0}|<|\lambda_{1}|<\cdots<|\lambda_{m}|,\quad\mbox{on
}\,D=\bigcup_{j=0}^{m}\Omega_{j}.$

They also behave asymptotically

(1.13)
$\begin{array}[]{l}\displaystyle\lambda_{0}(z)=\frac{1}{z}+\mathcal{O}\left(\frac{1}{z^{2}}\right)\quad\mbox{as}\,z\to\infty,\\\
\\\ \displaystyle\lim_{z\in
D,\,z\to\infty}\frac{c\,\lambda_{j}^{m}(z)}{z}=1,\quad
j=1,\ldots,m.\end{array}$

######  Theorem 1.3.

Consider the sequence
$\displaystyle\left\\{t_{r}\right\\}_{r\in\mathbb{Z}_{+}}$ whose elements are
the polynomials in (1.6). Then

(1.14) $\lim_{r\to\infty}\frac{t_{r}}{\lambda_{m}^{r}}=\frac{\displaystyle
1}{\displaystyle
cm-\frac{1}{\lambda_{m}^{m+1}}}\in\mathcal{H}\left(\mathbb{C}\setminus
S_{m}\right),$

uniformly in any compact subset of $\overline{\mathbb{C}}\setminus S_{m}$.

Since the functions $\lambda_{m}$ is a solution of the algebraic equation
(1.10), it does not vanish in $\mathbb{C}\setminus S_{m}$, hence the equality
(1.14) in Theorem 1.3 implies the set $S_{m}$ attracts the roots of the
polynomials $t_{r}$, as $r\to\infty$. A prescedent of this result can be found
in [5]. In the context of Nikishin systems of measures supported on the real
line (see [7]), the authors proved that all the zeros of type I polynomials
are lying in a set which is analogous to our $S_{m}$. They use the structure
of Nikishin systems to prove this statement about the zeros distribution. In
the proof of Theorem 1.3 in Section 4 we observe that this result comes from
the order stablished by the inequalities (1.12) between the solutions
corresponding to the algebraic equation (1.10), which is the one associated to
the recurrence relation in (1.2).

In Section 2 we prove Theorem 1.1, Theorem 1.2 in Section 3 and the proof of
Theorem 1.3 is in Section 4.

##  2\. Proof of Theorem 1.1

Let us consider the associated linear operator ${\bf T}$ acting on the space
of sequences $\ell^{2}$, which is determined by a infinite non-symmetric
matrix $\displaystyle{\bf T}=[\tau_{i,j}]_{(i,j)\in\mathbb{Z}_{+}^{2}}$ with
entries satisfying

$\tau_{i,j}=\left\\{\begin{array}[]{r c l}c&\mbox{if}&j=i-m,\\\ &&\\\
1&\mbox{if}&j=i+1,\\\ &&\\\
0&\mbox{if}&j\not\in\\{i+1,i-m\\},\end{array}\right.\quad(i,j)\in\mathbb{Z}_{+}^{2},$

with schematic form

(2.1) ${\bf T}=\left(\begin{array}[]{c c c c c c c c
c}0&1&0&\cdots&0&0&0&0&\cdots\\\ 0&0&1&&0&0&0&0&\\\
\vdots&\ddots&\ddots&\ddots&\vdots&&\vdots&\vdots&\\\ &&&&&&&&\\\
0&0&\cdots&0&0&1&0&0&\cdots\\\ c&0&\cdots&0&0&0&1&0&\ddots\\\
0&c&&0&0&0&0&1&\ddots\\\ \vdots&\ddots&\ddots&&\ddots&&\ddots&\ddots&\ddots\\\
\end{array}\right).$

We normally write the elements of $\ell^{2}$ as infinite column vectors. Set
$\\{{\bf e}_{j}\\}_{j\in\mathbb{Z}_{+}}$ the standard basis of $\ell^{2}$,
then the operator ${\bf T}$ is defined as follows

(2.2) ${\bf T}^{\top}{\bf e}_{j}=\left\\{\begin{array}[]{l l l}{\bf
e}_{j+1}&\mbox{if}&j\in\\{0,1,\ldots,m-1\\},\\\ &&\\\ {\bf e}_{j+1}+c{\bf
e}_{j-m}&\mbox{if}&j\geq m.\end{array}\right.$

