# Transverse momentum $p_{\mathrm{T}}$ spectra of strange particles production
in different collisions at $\mbox{$\sqrt{s_{NN}}$}=2.76,5.02,$ and $7$ TeV

Hayam Yassin Corresponding Author<EMAIL_ADDRESS>Eman R.
Abo Elyazeed<EMAIL_ADDRESS>Physics Department, Faculty of Women
for Arts, Science and Education, Ain Shams University, 11577 Cairo, Egypt

###### Abstract

We analyse the transverse momentum $p_{\mathrm{T}}$ spectra of strange
particles $K^{0}_{s}$, $\Lambda$, and $\Xi^{-}$produced in Pb+Pb collision at
$\mbox{$\sqrt{s_{NN}}$}=2.76$ TeV, p+Pb collision at
$\mbox{$\sqrt{s_{NN}}$}=5.02$ TeV, and p+p collision at
$\mbox{$\sqrt{s_{NN}}$}=7$ TeV in different multiplicity events measured by
the CMS experiment at the Large Hadron Collider. The $p_{\mathrm{T}}$ spectra
of strange particles are fitted by Tsallis statistics and Boltzmann
statistics, respectively. The fitting parameters are studied as a function of
the multiplicity events for all systems. The Tsallis temperature
($T_{\mathtt{Ts}}$), Boltzmann temperature ($T_{\mathtt{Boltz}}$), and radius
of the system ($R$) increase with both the mass and strangeness number of the
particle and also increase with the multiplicity events. The non-extensive
parameter ($q$) decreases with the increase in the mass of the particle and
also decrease with the increase in the multiplicity events which means that
the system tends to thermodynamic stabilization. The extracted temperatures
from the two statistics for the strange particles are exhibited a linear
correlation.

25.75.Dw; 25.75.-q; 24.10.Pa

##  1 Introduction

Quark-gluon plasma (QGP) was created at heavy-ion collisions at the
Relativistic Heavy-Ion Collider (RHIC) and the Large Hadron Collider (LHC) [1,
2, 3]. The heavy-ion experiments seek to study strongly interacting matter
under extreme conditions of high density and /or high temperatures [4]. QGP
was described as the deconfinement of the colliding hadrons which rapidly
expands and cools down [5]. ”Hadronization” or phase transition from QGP to
hadrons found at the combination of the quarks and gluons at special
temperatures defined as critical temperatures. After cooling again, chemical
freeze-out occur where the produced particles are stable [6]. One of the
signatures for the creation of QGP is the enhancement of the strangeness
number [7, 8]. Strange quarks are not produced into the reaction by the
colliding nuclei. Therefore, any strange quarks or anti-quarks seen in
experiments have been newly created from the kinetic energy of the colliding
nuclei [8, 9]. Also, another important indication for the formation of QGP is
the transverse momentum $p_{\mathrm{T}}$ spectra of the charged and strange
particles [10]. Because the $p_{\mathrm{T}}$ spectra can give information
about the chemical freezeout - chemical potential and temperature - by
utilizing many statistical models [11, 12, 13, 14, 15, 16, 17]. These
statistical models describe the experimental measurements over a wide range of
center-of-mass energies depending on different statistics.

Tsallis statistics [18, 19, 20, 21, 22, 23] success in fitting the
experimental data [24, 25] of the transverse momentum $p_{\mathrm{T}}$ spectra
in high energy physics. The $p_{\mathrm{T}}$ spectra of all identified
particles which measured in p+p collisions at RHIC and LHC energies was fitted
excellently by Tsallis statistics in Ref. [26, 27, 28]. The Tsallis statistics
with the transverse flow effect is used in the analysis of the
$p_{\mathrm{T}}$ spectra of charged and strange particles in Ref. [29, 30, 31,
22, 23, 32]. Tsallis and Boltzmann statistics describe the $p_{\mathrm{T}}$
spectra of all identified particles measured in Au+Au collision and Pb+Pb
collision at RHIC and LHC energies in Ref. [33]. Also, the obtained
temperatures from these previous fitting gave some useful correlations with
each other. The aim of this paper is to obtain the correlation between the
Tsallis and Boltzmann temperatures from the $p_{\mathrm{T}}$ spectra of
strange particles at different collisions and energies.

The paper is organized as follows. In Sect. 2.1, the Tsallis statistics is
presented, which is used to describe the particle spectra. In Sect. 2.2, we
present the Boltzmann statistics which is used to describe the particle
spectra. Then we discuss the results of the description of the
$p_{\mathrm{T}}$ spectra of strange particles at different collisions and
energies by using both statistics in Sect. 3. Also, the dependence of the
fitting parameters on the multiplicity events and particle mass is discussed
in Sect. 3. Finally, in Sect. 4 the conclusion of our results is presented.

##  2 Transverse momentum spectra

Transverse momentum $p_{\mathrm{T}}$ spectra of strange particles $K^{0}_{s}$,
$\Lambda$, and $\Xi^{-}$prod-uced in Pb+Pb collision at
$\mbox{$\sqrt{s_{NN}}$}=2.76$ TeV, p+Pb collision at
$\mbox{$\sqrt{s_{NN}}$}=5.02$ TeV, and p+p collision at
$\mbox{$\sqrt{s_{NN}}$}=7$ TeV which measured by the CMS experiment will be
discussed using Tsallis and Boltzmann statistics in the following Sects. 2.1
and 2.2.

###  2.1 Tsallis statistics

The experimental measurements of the $p_{\mathrm{T}}$ spectra in high-energy
collisions can be described by different formula of Tsallis statistics [18,
34, 35, 36]. The total number of particles is given by

$\displaystyle
N=\frac{gV}{(2\pi)^{3}}\int_{0}^{\infty}\left[1+(q-1)\left(\frac{E-\mu}{T_{\mathrm{Ts}}}\right)\right]^{\frac{q}{1-q}}d{p^{3}},$
(1)

where, $T_{\mathrm{Ts}}$ is the Tsallis temperature, $q$ is the non-extensive
parameter, $E$ is the energy, $p$ is the pressure, $g$ is the degeneracy
factor, $V$ is the system volume, and $\mu$ is the chemical potential. The
momentum distribution [37] can be obtained as

$\displaystyle
E\frac{d^{3}N}{dp^{3}}=\frac{gVE}{(2\pi)^{3}}\left[1+(q-1)\left(\frac{E-\mu}{T_{\mathrm{Ts}}}\right)\right]^{\frac{q}{1-q}}.$
(2)

In terms of the rapidity ($y$) and the transverse mass
($\;m_{\mathrm{T}}=\sqrt{\;p_{\mathrm{T}}^{2}+m^{2}}$), energy can be written
as $E=m_{\mathrm{T}}\cosh y$, so at mid-rapidity $y=0$ and $\mu\approx 0$, Eq.
2 [38] becomes

$\displaystyle\left.\frac{1}{2\pi
p_{\mathrm{T}}}\frac{d^{2}N}{dp_{\mathrm{T}}dy}\right|_{y=0}=\frac{gVm_{\mathrm{T}}}{(2\pi)^{3}}\left[1+(q-1)\left(\frac{m_{\mathrm{T}}}{T_{\mathrm{Ts}}}\right)\right]^{\frac{q}{1-q}}.$
(3)

###  2.2 Boltzmann statistics

According to Boltzmann statistics [39], the number of particles can be written
as

$\displaystyle
N=\frac{gV}{(2\pi)^{3}}\int_{0}^{\infty}\frac{d{p^{3}}}{\exp\left(\frac{E-\mu}{T_{\mathrm{Boltz}}}\right)},$
(4)

where $T_{\mathrm{Boltz}}$ is the Boltzmann temperature. The momentum
distribution [39, 40, 41] can be obtained as

$\displaystyle
E\frac{d^{3}N}{dp^{3}}=\frac{gVE}{(2\pi)^{3}}\exp\left(\frac{\mu-E}{T_{\mathrm{Boltz}}}\right).$
(5)

The momentum distribution at mid-rapidity $y=0$ and $\mu\approx 0$ can be
given by

$\displaystyle\left.\frac{1}{2\pi
p_{\mathrm{T}}}\frac{d^{2}N}{dp_{\mathrm{T}}dy}\right|_{y=0}=\frac{gVm_{\mathrm{T}}}{(2\pi)^{3}}\exp\left(-\frac{m_{\mathrm{T}}}{T_{\mathrm{Boltz}}}\right).$
(6)

Figure 1: (Color online) Transverse momentum distributions of the strange
particles $K^{0}_{s}$, $\Lambda$, and $\Xi^{-}$for Pb+Pb collision at
$\mbox{$\sqrt{s_{NN}}$}=2.76$ TeV measured in CMS [42] experiment (symbols)
are compared with calculations from Tsallis statistics (solid curves) using
Eq. 3 and with Boltzmann statistics (dashed curves) using Eq. 6 for different
multiplicity intervals.

Figure 2: (Color online) Transverse momentum distributions of the strange
particles $K^{0}_{s}$, $\Lambda$, and $\Xi^{-}$for p+Pb collision at
$\mbox{$\sqrt{s_{NN}}$}=5.02$ TeV measured in CMS [42] experiment (symbols)
are compared with calculations from Tsallis statistics (solid curves) using
Eq. 3 and with Boltzmann statistics (dashed curves) using Eq. 6 for different
multiplicity intervals.

Figure 3: (Color online) Transverse momentum distributions of the strange
particles $K^{0}_{s}$, $\Lambda$, and $\Xi^{-}$for p+p collision at
$\mbox{$\sqrt{s_{NN}}$}=7$ TeV measured in CMS [42] experiment (symbols) are
compared with calculations from Tsallis statistics (solid curves) using Eq. 3
and with Boltzmann statistics (dashed curves) using Eq. 6 for different
multiplicity intervals.

##  3 Results and Discussion

The transverse momentum distributions of the strange particles $K^{0}_{s}$,
$\Lambda$, and $\Xi^{-}$in Pb+Pb collision at $\mbox{$\sqrt{s_{NN}}$}=2.76$
TeV, p+Pb collision at $\mbox{$\sqrt{s_{NN}}$}=5.02$ TeV, and p+p collision at
$\mbox{$\sqrt{s_{NN}}$}=7$ TeV [42] in different multiplicity intervals are
fitted using two different kinds of statistics. Boltzmann statistics using Eq.
6 and Tsallis statistics using Eq. 3. These are shown in Figs. 1, 2, and 3.
The fitting parameters are listed in Tables 1, 2, and 3.

Figure 1 shows the $p_{\mathrm{T}}$ spectra of the strange particles, (a)
$K_{s}^{0}$, (b) $\Lambda$, and (c) $\Xi^{-}$, produced in Pb+Pb collision at
$\mbox{$\sqrt{s_{NN}}$}=2.76$ TeV with different multiplicity intervals. The
experimental data of the CMS experiment [42] are represented by symbols. The
experimental data are divided into classes based on the multiplicity intervals
$N_{\mathrm{trk}}^{\mathrm{offline}}$ in the mid-rapidity range $|y|<1.0$. The
corresponding averaged multiplicity
$<N_{\mathrm{track}}>=21,58,92,130,168,210,253$ and $299$ [43]. The solid and
dashed curves are the calculated results using the Tsallis statistics (Eq. 3)
and the Boltzmann statistics (Eq. 6), respectively. Here, we concentrate on
the smallest $p_{\mathrm{T}}$ region. The resulting fit parameters are given
in Table 1. We notice that our two results for $\Lambda$ and $\Xi^{-}$
particles are in good agreement with the experimental data of Pb+Pb collisions
at $\mbox{$\sqrt{s_{NN}}$}=2.76$ TeV for all multiplicity intervals. For
$K_{s}^{0}$ particle, also there is a good agreement only with the results
obtained from Tsallis statistics but with the results obtained from Boltzmann
statistics, the good agreement decreases with the increase in the multiplicity
classes.

Table 1: Parameters deduced for the fitting of Tsallis and Boltzmann transverse momentum distributions at different multiplicity intervals for Pb+Pb collision at $\sqrt{s_{\mathtt{NN}}}=2.76$ TeV, Fig. 1. Particle | $<N_{\mathrm{track}}>$ |  $T_{\mathtt{Ts}}$ [MeV] | $q$ |  $R$ [fm] | $\chi^{2}/\mathrm{dof}$ |  $T_{\mathtt{Boltz}}$ [MeV] |  $R$ [fm] | $\chi^{2}/\mathrm{dof}$
---|---|---|---|---|---|---|---|---
$K_{s}^{0}$ | $21$ | $87.169\pm 0.105$ | $1.15\pm 0.001$ | $0.154\pm 0.026$ | $0.023$ | $198.808\pm 1.672$ | $0.07\pm 0.021$ | $0.13$
| $58$ | $133.113\pm 0.378$ | $1.1224\pm 0.003$ | $0.211\pm 0.044$ | $0.047$ | $220.873\pm 1.47$ | $0.136\pm 0.036$ | $0.1$
| $92$ | $139.857\pm 0.131$ | $1.1189\pm 0.001$ | $0.358\pm 0.052$ | $0.015$ | $228.78\pm 1.46$ | $0.236\pm 0.062$ | $0.09$
| $130$ | $147.611\pm 0.283$ | $1.1158\pm 0.002$ | $0.589\pm 0.107$ | $0.03$ | $238.526\pm 1.749$ | $0.394\pm 0.107$ | $0.1$
| $168$ | $165.663\pm 0.418$ | $1.1017\pm 0.003$ | $0.894\pm 0.175$ | $0.038$ | $248.365\pm 1.7$ | $0.642\pm 0.169$ | $0.091$
| $210$ | $170.876\pm 0.345$ | $1.1007\pm 0.003$ | $1.421\pm 0.257$ | $0.03$ | $248.671\pm 1.624$ | $1.051\pm 0.273$ | $0.087$
| $253$ | $181.643\pm 0.294$ | $1.0941\pm 0.002$ | $2.226\pm 0.371$ | $0.023$ | $249.084\pm 1.053$ | $1.741\pm 0.391$ | $0.056$
| $299$ | $188.54\pm 0.499$ | $1.079\pm 0.004$ | $3.381\pm 0.622$ | $0.031$ | $256.769\pm 2.215$ | $2.65\pm 0.737$ | $0.108$
$\Lambda$ | $21$ | $91.274\pm 0.084$ | $1.11\pm 0.0003$ | $0.193\pm 0.034$ | $0.056$ | $249.745\pm 1.111$ | $0.046\pm 0.013$ | $0.233$
| $58$ | $160.973\pm 0.093$ | $1.09\pm 0.0003$ | $0.166\pm 0.024$ | $0.03$ | $289.72\pm 0.944$ | $0.082\pm 0.02$ | $0.15$
| $92$ | $187.226\pm 0.27$ | $1.08\pm 0.001$ | $0.242\pm 0.046$ | $0.07$ | $304.703\pm 1.244$ | $0.137\pm 0.036$ | $0.181$
| $130$ | $229.993\pm 0.164$ | $1.064\pm 0.001$ | $0.321\pm 0.047$ | $0.032$ | $309.954\pm 0.359$ | $0.235\pm 0.04$ | $0.05$
| $168$ | $274.28\pm 0.412$ | $1.0429\pm 0.001$ | $0.441\pm 0.082$ | $0.064$ | $329.929\pm 0.511$ | $0.366\pm 0.068$ | $0.064$
| $210$ | $322.756\pm 1.085$ | $1.0245\pm 0.003$ | $0.605\pm 0.143$ | $0.132$ | $335.32\pm 1.454$ | $0.597\pm 0.156$ | $0.178$
| $253$ | $343.331\pm 1.146$ | $1.0086\pm 0.003$ | $0.957\pm 0.179$ | $0.13$ | $340.261\pm 1.479$ | $0.982\pm 0.255$ | $0.176$
| $299$ | $362.632\pm 0.633$ | $1.0067\pm 0.002$ | $1.488\pm 0.279$ | $0.066$ | $360.228\pm 0.887$ | $1.515\pm 0.321$ | $0.095$
$\Xi^{-}$ | $21$ | $156.108\pm 0.513$ | $1.09\pm 0.001$ | $0.042\pm 0.011$ | $0.246$ | $290.041\pm 1.119$ | $0.019\pm 0.165$ | $0.284$
| $58$ | $188.975\pm 0.719$ | $1.07\pm 0.001$ | $0.081\pm 0.023$ | $0.282$ | $309.878\pm 1.633$ | $0.042\pm 0.12$ | $0.367$
| $92$ | $257.487\pm 1.258$ | $1.0599\pm 0.003$ | $0.085\pm 0.025$ | $0.312$ | $340.112\pm 2.124$ | $0.063\pm 0.075$ | $0.402$
| $130$ | $265.875\pm 1.243$ | $1.052\pm 0.002$ | $0.151\pm 0.043$ | $0.3$ | $349.961\pm 2.211$ | $0.11\pm 0.047$ | $0.394$
| $168$ | $286.238\pm 0.784$ | $1.0441\pm 0.002$ | $0.231\pm 0.055$ | $0.172$ | $379.967\pm 1.907$ | $0.164\pm 0.035$ | $0.292$
| $210$ | $365.428\pm 1.569$ | $1.0248\pm 0.003$ | $0.27\pm 0.072$ | $0.238$ | $390.23\pm 2.162$ | $0.257\pm 0.02$ | $0.314$
| $253$ | $377.162\pm 1.579$ | $1.0168\pm 0.003$ | $0.437\pm 0.116$ | $0.232$ | $400.073\pm 2.165$ | $0.414\pm 0.013$ | $0.301$
| $299$ | $425.295\pm 2.719$ | $1.0152\pm 0.005$ | $0.595\pm 0.176$ | $0.322$ | $470.406\pm 3.561$ | $0.533\pm 0.005$ | $0.368$
Table 2: Parameters deduced for the fitting of Tsallis and Boltzmann transverse momentum distributions at different multiplicity intervals for p+Pb collision collision at $\sqrt{s_{\mathtt{NN}}}=5.02$ TeV, Fig. 2. Particle | $<N_{\mathrm{track}}>$ |  $T_{\mathtt{Ts}}$ [MeV] | $q$ |  $R$ [fm] | $\chi^{2}/\mathrm{dof}$ |  $T_{\mathtt{Boltz}}$ [MeV] |  $R$[fm] | $\chi^{2}/\mathrm{dof}$
---|---|---|---|---|---|---|---|---
$K_{s}^{0}$ | $21$ | $133.898\pm 0.295$ | $1.1307\pm 0.002$ | $0.077\pm 0.015$ | $0.035$ | $228.653\pm 1.589$ | $0.049\pm 0.013$ | $0.098$
• | $57$ | $179.121\pm 0.646$ | $1.1147\pm 0.005$ | $0.118\pm 0.026$ | $0.051$ | $256.887\pm 1.525$ | $0.09\pm 0.023$ | $0.078$
• | $89$ | $199.756\pm 0.687$ | $1.1083\pm 0.005$ | $0.19\pm 0.04$ | $0.046$ | $269.206\pm 1.133$ | $0.153\pm 0.034$ | $0.054$
• | $125$ | $206.332\pm 0.489$ | $1.1047\pm 0.004$ | $0.32\pm 0.059$ | $0.031$ | $278.893\pm 1.433$ | $0.258\pm 0.06$ | $0.064$
• | $159$ | $218.697\pm 0.833$ | $1.1021\pm 0.006$ | $0.497\pm 0.106$ | $0.048$ | $288.99\pm 1.548$ | $0.409\pm 0.096$ | $0.065$
• | $195$ | $226.848\pm 1.085$ | $1.1005\pm 0.008$ | $0.81\pm 0.185$ | $0.06$ | $294.12\pm 1.769$ | $0.678\pm 0.165$ | $0.072$
• | $236$ | $255.94\pm 1.87$ | $1.0787\pm 0.013$ | $1.205\pm 0.312$ | $0.087$ | $309.36\pm 2.499$ | $1.063\pm 0.282$ | $0.093$
• | $280$ | $293.332\pm 1.425$ | $1.0676\pm 0.009$ | $1.869\pm 0.419$ | $0.056$ | $334.762\pm 1.591$ | $1.721\pm 0.382$ | $0.055$
$\Lambda$ | $21$ | $172.072\pm 0.124$ | $1.09\pm 0.0004$ | $0.067\pm 0.01$ | $0.036$ | $299.715\pm 0.865$ | $0.035\pm 0.008$ | $0.13$
• | $57$ | $297.482\pm 0.386$ | $1.0499\pm 0.001$ | $0.065\pm 0.011$ | $0.052$ | $369.708\pm 0.797$ | $0.053\pm 0.011$ | $0.083$
• | $89$ | $373.144\pm 0.253$ | $1.0249\pm 0.001$ | $0.089\pm 0.012$ | $0.024$ | $409.837\pm 0.47$ | $0.082\pm 0.013$ | $0.04$
• | $125$ | $436.177\pm 1.142$ | $1.0091\pm 0.003$ | $0.127\pm 0.026$ | $0.085$ | $430.532\pm 1.625$ | $0.131\pm 0.031$ | $0.127$
• | $159$ | $453.51\pm 0.919$ | $1.0065\pm 0.003$ | $0.19\pm 0.035$ | $0.064$ | $460.097\pm 1.122$ | $0.188\pm 0.037$ | $0.077$
• | $195$ | $482.8\pm 2.922$ | $1.0047\pm 0.008$ | $0.292\pm 0.077$ | $0.183$ | $470.728\pm 3.661$ | $0.302\pm 0.087$ | $0.243$
• | $236$ | $501.749\pm 2.302$ | $1.0034\pm 0.007$ | $0.465\pm 0.111$ | $0.135$ | $480.908\pm 2.991$ | $0.487\pm 0.13$ | $0.192$
• | $280$ | $531.914\pm 2.423$ | $1.0012\pm 0.007$ | $0.747\pm 0.176$ | $0.131$ | $535.578\pm 2.905$ | $0.744\pm 0.186$ | $0.157$
$\Xi^{-}$ | $21$ | $192.967\pm 0.508$ | $1.083\pm 0.001$ | $0.031\pm 0.008$ | $0.176$ | $339.7\pm 1.556$ | $0.016\pm 0.009$ | $0.277$
• | $57$ | $301.869\pm 1.753$ | $1.0711\pm 0.004$ | $0.034\pm 0.01$ | $0.306$ | $420.171\pm 1.799$ | $0.024\pm 0.006$ | $0.217$
• | $89$ | $396.942\pm 2.066$ | $1.0684\pm 0.005$ | $0.042\pm 0.011$ | $0.232$ | $480.405\pm 2.054$ | $0.036\pm 0.019$ | $0.194$
• | $125$ | $423.094\pm 4.797$ | $1.049\pm 0.013$ | $0.067\pm 0.023$ | $0.547$ | $531.027\pm 4.793$ | $0.054\pm 0.017$ | $0.38$
• | $159$ | $568.425\pm 26.71$ | $1.0287\pm 0.01$ | $0.084\pm 0.022$ | $0.236$ | $580.505\pm 3.308$ | $0.083\pm 0.022$ | $0.222$
• | $195$ | $610.815\pm 6.597$ | $1.0219\pm 0.02$ | $0.131\pm 0.041$ | $0.382$ | $630.846\pm 6.12$ | $0.128\pm 0.039$ | $0.317$
• | $236$ | $642.08\pm 7.933$ | $1.0189\pm 0.019$ | $0.202\pm 0.066$ | $0.437$ | $662.482\pm 9.601$ | $0.198\pm 0.069$ | $0.051$
• | $280$ | $660.022\pm 10.78$ | $1.0139\pm 0.04$ | $0.311\pm 0.111$ | $0.558$ | $672.726\pm 9.541$ | $0.308\pm 0.104$ | $0.488$