Let $p(x)=b_{\kappa}\,x^{\kappa}+\cdots+b_{1}\,x+b_{0}$ be a polynomial with
an arbitrary degree $\kappa$. Then $p({\bf T}^{\top})$ denotes the operator

$p({\bf T}^{\top}){\bf u}=\left[b_{\kappa}\left({\bf
T}^{\top}\right)^{\kappa}+\cdots+b_{1}\,{\bf T}^{\top}+b_{0}\right]{\bf
u}=b_{\kappa}\left({\bf T}^{\top}\right)^{\kappa}{\bf u}+\cdots+b_{1}\,{\bf
T}^{\top}{\bf u}+b_{0}{\bf u},\quad{\bf u}\in\ell^{2},$

where $\top$ denotes the operation of matrix transposition.

Set two sequences ${\bf
u}=\left(u_{0},u_{1},u_{2},\ldots\right)^{\top}\in\ell^{2}$ and ${\bf
v}=\left(v_{0},v_{1},v_{2},\ldots\right)^{\top}\in\ell^{2}$, we denote its
inner (dot) product by

${\bf u}\cdot{\bf v}={\bf u}^{\top}{\bf v}=\sum_{j=0}^{\infty}u_{j}\,v_{j}.$

From several works (see [2, 3, 4]) we have the following Lemma

######  Lemma 2.1.

Given a polynomial $p$, for each $\displaystyle
j\in\left\\{1,\ldots,m\right\\}$ we obtain that

${\bf e}_{j}\cdot p({\bf T}^{\top}){\bf e}_{0}={\bf e}_{j}^{\top}p({\bf
T}^{\top}){\bf e}_{0}=\int p(x)\rho_{j}(x)\rho(x)dx.$

The following result was proved in [4] in a more general situation. In order
to make easy the reading we write its proof focussing in our case of the
operator ${\bf T}$ and its corresponding sequence of polynomials
$\displaystyle\left\\{T_{n}\right\\}_{n\in\mathbb{Z}_{+}}$

######  Lemma 2.2.

Consider the sequence of polynomials
$\displaystyle\left\\{T_{n}\right\\}_{n\in\mathbb{Z}_{+}}$ satisfying the
recurrence relation (1.3). Then

${\bf e}_{n}=T_{n}({\bf T}^{\top})\,{\bf e}_{0},\quad n\in\mathbb{Z}_{+}.$

###### Proof.

Observe that for each $\displaystyle n\in\left\\{1,\ldots,m-1\right\\}$ we
have that

$T_{n}(x)=xT_{n-1}-c\,0=x(xT_{n-2}-c\,0)=\cdots=x(\cdots(xT_{0}-c\,0)\cdots)=x^{n}.$

This implies that $\displaystyle T_{n}({\bf T}^{\top})=T^{n}$, hence

$\begin{array}[]{r c l}T_{0}({\bf T}^{\top})\,{\bf e}_{0}&=&\displaystyle{\bf
T}^{\top 0}{\bf e}_{0}={\bf e}_{0},\\\ &&\\\ T_{1}({\bf T}^{\top})\,{\bf
e}_{0}&=&\displaystyle{\bf T}^{\top}{\bf e}_{0}={\bf e}_{1},\\\ &&\\\
T_{2}({\bf T}^{\top})\,{\bf e}_{0}&=&\displaystyle{\bf T}^{\top 2}{\bf
e}_{0}={\bf T}{\bf e}_{1}={\bf e}_{2},\\\ &&\\\ &\vdots&\\\ &&\\\ T_{m-1}({\bf
T}^{\top})\,{\bf e}_{0}&=&\displaystyle{\bf T}^{\top m-1}{\bf e}_{0}={\bf
T}^{\top m-2}{\bf e}_{1}=\cdots={\bf T}^{\top}{\bf e}_{m-2}={\bf
e}_{m-1}.\end{array}$

We now consider $n\geq m-1$. For each $k=1,\ldots,n$, assume that $T_{k}({\bf
T}^{\top})\,{\bf e}_{0}={\bf e}_{k}$. By the recurrence relation (1.3), we
have that

$T_{n+1}({\bf T}^{\top}){\bf e}_{0}={\bf T}^{\top}T_{n}({\bf T}^{\top}){\bf
e}_{0}-cT_{n-m}({\bf T}^{\top}){\bf e}_{0}={\bf T}^{\top}{\bf e}_{n}-c{\bf
T}^{\top}{\bf e}_{n-m}.$

hence

$T_{n+1}({\bf T}^{\top}){\bf e}_{0}={\bf e}_{n+1}+c\,{\bf e}_{j-m}-c\,{\bf
e}_{j-m}={\bf e}_{n+1}.$

This completes the proof by induction. ∎

######  Remark 2.3.

Combining Lemma 2.1 and Lemma 2.2 we obtain the equalities in (1.4).

We now study the sequence of polynomial vectors $\displaystyle\left\\{{\bf
t}_{r}=\left(t_{0,r},\ldots,t_{m-1,r}\right)\right\\}_{r\in\mathbb{Z}_{+}}$
which is defined by the recurrence relation (1.2). Set an arbitrary polynomial
vector $\displaystyle{\bf p}=\left(p_{0},\ldots,p_{m-1}\right)$, let us denote
for every system ${\bf U}=\left({\bf u}_{0},\ldots,{\bf u}_{m-1}\right)$ with
${\bf u}_{j}\in\ell^{2}$, $j=0,\ldots,m-1$ the ”dot” product

${\bf p}({\bf T})\bigodot{\bf U}=\sum_{j=0}^{m-1}p_{j}({\bf T}){\bf u}_{j}.$

We also make explicit

(2.3) ${\bf T}{\bf e}_{j}={\bf e}_{j-1}+a{\bf e}_{j+m},\quad
j\in\mathbb{Z}_{+}.$

######  Lemma 2.4.

Let $\displaystyle{\bf E}=\left({\bf e}_{0},\ldots,{\bf e}_{m-1}\right)$
denote. Then

${\bf e}_{r}={\bf t}_{r}({\bf T})\bigodot{\bf E},\quad r\in\mathbb{Z}_{+}.$

###### Proof.

We observe that for each $\displaystyle r\in\left\\{0,1,\ldots,m-1\right\\}$

${\bf t}_{r}({\bf T})\bigodot{\bf
E}=(\overset{r+1}{\overbrace{0,\ldots,0}},1,0,\ldots,0)\bigodot({\bf
e}_{0},\ldots,{\bf e}_{m-1})={\bf e}_{r}.$

Let us consider $r\geq m-1$. For each $k=0,\ldots,r-1$, let us assume that
$\displaystyle{\bf t}_{k}({\bf T})\bigodot{\bf E}={\bf e}_{k}$. By the
recurrence relation (1.2), we have that

${\bf t}_{r}({\bf T})\bigodot{\bf E}=\frac{1}{c}\left[{\bf T}{\bf
t}_{r-m}({\bf T})\bigodot{\bf E}-{\bf t}_{r-m-1}({\bf T})\bigodot{\bf
E}\right]$ $=\frac{1}{c}\left[{\bf T}{\bf e}_{r-m}-{\bf
e}_{r-m-1}\right]=\frac{1}{c}\left[c{\bf e}_{r}+{\bf e}_{r-m-1}-{\bf
e}_{r-m-1}\right]={\bf e}_{r}.$

The proof is completed by induction. ∎

In $\ell^{2}$ we consider the regular inner product. This means that given two
sequences ${\bf u}=\left\\{u_{j}\in\mathbb{R}\right\\}_{j\in\mathbb{Z}_{+}}$
and ${\bf v}=\left\\{v_{j}\in\mathbb{R}\right\\}_{j\in\mathbb{Z}_{+}}$ of
$\ell^{2}$

${\bf u}\cdot{\bf v}=\sum_{j=0}^{\infty}u_{j}v_{j}={\bf u}{\bf v}^{\top},$

where ${\bf v}^{\top}$ represents the transposition operator of ${\bf v}$.