Figure 2 represents the $p_{\mathrm{T}}$ spectra of the strange particles, (a)
$K_{s}^{0}$, (b) $\Lambda$, and (c) $\Xi^{-}$, produced in p+Pb collisions at
$\mbox{$\sqrt{s_{NN}}$}=5.02$ TeV with different multiplicity intervals. The
experimental data of the CMS experiment [42] are represented by symbols. The
experimental data are divided into classes based on the multiplicity intervals
$N_{\mathrm{trk}}^{\mathrm{offline}}$ in the mid-rapidity range $|y|<1.0$. The
corresponding averaged multiplicity
$<N_{\mathrm{track}}>=21,57,89,125,159,195,236$ and $280$ [43]. The solid and
dashed curves are the calculated results using the Tsallis statistics (Eq. 3)
and the Boltzmann statistics (Eq. 6), respectively. Here, we concentrate on
the smallest $p_{\mathrm{T}}$ region. The resulting fit parameters are given
in Table 2. We notice that our two results for $\Lambda$ and $\Xi^{-}$
particles are in good agreement with the experimental data of p+Pb collisions
at $\mbox{$\sqrt{s_{NN}}$}=5.02$ TeV for all multiplicity intervals. For
$K_{s}^{0}$ particle, also there is a good agreement only with the results
obtained from Tsallis statistics but with the results obtained from Boltzmann
statistics, the good agreement decrease with the increasing of the
multiplicity classes.

Figure 3 depicts the $p_{\mathrm{T}}$ spectra of the strange particles, (a)
$K_{s}^{0}$, (b) $\Lambda$, and (c) $\Xi^{-}$, produced in p+p collision at
$\mbox{$\sqrt{s_{NN}}$}=7$ TeV with different multiplicity intervals. The
experimental data of the CMS experiment [42] are represented by symbols. The
experimental data are divided into classes based on the multiplicity intervals
$N_{\mathrm{trk}}^{\mathrm{offline}}$ in the mid-rapidity range $|y|<1.0$. The
corresponding averaged multiplicity $<N_{\mathrm{track}}>=14,50,79,111,135$
and $158$ [43]. The solid and dashed curves are the calculated results using
the Tsallis statistics (Eq. 3) and the Boltzmann statistics (Eq. 6),
respectively. Here, we concentrate on the smallest $p_{\mathrm{T}}$ region.
The resulting fit parameters are given in Table 3. We notice that our two
results for $\Lambda$ and $\Xi^{-}$ particles are in good agreement with the
experimental data of p+p collisions at $\mbox{$\sqrt{s_{NN}}$}=7$ TeV for all
multiplicity intervals. For $K_{s}^{0}$ particle, also there is a good
agreement only with the results obtained from Tsallis statistics but with the
results obtained from Boltzmann statistics, the good agreement decrease with
the increase in the multiplicity intervals.

As seen from Tables 1, 2, and 3, the value of $\chi^{2}/\mathrm{dof}$ is so
small which represents the good quality of the fitting. Especially, the
Tsallis statistics that gives excellent agreement with the experimental
measurements for all multiplicity classes. Also, the fitting results by
Tsallis statistics are better than those of Boltzmann statistics, especially
at high range of $p_{\mathrm{T}}$.

Furthermore, we have extracted the fitting parameters the non-extensive
parameter $q$, the Tsallis temperature parameter $T_{\mathrm{Ts}}$, and the
Boltzmann temperature parameter $T_{\mathrm{Boltz}}$ which inform us about the
variations between Tsallis statistics and Boltzmann statistics. Also, there is
another fitting parameter known as the radius $R$ as we assume the geometry of
the fireball is spherical so $R=(3V/4\pi)^{1/3}$ which signifies the dimension
of the system and is related to the normalization in the statistical
distribution function which used in describing the particle yield or spectra
[44].

Figure 4 (a-c) depict the fitting parameter $q$ for the strange particles
$K^{0}_{s}$, $\Lambda$, and $\Xi^{-}$as a function of $<N_{\mathrm{track}}>$
produced in the $\textsf{Pb+Pb},\textsf{p+Pb},$ and p+p collisions at
$\sqrt{s_{\mathtt{NN}}}=2.76,5.02,$ and $7$ TeV, respectively. Results are
obtained using Tsallis statistics and modified Tsallis statistics [30]. The
value of $q$ (our calculations) decreases with the increasing of both the
particle mass and strangeness number and also with the increasing in the
multiplicity events for all systems. Our results different from modified one
in the effect of multiplicity on the non-extensivity parameter. This
decreasing indicate that the system tend to be in equilibrium
(thermodynamically).

Figure 5 (a-c) show the fitting parameter $R$ for the strange particles
$K^{0}_{s}$, $\Lambda$, and $\Xi^{-}$as a function of $<N_{\mathrm{track}}>$
produced in the $\textsf{Pb+Pb},\textsf{p+Pb},$ and p+p collisions at
$\sqrt{s_{\mathtt{NN}}}=2.76,5.02,$ and $7$ TeV, respectively. The value of
$R$ increases with the increasing in the multiplicity classes but decreases
with the increase in the particle mass. So the volume of the system increases
with the increase in the multiplicity events as expected.