######  Lemma 2.5.

Set ${\bf E}=({\bf e}_{0},\ldots,{\bf e}_{m-1})$. Given an order pair
$(n,r)\in\mathbb{Z}_{+}^{2}$, the polynomial ${\bf t}_{r}$ in (1.2) and
$T_{n}$ in (1.3) satisfy the orthogonality relations

(2.4) ${\bf t}_{r}({\bf T})\bigodot{\bf E}\cdot T_{n}({\bf T}^{\top}){\bf
e}_{0}=\delta_{n,r}.$

###### Proof.

From Lemma 2.2 and Lemma 2.4 we have immediately

${\bf t}_{r}({\bf T})\bigodot{\bf E}\cdot T_{n}({\bf T}^{\top}){\bf
e}_{0}={\bf e}_{r}\cdot{\bf e}_{n}=\delta_{r,n}.$

This proves (2.4). ∎

The proof of Theorem 1.1 is completed combining Lema 2.1 and Lemma 2.5.

##  3\. Proof of Theorem 1.2

We stat by proving the equalities in (1.6). Fix $j\in\\{0,1,\ldots,m\\}$. The
recurrence relations (1.2) can be written as

(3.1) $t_{j,r}\equiv 0,\quad
r\in\\{-1,0,\ldots,j-1,j+1,\ldots,m-1\\}\quad\mbox{and}\quad t_{j,j}(z)=1,$

and

$c\,t_{j,r}=x\,t_{j,r-m}-t_{j,r-m-1},\quad
r\in\mathbb{Z}_{+}\setminus\\{0,1,\ldots,m-1\\}.$

We show that the expressions above are equivalent to the following

(3.2) $t_{j,r}\equiv 0,\quad r\in\\{-m+j-1,\ldots,j-1\\}\quad\mbox{and}\quad
t_{j,j}(z)=1,$

and

(3.3) $c\,t_{j,r}=x\,t_{j,r-m}-t_{j,r-m-1},\quad
r\in\mathbb{Z}_{+}\setminus\\{-m+j-1,\ldots,j-1\\}.$

Given $\displaystyle j\in\left\\{-1,0,1,\ldots,m-1\right\\}$, the conditions
(3.1) and (3.2) coincide in

$t_{j,r}=0,\quad r\in\\{-1,0,\ldots,j-1\\}\quad\mbox{and}\quad t_{j,j}(z)=1.$

We declare that

$t_{j,r}=0,\quad r\in\\{-m+j-1,\ldots,-2\\},$

which does not contradict (3.1). Taking into account (3.3) we obtain that

$c\,t_{j,r}=x\,t_{j,r-m}-t_{j,r-m-1}=0,\quad r\in\\{j+1,\ldots,m+j-1\\}.$

This shows the equalities in (3.2). Hence given $j\in\\{0,\ldots,m-2\\}$ the
sequences $\displaystyle\left\\{t_{j,r}\right\\}_{r\in\mathbb{Z}_{+}}$ and
$\displaystyle\left\\{t_{j+1,r}\right\\}_{r\in\mathbb{Z}_{+}}$ satisfy the
same recurrence relation (3.3) and according to (3.2) differ in the initial
conditions with the shift

$t_{j,r}=t_{j+1,r+1}=0,\quad r\in\\{-m+j-1,\ldots,j-1,j+1,\ldots,m+j-1\\},$

and

$t_{j,j}=t_{j+1,j+1}=1.$

This proves (1.6).

From (1.6) we conclude that any sequence
$\displaystyle\left\\{t_{j,r}\right\\}_{r\in\mathbb{Z}_{+}}$,
$j\in\\{0,1,\ldots,m-1\\}$ contains all the information of the polynomial
vectors $\displaystyle\left\\{{\bf t}_{r}\right\\}_{r\in\mathbb{Z}_{+}}$. We
then focus on
$\displaystyle\left\\{t_{0,r}=t_{r}\right\\}_{r\in\mathbb{Z}_{+}},$ whose
elements, according to (1.2) satisfy the recurrence relation

(3.4) $ct_{r}(z)=zt_{r-m}(z)-t_{r-m-1},\quad r\in\mathbb{Z}_{+}.$

According to the initial conditions in the recurrence relations (1.2) we have
that ${\bf t}_{0}=\left(1,0,\ldots,0\right)$. Taking now into account (1.6) we
obtain that $t_{0}=1$ and $t_{j}=0$, $j=1,\ldots,m-1$, which proves (1.7).
When $r=0$ $d=k=0$, hence $\tau=0$ and $\ell=0$, which is coherent with the
fact $\deg t_{0}=0$. We observe that if $r\in\\{1,\ldots,m-1\\}$, then $d=0$,
$r=k$ and $d-k=-k=-(m+1)+\ell$, hence $\tau=-1$, which agrees with $h_{r}=0$,
$r\in\\{1,\ldots,m-1\\}$. We also see that when $r=0=m\cdot 0+0$,
$t_{0}=1=(-1)^{0}h_{0}$ with $h_{0}=1$.