We compare our obtained values of $R$ with Hanbury-Brown–Twiss (HBT) radii
[45, 46, 47] at different center-of-mass energies $\sqrt{s_{\mathtt{NN}}}$ as
shown in Fig. 6. The fitting parameter $R$ for the strange particles
$K^{0}_{s}$, $\Lambda$, and $\Xi^{-}$and from HBT radii obtained at most
central collisions and also at mid-rapidity (for details on HBT radius
parameters, see Refs. [48, 49, 50, 51, 52]). The obtained values of $R$ were
in the same behaviour as $R$ from HBT especially $R_{\mathrm{long}}$.

Figure 7 (a-c) depict the fitting parameter $T$ for the strange particles
$K^{0}_{s}$, $\Lambda$, and $\Xi^{-}$as a function of $<N_{\mathrm{track}}>$
produced in the $\textsf{Pb+Pb},\textsf{p+Pb},$ and p+p collisions at
$\sqrt{s_{\mathtt{NN}}}=2.76,5.02,$ and $7$ TeV, respectively. The value of
both temperatures (Tsallis and Boltzmann) increase with the increasing in both
the particle mass and the strangeness number and also with the multiplicity
classes for all systems. The direct relationship between the temperature and
the strangeness number of hadrons is unlike the known behaviour obtained
previously in extensive models (particle yields and ratios) [53, 54], so we
will work on the confirmation of the behaviour of this dependence in extensive
and nonextensive particle yields and ratios in future work. Our calculations
are compared with the modified Tsallis results [30] and agreed with each
other. The freeze-out of particles with large mass occur earlier than that of
small mass. When volume is small, the particle with large mass freeze-out
early [4]. Also, the Boltzmann temperature always greater than the Tsallis one
which are independent on the type of particles or systems so
$T_{\mathrm{Boltz}}>T_{\mathrm{Ts}}$.

Table 3: Parameters deduced for the fitting of Tsallis and Boltzmann transverse momentum distributions at different multiplicity intervals for p+p collision at $\sqrt{s_{\mathtt{NN}}}=7$ TeV, Fig. 3. Particle | $<N_{\mathrm{track}}>$ |  $T_{\mathtt{Ts}}$ [MeV] | $q$ |  $R$ [fm] | $\chi^{2}/\mathrm{dof}$ |  $T_{\mathtt{Boltz}}$ [MeV] |  $R$ [fm] | $\chi^{2}/\mathrm{dof}$
---|---|---|---|---|---|---|---|---
$K_{s}^{0}$ | $14$ | $91.407\pm 0.129$ | $1.154\pm 0.001$ | $0.282\pm 0.049$ | $0.026$ | $213.472\pm 2.176$ | $0.126\pm 0.04$ | $0.159$
• | $50$ | $155.351\pm 0.397$ | $1.1328\pm 0.003$ | $0.337\pm 0.066$ | $0.037$ | $258.135\pm 2.336$ | $0.225\pm 0.065$ | $0.123$
• | $79$ | $234.716\pm 1.78$ | $1.1047\pm 0.013$ | $0.536\pm 0.142$ | $0.093$ | $294.827\pm 2.402$ | $0.463\pm 0.125$ | $0.097$
• | $111$ | $235.749\pm 1.784$ | $1.1028\pm 0.013$ | $0.673\pm 0.178$ | $0.093$ | $308.689\pm 2.798$ | $0.561\pm 0.155$ | $0.105$
• | $135$ | $252.973\pm 1.445$ | $1.1082\pm 0.011$ | $1.042\pm 0.247$ | $0.067$ | $333.049\pm 2.631$ | $0.869\pm 0.226$ | $0.088$
• | $158$ | $313.052\pm 1.92$ | $1.0691\pm 0.011$ | $1.472\pm 0.355$ | $0.07$ | $359.903\pm 2.047$ | $1.355\pm 0.317$ | $0.064$
$\Lambda$ | $14$ | $152.631\pm 0.284$ | $1.0945\pm 0.0008$ | $0.183\pm 0.039$ | $0.094$ | $289.682\pm 1.218$ | $0.082\pm 0.023$ | $0.209$
• | $50$ | $259.374\pm 0.211$ | $1.08\pm 0.001$ | $0.186\pm 0.028$ | $0.033$ | $379.636\pm 1.303$ | $0.127\pm 0.03$ | $0.136$
• | $79$ | $321.4\pm 1.805$ | $1.0714\pm 0.005$ | $0.241\pm 0.066$ | $0.202$ | $419.733\pm 2.041$ | $0.193\pm 0.05$ | $0.178$
• | $111$ | $413.873\pm 3.627$ | $1.0688\pm 0.011$ | $0.309\pm 0.093$ | $0.268$ | $490.01\pm 3.286$ | $0.278\pm 0.077$ | $0.216$
• | $135$ | $459.049\pm 2.757$ | $1.0519\pm 0.008$ | $0.452\pm 0.118$ | $0.176$ | $500.921\pm 2.72$ | $0.434\pm 0.113$ | $0.175$
• | $158$ | $501.352\pm 3.193$ | $1.0336\pm 0.009$ | $0.661\pm 0.173$ | $0.18$ | $541.774\pm 4.08$ | $0.627\pm 0.174$ | $0.213$
$\Xi^{-}$ | $14$ | $155.92\pm 0.093$ | $1.099\pm 0.0002$ | $0.093\pm 0.014$ | $0.045$ | $339.888\pm 0.995$ | $0.032\pm 0.008$ | $0.203$
• | $50$ | $274.655\pm 0.151$ | $1.076\pm 0.0003$ | $0.096\pm 0.013$ | $0.034$ | $409.925\pm 1.02$ | $0.062\pm 0.014$ | $0.146$
• | $79$ | $419.894\pm 1.824$ | $1.0299\pm 0.003$ | $0.098\pm 0.026$ | $0.229$ | $489.854\pm 3.369$ | $0.084\pm 0.025$ | $0.345$
• | $111$ | $516.464\pm 1.844$ | $1.028\pm 0.004$ | $0.128\pm 0.03$ | $0.159$ | $520.321\pm 2.446$ | $0.131\pm 0.034$ | $0.225$
• | $135$ | $528.961\pm 2.024$ | $1.027\pm 0.004$ | $0.204\pm 0.048$ | $0.164$ | $540.85\pm 4.672$ | $0.205\pm 0.065$ | $0.403$
• | $158$ | $539.162\pm 3.965$ | $1.0204\pm 0.008$ | $0.333\pm 0.099$ | $0.326$ | $554.286\pm 5.984$ | $0.329\pm 0.111$ | $0.485$

Fig. 8 shows the relation between $T_{\mathrm{Ts}}$ and $T_{\mathrm{Boltz}}$,
where all values of temperatures obtained from Figs. 1, 2, and 3 which listed
in Tables 1, 2, and 3 are given by symbols. The fitted result for all strange
particles at different collisions is given by the solid line which is
described by

$T_{\mathrm{Ts}}=a\;\;T_{\mathrm{Boltz}}+b,$

where $a=1.2465\pm 0.0138$ and $b=-160.499\pm 5.386$ are constants with
$\chi^{2}/\mathrm{dof}=0.0528$. But these values are different for different
particles. So this linear relations is the same for all particles but the
value of constants different depending on the mass of the particle. The
dependence of the constants on the particle type is listed in Table 4.

Table 4: Constants values deduced from the fitting of linear relation between Tsallis and Boltzmann temperatures, Fig. 8. Particle | $a$ | $b$ | $\chi^{2}/\mathrm{dof}$
---|---|---|---
$K_{s}^{0}$ | $(1.3714\pm 0.0092)$ | $(-177.514\pm 2.503)$ | $0.0319$
$\Lambda$ | $(1.3856\pm 0.0255)$ | $(-209.84\pm 10.2)$ | $0.0876$
$\Xi^{-}$ | $(1.39513\pm 0.0212)$ | $(-249.213\pm 9.963)$ | $0.0722$

As can be seen from Table 4, the value of constant $a$ increase with the
increasing of the particle mass but constant $b$ inversely proportional to the
particle mass. A similar linear relation between $T_{\mathrm{Ts}}$ and
$T_{\mathrm{Boltz}}$ for charged (but not strange) particles at various
collisions was found in Ref. [33]. The values of constants $a$ and $b$ for the
charged non-strange particles confirm our concept about the dependence of the
constants on the particle mass.

Therefore, we can find a direct relation between any temperature and Boltzmann
temperature and considering Boltzmann temperature as the ”reference” for all
other temperatures [33]. However, Boltzmann statistics fails to fit some
experimental transverse momentum spectra especially at high $p_{\mathrm{T}}$
region as a result of using the simplest form of Boltzmann statistics [16, 41,
55, 33]. On the contrary, Tsallis statistics succeeded in depicting all
experimental transverse momentum spectra used in the present work and also at
high region of $p_{\mathrm{T}}$ [30, 33, 35, 56, 57, 58, 59, 60, 61]. We also
compare our Tsallis results with modified Tsallis one [30] (with a radial
flow). Our Tsallis temperatures are greater than modified one [30]. The reason
for this increasing gets from neglecting the effect of the radial flow. We
neglect this parameter as a result of the considered low range of transverse
momentum [33].

Figure 4: (Color online) The fitting parameter $q$ as a function of the event
multiplicity $<N_{\mathrm{track}}>$ for the strange particles $K^{0}_{s}$,
$\Lambda$, and $\Xi^{-}$which fitted using Tsallis and Boltzmann statistics
for (a) Pb+Pb collision at $\mbox{$\sqrt{s_{NN}}$}=2.76$ TeV, (b) p+Pb
collision at $\mbox{$\sqrt{s_{NN}}$}=5.02$ TeV, and (c) p+p collision at
$\mbox{$\sqrt{s_{NN}}$}=7$ TeV.

Figure 5: (Color online) The fitting parameter $R$ as a function of the event
multiplicity $<N_{\mathrm{track}}>$ for the strange particles $K^{0}_{s}$,
$\Lambda$, and $\Xi^{-}$which fitted using Tsallis and Boltzmann statistics
for (a) Pb+Pb collision at $\mbox{$\sqrt{s_{NN}}$}=2.76$ TeV, (b) p+Pb
collision at $\mbox{$\sqrt{s_{NN}}$}=5.02$ TeV, and (c) p+p collision at
$\mbox{$\sqrt{s_{NN}}$}=7$ TeV.
Figure 6: (Color online) The parameter $R$ as a function of the center-of-mass
energies $\sqrt{s_{NN}}$ for the obtained strange particles $K^{0}_{s}$,
$\Lambda$, and $\Xi^{-}$which fitted using Tsallis and Boltzmann statistics
and values of HBT radii
($R_{\mathrm{out}},R_{\mathrm{long}},R_{\mathrm{side}}$) [45, 46, 47].

Figure 7: (Color online) The fitting parameter $T$ as a function of the event
multiplicity $<N_{\mathrm{track}}>$ for the strange particles $K^{0}_{s}$,
$\Lambda$, and $\Xi^{-}$which fitted using Tsallis and Boltzmann statistics-
for (a) Pb+Pb collision at $\mbox{$\sqrt{s_{NN}}$}=2.76$ TeV, (b) p+Pb
collision at $\mbox{$\sqrt{s_{NN}}$}=5.02$ TeV, and (c) p+p collision at
$\mbox{$\sqrt{s_{NN}}$}=7$ TeV.
Figure 8: (Color online) The Tsallis temperature $T_{\mathrm{Ts}}$ as a
function of Boltzmann temperature $T_{\mathrm{Boltz}}$. All values of
temperatures obtained from Figs. 1, 2, and 3 which listed in Tables 1, 2, and
3 are given by symbols. The solid line represents the fitted result that given
by Eq. 3.

##  4 Conclusion

We analysed the transverse momentum $p_{\mathrm{T}}$ spectra of the strange
particles $K^{0}_{s}$, $\Lambda$, and $\Xi^{-}$in different multiplicity
events produced in Pb+Pb collision at $\mbox{$\sqrt{s_{NN}}$}=2.76$ TeV, p+Pb
collision at $\mbox{$\sqrt{s_{NN}}$}=5.02$ TeV, and p+p collision at
$\mbox{$\sqrt{s_{NN}}$}=7$ TeV using Tsallis and Boltzmann statistics. In all
cases nearly, our Tsallis and Boltzmann results are in excellent agreement
with the experimental data of CMS collaboration at the LHC. Except for
$K_{s}^{0}$, the goodness of our Boltzmann result decreases with the increase
in the multiplicity events.

In all studied collisions, for all particles in different multiplicity events,
the Tsallis $T_{\mathrm{Ts}}$ and Boltzmann $T_{\mathrm{Boltz}}$ temperatures
values increase with the increasing in the mass of the mentioned particle.
Also, there is a direct relationship between the values of the temperatures
and the multiplicity events. We observe that
$T_{\mathrm{Ts}}<T_{\mathrm{Boltz}}$, the value of $q$ is proportional
inversely with the multiplicity events in all studied systems, and the value
of $R$ in both statistics increases with the increase in the multiplicity
events. As the direct dependence of the multiplicity on the size of the volume
of the system.

There is a linear relation between the Tsallis temperature $T_{\mathrm{Ts}}$
and the Boltzmann temperature $T_{\mathrm{Boltz}}$. We have
$T_{\mathrm{Ts}}=(1.3714\pm 0.0092)\;\;T_{\mathrm{Boltz}}+(-177.514\pm 2.503)$
for $K_{s}^{0}$, $T_{\mathrm{Ts}}=(1.3856\pm
0.0255)\;\;T_{\mathrm{Boltz}}+(-209.84\pm 10.2)$ for $\Lambda$, and
$T_{\mathrm{Ts}}=(1.39513\pm 0.0212)\;\;T_{\mathrm{Boltz}}+(-249.213\pm
9.963)$ for $\Xi^{-}$. Also, we have $T_{\mathrm{Ts}}=(1.2465\pm
0.0138)\;\;T_{\mathrm{Boltz}}+(-160.499\pm 5.386)$ for all studied strange
particles. In all cases, $T_{\mathrm{Ts}}<T_{\mathrm{Boltz}}$ as the Boltzmann
temperature is considered as the base for all other temperatures, and both
values of temperatures increase with the increasing in the particle mass at
vanishing chemical potential.

## References

  * [1] J.D. Bjorken, Phys. Rev. D 27, 140 (1983).
  * [2] T. Ullrich, B. Wyslouch, J.W. Harris, Nucl. Phys. A 904-905, 1c (2013).
  * [3] M. Gyulassy, L. McLerran, Nucl. Phys. A 750, 30 (2005).
  * [4] A. Khuntia, S. Tripathy, R. Sahoo, J. Cleymans, Eur. Phys. J. A 53, 103 (2017).
  * [5] S. Ahmad, A. Ahmad, A. Chandra, M. Zafar, M. Irfan, Adv. High Energy Phys. 2013, 836071 (2013).
  * [6] A. Tawfik, Int. J. Mod. Phys. A 29, 1430021 (2014).
  * [7] J. Letessier, J. Rafelski, A. Tounsi, Phys. Rev. C 50, 406 (1994).
  * [8] J. Adam et al. (ALICE collaboration), Nature Phys. 13, 535 (2017).
  * [9] J. Rafelski, B. Muller, Phys. Rev. Lett. 48, 1066 (1982). Erratum: Phys. Rev. Lett. 56, 2334 (1986).
  * [10] H-C Song, S.A. Bass, U.W. Heinz, T. Hirano, C. Shen, Phys. Rev. Lett. 106, 192301 (2011), Erratum: Phys. Rev. Lett. 109, 139904 (2012).
  * [11] J. Adams et al. (STAR Collaboration), Phys. Rev. Lett. 92, 112301 (2004).
  * [12] F. Becattini, J. Phys. Conf. Ser. 5, 175 (2005).
  * [13] A.H. Mueller, Nucl. Phys. B 415, 373 (1994).
  * [14] A.H. Mueller, B. Patel, Nucl. Phys. B 425, 471 (1994).
  * [15] A.H. Mueller, Nucl. Phys. B 437, 107 (1995).
  * [16] E. Schnedermann, J. Sollfrank, U.W. Heinz, Phys. Rev. C 48, 2462 (1993).
  * [17] A.N. Tawfik, H. Yassin, E.R. Abo Elyazeed, Chin. Phys. C 41, 053107 (2017).
  * [18] C. Tsallis, J. Statist. Phys. 52, 479 (1988).
  * [19] T.S. Biro, G. Purcsel, K. Urmossy, Eur. Phys. J. A 40, 325 (2009).
  * [20] T. Bhattacharyya, J. Cleymans, A. Khuntia, P. Pareek, R. Sahoo, Eur. Phys. J. A 52, 30 (2016).
  * [21] H. Zheng, L. Zhu, Adv. High Energy Phys. 2015, 180491 (2015).
  * [22] Z. Tang, Y. Xu, L. Ruan, G. van Buren, F. Wang, Z. Xu, Phys. Rev. C 79, 051901 (2009).
  * [23] B. De, Eur. Phys. J. A 50, 138 (2014).
  * [24] A. Adare et al. (PHENIX Collaboration), Phys. Rev. C 83, 064903 (2011).
  * [25] V. Khachatryan et al. (CMS Collaboration), JHEP 1105, 064 (2011).
  * [26] P.K. Khandai, P. Sett, P. Shukla, V. Singh, Int. J. Mod. Phys. A 28, 1350066 (2013).
  * [27] A. Adare et al. (PHENIX Collaboration), Phys. Rev. D 83, 052004 (2011).
  * [28] P. Sett, P. Shukla, Adv. High Energy Phys. 2014, 896037 (2014).
  * [29] P.K. Khandai, P. Sett, P. Shukla, V. Singh, J. Phys. G 41, 025105 (2014).
  * [30] K. Saraswat, P. Shukla, V. Kumar, V. Singh, Eur. Phys. J. A 53, 5 (2017).
  * [31] J. Cleymans, D. Worku, J. Phys. G 39, 025006 (2012).
  * [32] P. Sett, P. Shukla, Int. J. Mod. Phys. E 24, 1550046 (2015).
  * [33] Y-Q Gao, F-H Liu, Indian J. Phys. 90, 319 (2016).
  * [34] C. Tsallis, R.S. Mendes, A.R. Plastino, Physica A 261, 534 (1998).
  * [35] J. Cleymans, D. Worku, Eur. Phys. J. A 48, 160 (2012).
  * [36] J. Cleymans, M.D. Azmi, Eur. Phys. J. C 75, 430 (2015).
  * [37] J. Cleymans, M.D. Azmi, J. Phys. Conf. Ser. 668, 012050 (2016).
  * [38] B.C. Li, Z. Zhang, J.H. Kang, G.X. Zhang, F.H. Liu, Adv. High Energy Phys. 2015, 741816 (2015).
  * [39] P-Z Ning, L. Li, D-F Min, Foundation of Nuclear Physics: Nucleons and Nuclei, (Higher Education Press, Beijing, China, 2003).
  * [40] C.D. Dermer, Astrophys. J. 280, 328 (1984).
  * [41] C-R Meng, Chin. Phys. Lett. 26, 102501 (2009).
  * [42] V. Khachatryan et al. (CMS Collaboration), Phys. Lett. B 768, 103 (2017).
  * [43] S. Chatrchyan et al. (CMS Collaboration), Phys. Lett. B 724, 213 (2013).
  * [44] J. Cleymans, G.I. Lykasov, A.S. Parvan, A.S. Sorin, O.V. Teryaev, D. Worku, Phys. Lett. B 723, 351 (2013).
  * [45] K. Aamodt et al. (ALICE Collaboration), Phys. Lett. B 696, 328 (2011).
  * [46] V.A. Okorokov, Adv. High Energy Phys. 2015, 790646 (2015).
  * [47] A. Bialas, W. Florkowski, K.Zalewski, J. Phys. G 42, no. 4, 045001 (2015).
  * [48] A.N. Makhlin, Yu.M. Sinyukov, Z. Phys. C 39, 69 (1988).
  * [49] D. Boal, C.G. Gelbke, B. Jennings, Rev. Mod. Phys. 62, 553 (1990).
  * [50] U. Mayer, E. Schnedermann, U. Heinz, Phys. Lett. B 294, 69 (1992).
  * [51] S.V. Akkelin, Yu.M. Sinyukov, Phys. Lett. B 356, 525 (1995).
  * [52] S. Chapman, P. Scotto, U. Heinz, Phys. Rev. Lett. 74, 440 (1995).
  * [53] P. Castorina, H. Satz, Adv. High Energy Phys. 2014, 376982 (2014).
  * [54] A.N. Tawfik, H. Yassin, E.R.A. Elyazeed, Int. J. Mod. Phys. E 26, no. 03, 1750001 (2017).
  * [55] H. Zhao, F-H Liu, Adv. High Energy Phys. 2014, 742193 (2014).
  * [56] C-Y Wong, G. Wilk, Acta. Phys. Pol. B 43, 2047 (2012).
  * [57] K. Urmosy, G.G. Barnafoldi, T.S. Biro, Phys. Lett. B 718, 125 (2012).
  * [58] G. Wilk, Z. Wldarczyk, Eur. Phys. J. A 48, 161 (2012).
  * [59] L. Marques, J. Cleymans, A. Deppman, Phys. Rev. D 91, 054025 (2015).
  * [60] L. Marques, E. Andrade-II, A. Deppman, Phys. Rev. D 87, 114022 (2013).
  * [61] M.D. Azmi, J. Cleymans, J. Phys. G 41, 065001 (2014).