In oder to prove (1.8) we use the induction method. Fix $r=dm+k\geq m$ which
implies that $r-m=(d-1)m+k$, $k\in\\{0,1,\ldots,m-1\\}$ and

$r-m-1=\left\\{\begin{array}[]{l c l}(d-2)m+m-1&\mbox{if}&k=0,\\\ &&\\\
(d-1)m+k-1&\mbox{if}&k\in\\{1,\ldots,m-1\\}.\end{array}\right.$

Set $d-k=(m+1)\tau+\ell$. We obtain that

$(d-1)-k=\left\\{\begin{array}[]{l c l}(m+1)(\tau-1)+m&\mbox{if}&\ell=0,\\\
&&\\\ (m+1)\tau+\ell-1&\mbox{if}&\ell\in\\{1,\ldots,m\\},\end{array}\right.$

If $k\not=0$ we are in the situation of $(d-1)-(k-1)=d-k=(m+1)\tau+\ell$, and
when $k=0$ we have that $d-2-m+1=d-(m+1)=(m+1)(\tau-1)+\ell,$
$\ell\in\\{0,1,\ldots,m\\}$. As induction hypothesis we assume that

$t_{r-m}(z)=\left\\{\begin{array}[]{l l l}(-1)^{k}z^{m}h_{r-m}(z^{m+1}),&\deg
h_{r-m}=\tau-1,&\ell=0.\\\ &&\\\ (-1)^{k}z^{\ell-1}h_{r-m}(z^{m+1}),&\deg
h_{r-m}=\tau,&\ell\in\\{1,\ldots,m\\}.\end{array}\right.$

and

$t_{r-m-1}(z)=\left\\{\begin{array}[]{l l
l}(-1)^{m-1}z^{\ell}h_{r-m-1}(z^{m+1})&\mbox{if}&k=0,\\\ &&\\\
(-1)^{k-1}z^{\ell}h_{r-m-1}(z^{m+1})&\mbox{if}&k\not=0,\end{array}\right.\quad\ell\in\\{0,1,\ldots,m\\},$
$\deg h_{r-m-1}=\left\\{\begin{array}[]{l l l}\tau-1&\mbox{if}&k=0,\\\ &&\\\
\tau&\mbox{if}&k\not=0.\end{array}\right.$

We divide the proof in four cases considering the different values of $k$ and
$\ell$.

  1.   2. (1)

Case 1 $k=0$ and $\ell=0$ According to (3.4) we obtain the following chain of
equalities

$c\,t_{r}(z)=zt_{r-m}(z)-t_{r-m-1}=$
$=(-1)^{0}zz^{m}h_{r-m}(z^{m+1})-(-1)^{m-1}h_{r-m-1}(z^{m+1})$
$=z^{m+1}h_{r-m}(z^{m+1})+(-1)^{m-1}h_{r-m-1}(z^{m+1}),$

We observe that we can write $at_{r}(z)=ah_{r}(z^{m+1})$ which satisfies that

$c\,h_{r}(z)=zh_{r-m}(z)-(-1)^{m-1}h_{r-m-1}(z).$

Since $\deg h_{r-m}=\deg h_{r-m-1}=\tau-1$ with $\deg h_{r}=1+\deg
h_{r-m}=1+\tau-1=\tau$. The statement is satisfied in this case.

  3.   4. (2)

Case 2 $k=0$ and $\ell\not=0$ Let us consider (3.4) again.

$c\,t_{r}(z)=zt_{r-m}(z)-t_{r-m-1}=$
$=(-1)^{0}zz^{\ell-1}h_{r-m}(z^{m+1})-(-1)^{m-1}z^{\ell}h_{r-m-1}(z^{m+1})$
$=z^{\ell}\left[h_{r-m}(z^{m+1})-(-1)^{m-1}h_{r-m-1}(z^{m+1})\right]$

This implies that $t_{r}=z^{\ell}h_{r}(z^{m+1})$, with

$c\,h_{r}(z)=h_{r-m}(z)-(-1)^{m-1}h_{r-m-1}(z)$

Since $\deg h_{r-m}=\tau$ and $\deg h_{r-m-1}=\tau-1$, the polynomial $h_{r}$
has degree $\tau$.

  5.   6. (3)

Case 3 $k\not=0$ and $\ell=0$ From (3.4) we have that

$ct_{r}(z)=zt_{r-m}(z)-t_{r-m-1}$
$=z(-1)^{k}z^{m}h_{n-m}(z^{m+1})-(-1)^{k-1}h_{r-m-1}(z^{m+1})$
$(-1)^{k}\left[z^{m+1}h_{r-m}(z^{m+1})+h_{r-m-1}(z^{m+1})\right]$

We write $t_{r}=(-1)^{k}h_{r}(z^{m+1})$ where

$ch_{r}(z)=zh_{r-m}(z)+h_{r-m-1}(z)].$

Then $\deg h_{r}=\deg zh_{r-m}=\deg h_{r-m-1}=\tau$.

  7.   8. (4)

Case 4 $k\not=0$ and $\ell\not=0$ In this last case we also star by
considering the recurrence relation (3.4), hence

$ct_{r}(z)=zt_{r-m}(z)-t_{r-m-1}$
$=z(-1)^{k}z^{\ell-1}h_{r-m}(z^{m+1})-(-1)^{k-1}z^{\ell}h_{r-m-1}(z^{m+1})$
$=(-1)^{k}z^{\ell}\left[h_{r-m}(z^{m+1})+h_{r-m-1}(z^{m+1})\right],$

hence $t_{r}=(-1)^{k}z^{\ell}h_{r}(z^{m+1})$ with

$c\,h_{r}(z)=h_{r-m}(z)+h_{r-m-1}(z).$

Since $\deg h_{r+m}=\deg h_{r+m}=\tau$, then $\deg h_{r}=\tau$.

  9.

Taking into account the four cases, we complete the proof of Theorem 1.2.
$\Box$

##  4\. Proof of Theorem 1.3

Let us observe that the following function satisfies the recurrence relation
(3.4)

(4.1)
$t_{r}=b_{0}\lambda_{0}^{r+m}+b_{1}\,\lambda_{1}^{r+m}+\cdots+b_{m}\lambda_{m}^{r+m},\quad
r\in\mathbb{Z}_{+}$

The coefficients $b_{0}(z),\ldots,b_{m}(z)$ are defined from the initial
conditions

$t_{r}=0,\quad r=-m,-m+1,\ldots,-1,\quad\mbox{and}\quad t_{0}=1,$

which implies that

$\left\\{\begin{array}[]{l l}b_{0}+b_{1}+b_{2}+\cdots+b_{m}&=0\\\ &\\\
\displaystyle
b_{0}\lambda_{0}+b_{1}\lambda_{1}+b_{2}\lambda_{2}+\cdots+b_{m}\lambda_{m}&=0\\\
&\\\ \cdots\cdots\cdots\cdots\cdots\cdots&\\\ &\\\ \displaystyle
b_{0}\lambda_{0}^{m-1}+b_{1}\lambda_{1}^{m-1}+b_{2}\lambda_{2}^{m-1}+\cdots+b_{m}\lambda_{m}^{m-1}&=1.\end{array}\right.$

Set ${\bf L}\,{\bf b}={\bf e}_{m}$ the matrix expression of the linear system
of equations above, where components of the vector ${\bf b}$ has the
coefficients $b_{j}$, $\displaystyle j\in\\{0,1,\ldots,m\\}$ and ${\bf W}$ is
a $(m+1)\times(m+1)$ matrix of ${\bf D}$ with forms

${\bf L}=\left(\begin{array}[]{c c c c}1&1&\cdots&1\\\
\lambda_{0}&\lambda_{1}&\cdots&\lambda_{m}\\\ \vdots&\vdots&\ddots&\vdots\\\
\lambda_{0}^{m-1}&\lambda_{1}^{m-1}&\cdots&\lambda_{m}^{m-1}\end{array}\right)\quad{\bf
b}=\left(\begin{array}[]{ c}b_{0}\\\ b_{1}\\\ \vdots\\\
b_{m}\end{array}\right),\quad\mbox{and}\quad{\bf e}_{m}=\left(\begin{array}[]{
c}0\\\ \vdots\\\ 0\\\ 1\end{array}\right).$