# Approximation properties of $q$-Bernoulli polynomials

M. Momenzadeh and I. Y. Kakangi
Near East University
Lefkosa, TRNC, Mersiin 10, Turkey
Email<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

We study the $q-$analogue of Euler-Maclaurin formula and by introducing a new
$q$-operator we drive to this form. Moreover, approximation properties of
$q$-Bernoulli polynomials is discussed. We estimate the suitable functions as
a combination of truncated series of $q$-Bernoulli polynomials and the error
is calculated. This paper can be helpful in two different branches, first
solving differential equations by estimating functions and second we may apply
these techniques for operator theory.

##  1 Introduction

The present study has sough to investigate the approximation of suitable
function $f(x)$ as a linear combination of $q-$Bernoulli polynomials. This
study by using $q-$operators through a parallel way has achived to the kind of
Euler expansion for $f(x)$, and expanded the function in terms of
$q-$Bernoulli polynomials. This expansion offers a proper tool to solve
$q$-difference equations or normal differential equation as well. There are
many approaches to approximate the capable functions. According to properties
of $q$-functions many $q$-function have been used in order to approximate a
suitable function.For example, at [28], some identities and formulae for the
q-Bernstein basis function, including the partition of unity property,
formulae for representing the monomials were studied. In addition, a kind of
approximation of a function in terms of Bernoulli polynomials are used in
several approaches for solving differential equations, such as [7, 20, 21].
This paper gives conditions to approximate capable functions as a linear
combination of $q$-Bernoulli polynomials as well as related examples. Also,
introduces the new $q-$operator to reach the $q-$analogue of Euler-Maclaurin
formula.

This study, first, introduces some $q$-calculus concepts. There are several
types of $q-$Bernoulli polynomials and numbers that can be generated by
different $q-$exponential functions. Carlitz [2] is pionier of introducing
$q$-analogue of the Bernoulli numbers, he applied a sequence
$\left\\{\beta_{m}\right\\}_{m\geq 0}$ in the middle of the 20th century;

$\sum_{k=0}^{m}\left(\begin{array}[c]{c}m\\\
k\end{array}\right)\beta_{k}q^{k+1}-\beta_{m}=\left\\{\begin{tabular}[c]{ll}$1,$&$m=1,$\\\
$0,$&$m>1.$\end{tabular}\ \ \ \ \ \ \ \right.$ (1)

First, this study makes an assumption that $\left|q\right|<1,$ and this
assumption is going to apply in the rest of the paper as well. If $q$ tends to
one from a left side the ordinary form would be reached.

Since Carlitz, there have been many distinct $q$-analogue of Bernoulli numbers
arising from varying motivations. In [3], they used improved $q-$exponential
functions to introduce a new class of $q-$Bernoulli polynomials. In addition,
they investigate some properties of these $q$-Bernoulli polynomials.

Let us introduce a class of $q-$Bernoulli polynomials in a form of generating
function as follows;

$\frac{ze_{q}(zt)}{e_{q}(z)-1}=\sum_{n=0}^{\infty}\mathit{\beta}_{n,q}(t)\frac{z^{n}}{\left[n\right]_{q}!},\text{
\ \ \ \ \ \ \ \ \ }\left|z\right|<2\pi.$ (2)

where $[n]_{q}$ is $q-$number and $q$-numbers factorial is defined by;

$\left[a\right]_{q}=\frac{1-q^{a}}{1-q}\ \ \ \left(q\neq 1\right);\ \ \
\left[0\right]_{q}!=1;\ \ \ \
\left[n\right]_{q}!=\left[n\right]_{q}\left[n-1\right]_{q}\ \ \ \
n\in\mathbb{N},\ \ a\in\mathbb{C}.$

The $q$-shifted factorial and $q$-polynomial coefficient are defined by the
following expressions respectively;

$\displaystyle\left(a;q\right)_{0}$ $\displaystyle=1,\ \ \
\left(a;q\right)_{n}={\displaystyle\prod\limits_{j=0}^{n-1}}\left(1-q^{j}a\right),\
\ \ n\in\mathbb{N},$ $\displaystyle\left(a;q\right)_{\infty}$
$\displaystyle={\displaystyle\prod\limits_{j=0}^{\infty}}\left(1-q^{j}a\right),\
\ \ \ \left|q\right|<1,\ \ a\in\mathbb{C}.$ $\left(\begin{array}[c]{c}n\\\
k\end{array}\right)_{q}=\frac{\left(q;q\right)_{n}}{\left(q;q\right)_{n-k}\left(q;q\right)_{k}},$

$q$-standard terminology and notation can be found at [1] and [6]. We call
$\mathit{\beta}_{n,q}$ Bernoulli number and
$\mathit{\beta}_{n,q}=\mathit{\beta}_{n,q}(0)$. In addition, $q$-analogue of
$\left(x-a\right)^{n}$ is defined as;

$\left(x-a\right)_{q}^{n}=\left\\{\begin{tabular}[c]{ll}$1,$&$n=0,$\\\
$(x-a)(x-aq)...(x-aq^{n-1}),$&$n\neq 0.$\end{tabular}\ \ \ \ \ \right.,$

In the standard approach to the $q-$calculus, two exponential function are
used.These $q-$exponentials are defined by;

$\displaystyle e_{q}\left(z\right)$
$\displaystyle=\sum_{n=0}^{\infty}\frac{z^{n}}{\left[n\right]_{q}!}=\prod_{k=0}^{\infty}\frac{1}{\left(1-\left(1-q\right)q^{k}z\right)},\
\ \ 0<\left|q\right|<1,\ \left|z\right|<\frac{1}{\left|1-q\right|},\ \ \ $
$\displaystyle\ \ E_{q}(z)$
$\displaystyle=e_{1/q}\left(z\right)=\sum_{n=0}^{\infty}\frac{q^{\frac{1}{2}n\left(n-1\right)}z^{n}}{\left[n\right]_{q}!}=\prod_{k=0}^{\infty}\left(1+\left(1-q\right)q^{k}z\right),\
\ \ \ \ \ \ 0<\left|q\right|<1,\ z\in\mathbb{C},$

$q-$shifted factorial can be expressed by Heine’s binomial formula as follow;

$\left(a;q\right)_{n}=\sum_{k=0}^{n}\left(\begin{array}[c]{c}n\\\
k\end{array}\right)_{q}q^{\frac{k(k-1)}{2}}\left(-1\right)^{k}a^{k}$

Let for some $0\leq\alpha<1,$ the function $|f(x)x^{\alpha}|$ is bounded on
the interval $\left(0,A\right]$, then Jakson integral defines as [1];

$\int\mathit{f}(x)d_{q}x=(1-q)x\sum_{i=0}^{\infty}q^{i}f(q^{i}x)$

Above expression converges to a function $F(x)$ on $\left(0,A\right],$ which
is a $q-$antiderivative of $f(x)$. Suppose $0<a<b$, the definite $q-$integral
is defined as;

$\displaystyle\int\limits_{0}^{b}\mathit{f}(x)d_{q}x$
$\displaystyle=(1-q)b\sum_{i=0}^{\infty}q^{i}f(q^{i}b)$
$\displaystyle\int\limits_{a}^{b}\mathit{f}(x)d_{q}x$
$\displaystyle=\int\limits_{0}^{b}\mathit{f}(x)d_{q}x-\int\limits_{0}^{a}\mathit{f}(x)d_{q}x$

We need to apply some properties of the $q$-Bernoulli polynomials to prepare
the approximation conditions. These properties are listed below as a lemma;

###### Lemma 1

Following statements holds true:

$\displaystyle a)\text{ }D_{q}\left(\mathit{\beta}_{n,q}(t)\right)$
$\displaystyle=\left[n\right]_{q}\mathit{\beta}_{n-1,q}(t),$ (3)
$\displaystyle b)\mathit{\beta}_{n,q}(t)$
$\displaystyle=\sum_{k=0}^{n}\binom{n}{k}_{q}\mathit{\beta}_{k,q}t^{n-k},$ (4)
$\displaystyle
c)\sum_{k=0}^{n}\binom{n}{k}_{q}\frac{\mathit{\beta}_{k,q}}{\left[n-k+1\right]_{q}}$
$\displaystyle=\delta_{0,n}\text{ where }\delta\text{ is kronecker delta}$ (5)
$\displaystyle d)\int\limits_{0}^{1}\mathit{\beta}_{n,q}(t)d_{q}t$
$\displaystyle=0\text{ \ \ \ }n\neq 0.$ (6)

Proof.

taking $q-$derivative from both side of $\left(2\right)$ to prove part (a). If
we apply Cauchy product for generating formula $(4)$, it leads to (b). Put
$x=0$ at part (b) and change boundary of summation to reach (c). If we take
Jackson integral directly from $\mathit{\beta}_{n,q}(t)$ with the aid of part
(c), we can reach (d).

A few numbers of these polynomials and related $q-$Bernoulli numbers can be
expressed as follows

$\begin{array}[c]{cc}\mathit{\beta}_{0,q}=1&\mathit{\beta}_{0,q}(t)=1\\\
\mathit{\beta}_{1,q}=-\frac{1}{[2]_{q}}&\mathit{\beta}_{1,q}(t)=t-\frac{1}{[2]_{q}}\\\
\mathit{\beta}_{2,q}=\frac{q^{2}}{[3]_{q}!}&\mathit{\beta}_{2,q}(t)=t^{2}-t+\frac{q^{2}}{[3]_{q}!}\\\
\mathit{\beta}_{3,q}=\text{
}\frac{q^{3}(1-q)}{[5]_{q}+[4]_{q}-1}&\mathit{\beta}_{3,q}(t)=t^{3}-\frac{[3]_{q}}{[2]_{q}}t^{2}+\frac{q^{2}}{[2]_{q}}t+\frac{q^{3}(1-q)}{[5]_{q}+[4]_{q}-1}\end{array}$

###### Definition 2

$q-$Bernoulli polynomials of two variables are defined as a generating
function as follow

$\frac{te_{q}(xt)e_{q}(yt)}{e_{q}(t)-1}=\sum_{n=0}^{\infty}\mathit{\beta}_{n,q}(x,y)\frac{t^{n}}{\left[n\right]_{q}!},\text{
\ \ \ \ \ \ \ \ \ }\left|t\right|<2\pi.$ (7)

A simple calculation of generating function leads us to

$te_{q}(xt)+\frac{t}{e_{q}(t)-1}e_{q}(tx)=\frac{te_{q}(xt)}{e_{q}(t)-1}e_{q}(t)=\sum_{n=0}^{\infty}\mathit{\beta}_{n,q}(x,1)\frac{t^{n}}{\left[n\right]_{q}!},\text{
\ \ \ \ \ \ \ \ \ }\left|t\right|<2\pi.$ (8)

The LHS can be written as

$\sum_{n=0}^{\infty}\frac{t^{n+1}x^{n}}{\left[n\right]_{q}!}+\sum_{n=0}^{\infty}\mathit{\beta}_{n,q}(x)\frac{t^{n}}{\left[n\right]_{q}!}\text{
\ \ \ \ \ \ \ \ }$ (9)

By comparing the nth coefficients, we drive to difference equations as follow

$\mathit{\beta}_{n,q}(x,1)-\mathit{\beta}_{n,q}(x)=\left[n\right]_{q}!x^{n-1}\text{
\ \ \ \ \ \ \ \ \ }n\geq 1.$ (10)

If we put $x=0,$ then $\mathit{\beta}_{n,q}(0,1)-\mathit{\beta}_{n,q}(0)=$
$\delta_{n,1}.$

##  2
q-analogue of Euler-Maclaurin formula

This section introduces $q-$operator to find $q-$analogue of Euler-Maclaurin
formula. The $q-$analogue of Euler-Maclaurin formula has been studied in [29].
They applied $q$-integral by parts to reach their formula. We can not apply
that approach to approximation, because it was written in term of
$p(x)=\mathit{\beta}_{n,q}(x-\left[x\right])$. Moreover, the errors has not
been studied and our $q-$operator which is totally new, leads us to the more
applicable function. That is why, this study situate $q-$Taylor theorem
first[6]

###### Theorem 3

If the function $f(x)$ is capable of expansion as a convergent power series
and if $q\neq$ root of unity, then

$f(x)=\sum_{n=0}^{\infty}\frac{\left(x-a\right)_{q}^{n}}{\left[n\right]_{q}!}\left(D_{q}^{n}f\right)(a)\
$ (11)

where $D_{q}(f(x))=\frac{f(xq)-f(x)}{x(q-1)}$ is $q-$derivative of $f(x),$ for
$x\neq 0,$ we can define it at $x=0$ as a normal derivative.

###### Definition 4

We define $H_{q}$ and $D_{h}$ operators as follow

$\displaystyle H_{q}^{n}(x)$
$\displaystyle=\frac{h}{1-q}\left(\frac{h}{1-q}+x\right)\left(\frac{h}{1-q}+\left[2\right]_{q}x\right)...\left(\frac{h}{1-q}+\left[n-1\right]_{q}x\right)\left(1-q\right)^{n}$
(12)
$\displaystyle=\left(x+h\right)^{n}\left(\frac{x}{x+h};n\right)_{q}=\sum_{k=0}^{n}\left(\begin{array}[c]{c}n\\\
k\end{array}\right)_{q}q^{\frac{k(k-1)}{2}}\left(x+h\right)^{n-k}\left(-x\right)^{k}\
$ (15) $\displaystyle D_{h}(F(x))$ $\displaystyle=\frac{F(x+h)-F(x)}{h}=f(x)$
(16)

In this definition we assume that $n\in\mathbb{N}$ and the functions and
values are well-defined $\left(h\neq 0,q\neq 1\right).$ The fist equality is
hold because of Heine’s binomial formula.

For a long time, mathematician have worked on the area of operators and they
solved several types of differential equations by using shifted-operators. The
$H-$operators rules like a bridge between the ordinary expansions and
$q$-analogue of these expansions. We may rewrite several formulae of these
area to the form of $q-$calculus such as [23, 24, 25, 26, 27]. Actually we may
write $q-$expansion of these functions. In a letter, Bernoulli concerned the
importence of this expansion for Leibnitz by these words” Nothing is more
elegent than the agreement, which you have observed between the numerical
power of the binomial and differetial expansions, there is no doubt that
something is hidden there” [26]

###### Theorem 5

(Fundemental Theorem of $h$-Calculus) If $F(x)$ is an $h$-derivative of $f(x)$
and $b-a\in h\mathbb{Z}$, we have[1]

$\int f(x)d_{h}x=F(b)-F(a)$ (17)

where we define $h$-integral as follows

$\int
f(x)d_{h}x=\left\\{\begin{array}[c]{c}h\left(f(a)+f(a+h)+...+f(b-h)\right)\ \
\ \ \text{if }a<b\\\ 0\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if }a=b\\\
-h\left(f(b)+f(b+h)+...+f(a-h)\right)\ \text{ \ if }a>b\end{array}\right.\ $
(18)

Now in the aid of $q-$Taylor expansion, we may write $F(x+h)$ as follows;

$F(x+h)=\sum_{j=0}^{\infty}\frac{\left((x+h)-x\right)_{q}^{j}}{\left[j\right]_{q}!}\left(D_{q}^{j}F\right)(x)\
=\sum_{j=0}^{\infty}\frac{D_{q}^{j}H_{q}^{j}}{\left[j\right]_{q}!}\left(F(x)\right)=e_{q}\left(D_{q}H_{q}\right)\left(F(x)\right)\
$ (19)

Here, $e_{q}$ is $q-$shifted operator and can be expressed as expansion of
$D_{q}$ and $H_{q}.$ We may assume that $D_{h}\left(F(x)\right)=f(x)$ and then

$f(x)=\frac{F(x+h)-F(x)}{h}=\frac{e_{q}\left(D_{q}H_{q}\right)-1}{H_{q}}F(x)$
(20)

Since the Jackson integral is $q-$antiderivative, we have

$\frac{H_{q}D_{q}}{e_{q}\left(D_{q}H_{q}\right)-1}\int f(x)d_{q}x=F(x)$ (21)

And in the aid of (2), the left hand side of the equation can be expressed as
a $q$-Bernoulli numbers, therefore

$\displaystyle F(x)$
$\displaystyle=\sum_{n=0}^{\infty}\mathit{\beta}_{n,q}\frac{\left(H_{q}D_{q}\right)^{n}}{\left[n\right]_{q}!}\int
f(x)d_{q}x$ (22) $\displaystyle=\int
f(x)d_{q}x-\frac{h}{\left[2\right]_{q}}f(x)+\sum_{n=1}^{\infty}\frac{\mathit{\beta}_{n,q}}{\left[n\right]_{q}!}\left(D_{q}^{n-1}f(x)\right)H_{q}^{n}(x)$
(23) $\displaystyle=\int
f(x)d_{q}x-\frac{h}{\left[2\right]_{q}}f(x)+\sum_{n=1}^{\infty}\left(\sum_{k=0}^{n}\left(\begin{array}[c]{c}n\\\
k\end{array}\right)_{q}q^{\frac{k(k-1)}{2}}\left(h+x\right)^{n-k}\left(-x\right)^{k}\right)\frac{\left(D_{q}^{n-1}f(x)\right)\mathit{\beta}_{n,q}}{\left[n\right]_{q}!}$
(26)