From Cramer’s rule we find

(4.2)
$b_{j}=\frac{1}{\displaystyle\lambda_{j}\prod_{k=0,k\not=j}^{m}(\lambda_{j}-\lambda_{k})},\qquad
j=0,\ldots,m.$

Let us denote the polynomial in $\zeta\in\mathbb{C}$:

$P_{z}(\zeta)=c\prod_{k=0}^{m}(\zeta-\lambda_{k})=c\zeta^{m+1}-z\,\zeta+1,$

hence for each $\displaystyle j\in\left\\{0,1,\ldots,m\right\\}$

$P_{z}^{\prime}\left(\lambda_{j}\right)=\prod_{k=0,k\not=j}^{m}\left(\lambda_{j}-\lambda_{k}\right)=c\,(m+1)\,\lambda_{j}^{m}-z.$

Since

$P_{z}(\lambda_{j})=c\prod_{k=0}^{m}(\lambda_{j}-\lambda_{k})=c\lambda_{j}^{m+1}-z\,\lambda_{j}+1=0,$

we have that

(4.3)
$\lambda_{j}P_{z}^{\prime}\left(\lambda_{j}\right)=c\,(m+1)\,\lambda_{j}^{m+1}-z\lambda_{j}=cm\lambda_{j}^{m+1}-1.$

Observe that $\lambda_{j}P_{z}^{\prime}\left(\lambda_{j}\right)$ never
vanishes in $\mathbb{C}\setminus S_{j}$. This the reason why in the statement
of Theorem 1.3 we write that

$\frac{1}{\displaystyle
cm-\frac{1}{\lambda_{m}^{m+1}}}\in\mathcal{H}\left(\mathbb{C}\setminus
S_{m}\right).$

We substitute the expression (4.3) in (4.2)

$b_{j}=\frac{\lambda_{m}}{\displaystyle cm\lambda_{j}^{m+1}-1},\qquad
j=0,\ldots,m.$

Considering (4.1) we obtain that

$t_{r}=\sum_{j=0}^{m}\frac{\displaystyle\lambda_{m}\,\lambda_{j}^{r+m}}{\displaystyle
cm\lambda_{j}^{m+1}-1}.$

We observe that

$\frac{t_{r}}{\lambda_{m}^{r}}=\sum_{j=0}^{m}\frac{\displaystyle}{\displaystyle
cm-\frac{1}{\lambda_{m}^{m+1}}}\left(\frac{\lambda_{j}}{\lambda_{m}}\right)^{r-1},$

hence

$\lim_{r\to\infty}\frac{t_{r}}{\lambda_{m}^{r}}=\frac{\displaystyle
1}{\displaystyle cm-\frac{1}{\lambda_{m}^{m+1}}},$

uniformly in any compact subset of $\overline{\mathbb{C}}\setminus S_{m}$. The
proof of Theorem 1.3 is then completed. $\Box$

## References

  * [1] M. Abramowitz, and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York, 9th Edition, (1964).
  * [2] A.I. Aptekarev, V. A. Kalyagin, and E.B. Saff. Higher-Order Three-Term Recurrences and Asymptotics of Multiple Orthogonal Polynomials. Constr. Approx. 30 (2009), 175-223.
  * [3] A.I. Aptekarev, V. A. Kalyagin, and V.A. Van Iseghem. The genetic sum’s representation for the moments of a system of Stieltjes functions and its application. Constr. Approx. 16 (2000), 487-524.
  * [4] U. Fidalgo, A multi-orthogonal polynomial’s approach to bulk queueing theory (submitted)
  * [5] G. López Lagomasino and S. Medina Peralta. On the convergence of type I Hermite-Padé approximants. Advances in Math. 273 (2015), 124-148.
  * [6] K. Mahler. Perfect systems. Compos. Math. 19 (1968), 95-166.
  * [7] E.M. Nikishin. On simultaneous Padé approximants. Math. USSR Sb. 41 (1982), 409–425.