Thus we can state the following $q-$analogue of Euler-Maclaurian formula

###### Theorem 6

If the function $f(x)$ is capable of expansion as a convergent power series
and $f(x)$ decrease so rapidly with $x$ such that all normal derivatives
approach zero as $x\rightarrow\infty$, then we can express the series of
function $f(x)$ as follow

$\displaystyle\sum_{n=a}^{\infty}f(n)$
$\displaystyle=\int_{a}^{\infty}f(x)d_{q}x+\frac{1}{\left[2\right]_{q}}\left(f(a)\right)-$
(27)
$\displaystyle\lim_{b\rightarrow\infty}\sum_{n=1}^{\infty}\left(\sum_{k=0}^{n}\left(\begin{array}[c]{c}n\\\
k\end{array}\right)_{q}q^{\frac{k(k-1)}{2}}\left(\left(1+b\right)^{n-k}\left(-b\right)^{k}-\left(1+a\right)^{n-k}\left(-a\right)^{k}\right)\right)\frac{h^{n}\left(D_{q}^{n-1}f(a)\right)\mathit{\beta}_{n,q}}{\left[n\right]_{q}!}$
(30)

Proof. In the aid of theorem (5) and (18), If we suppose that $h=1$ and
$b-a\in\mathbb{N},$ we have

$\displaystyle\sum_{n=a}^{b-1}f(n)$
$\displaystyle=\int\limits_{a}^{b}f(x)d_{1}x=F(b)-F(a)$ (31)
$\displaystyle=\int_{a}^{b}f(x)d_{q}x-\frac{1}{\left[2\right]_{q}}\left(f(b)-f(a)\right)$
(32)
$\displaystyle+\sum_{n=1}^{\infty}\left(\sum_{k=0}^{n}\left(\begin{array}[c]{c}n\\\
k\end{array}\right)_{q}q^{\frac{k(k-1)}{2}}\left(\left(1+b\right)^{n-k}\left(-b\right)^{k}-\left(1+a\right)^{n-k}\left(-a\right)^{k}\right)\right)\frac{\left(D_{q}^{n-1}f(b)-D_{q}^{n-1}f(a)\right)\mathit{\beta}_{n,q}}{\left[n\right]_{q}!}$
(35)

Let $f(x)$ decrease so rapidly with $x$ then $D_{q}f(x)$ can be estimated by
$\frac{-f(x)}{x(q-1)},$ tend $x$ to infinity and use L’Hopital to reach
$\frac{-f^{\prime}(x)}{(q-1)}.$Now If normal derivatives of $f(x)$ is tend to
zero, then $q-$derivatives is also tending zero. The same discussion make
$D_{q}^{n-1}f(b)=0$ for big enough $b.$ Therefore

$\displaystyle\sum_{n=a}^{\infty}f(n)$
$\displaystyle=\int_{a}^{\infty}f(x)d_{q}x+\frac{1}{\left[2\right]_{q}}\left(f(a)\right)-$
(36)
$\displaystyle\lim_{b\rightarrow\infty}\sum_{n=1}^{\infty}\left(\sum_{k=0}^{n}\left(\begin{array}[c]{c}n\\\
k\end{array}\right)_{q}q^{\frac{k(k-1)}{2}}\left(\left(1+b\right)^{n-k}\left(-b\right)^{k}-\left(1+a\right)^{n-k}\left(-a\right)^{k}\right)\right)\frac{h^{n}\left(D_{q}^{n-1}f(a)\right)\mathit{\beta}_{n,q}}{\left[n\right]_{q}!}$
(39)

###### Example 7

Let $f(x)=x^{s}$ where $s$ is a positive integer, then we can apply this
function at (22) where $a=0$ and $b-1\in\mathbb{N}$

$\displaystyle\sum_{n=0}^{b-1}n^{s}$
$\displaystyle=\frac{b^{s+1}}{\left[s+1\right]_{q}}-\left(\frac{b^{s}}{\left[2\right]_{q}}\right)$
(40)
$\displaystyle+\sum_{n=2}^{s+1}\left(\sum_{k=0}^{n}\left(\begin{array}[c]{c}n\\\
k\end{array}\right)_{q}q^{\frac{k(k-1)}{2}}\left(\left(1+b\right)^{n-k}\left(-b\right)^{k}\right)\right)\left(\begin{array}[c]{c}s\\\
n-1\end{array}\right)_{q}\frac{b^{s-n+1}\mathit{\beta}_{n,q}}{\left[n\right]_{q}}$
(45)

This relation shows the sum of power in the combination of $q-$Bernoulli
numbers. when $q\rightarrow 1$ from the right side, we have an ordinary form
of this relation. For the another forms of sum of power, see [22]

###### Example 8

Let $f(x)=e_{q}^{-x}=\frac{1}{E_{q}^{x}}$ then this function decreases so
rapidly with $x$ such that all normal derivatives approach zero as
$x\rightarrow\infty.$ For comfirming this, let us mention that $E_{q}^{x},$
for some fixed $0<|q|<1$ and $|-x|<\frac{1}{|1-q|}$ is increasing rapidly,
since
$\frac{d}{dx}\left(E_{q}^{x}\right)=\sum\limits_{j=1}^{\infty}(1-q)q^{j}\prod\limits_{\begin{subarray}{c}k=0\\\
k\neq j\end{subarray}}^{\infty}\left(1+(1-q)q^{k}x\right)>0.$

$\sum_{n=0}^{\infty}e_{q}^{-n}=1+\frac{1}{\left[2\right]_{q}}-\lim_{b\rightarrow\infty}\sum_{n=1}^{\infty}\left(\sum_{k=0}^{n}\left(\begin{array}[c]{c}n\\\
k\end{array}\right)_{q}q^{\frac{k(k-1)}{2}}\left(\left(1+b\right)^{n-k}\left(-b\right)^{k}\right)\right)\frac{\mathit{\beta}_{n,q}}{\left[n\right]_{q}!}$
(46)

Moreover, this is $q$-analogue of
$\sum_{n=0}^{\infty}e^{-n}=\frac{-1}{e^{-1}-1}=\frac{3}{2}+\sum_{n=1}^{\infty}\frac{\mathit{\beta}_{2n}}{\left(2n\right)!},$
where $\mathit{\beta}_{2n}$ is a normal Bernoulli numbers that is generated by
$\frac{-1}{e^{-1}-1}.$

##  3 approximation by q-Bernoulli polynomial

We know that the class of $q$-Bernoulli polynomials are not in a form of
orthogonal polynomials. In addition, we may apply Gram-Schmit algorithm to
make these polynomials orthonormal then, according to Theorem 8.11 [19] we
have the best approximation in the form of Fourier series. Now, Instead of
using that algorithm, we apply properties of lemma (1) to achieve an
approximation. Let
$H=L_{q}^{2}\left[0,1\right]=\left\\{f_{q}:\left[0,1\right]\rightarrow\left[0,1\right]|\int\limits_{0}^{1}\left|f_{q}^{2}(t)\right|d_{q}t<\infty\right\\}$
be the Hilbert Space[18], then $\mathit{\beta}_{n,q}(x)\in H$ for $n=0,...,N$
and
$Y=Span\left\\{\mathit{\beta}_{0,q}(x),\mathit{\beta}_{1,q}(x),...,\mathit{\beta}_{N,q}(x)\right\\}$
is finite dimensional vector subspace of $H$. Unique best approximation for
any arbitrary elements of $H$ like $h,$ is $\widehat{h}\in Y$ such that for
any $y\in Y$ the inequality $\left\|h-\widehat{h}\right\|_{2,q}\leq$
$\left\|h-y\right\|_{2,q},$ where the norm is defined by
$\left\|f\right\|_{2,q}:=\left(\int\limits_{0}^{1}\left|f_{q}^{2}(t)\right|d_{q}t\right)^{\frac{1}{2}}.$
Following proposition determine the coefficient of $q-$Bernoulli polynomials,
when we estimate any $f\in L_{q}^{2}\left[0,1\right]$ by truncated
$q-$Bernoulli series. In fact, in order to see how well a certain partial sum
approximates the actual value, we would like to drive a formula similar to
(19), but with the infinite sum on the right hand side replaced by the $N$th
partial sum $S_{N\text{,}}$ plus an additional term $R_{N.}$

Sippose $a\in\mathbb{Z}$ and $b=a+1.$Consider the Nth partial sum

$S_{N}=\sum_{k=0}^{N}\frac{\mathit{\beta}_{k,q}}{\left[k\right]_{q}!}\left(H_{q}^{k}D_{q}^{k}\left(f\right)(a+1)-H_{q}^{k}D_{q}^{k}\left(f\right)(a)\right)$

Let $g(x,y)=\frac{\mathit{\beta}_{N,q}(x,y)}{\left[N\right]_{q}!},$ then in
the aid of lemma (1) we have
$D_{q}^{n-k}g(x,y)=\frac{\mathit{\beta}_{k,q}(x,y)}{\left[k\right]_{q}!}$ and
we have

$\displaystyle S_{N}$
$\displaystyle=\sum_{k=0}^{N}\left(\frac{\mathit{\beta}_{k,q}(0,0)}{\left[k\right]_{q}!}H_{q}^{k}D_{q}^{k}\left(f\right)(a+1)-\frac{\mathit{\beta}_{k,q}(0,1)}{\left[k\right]_{q}!}H_{q}^{k}D_{q}^{k}\left(f\right)(a)\right)-D_{q}\left(f\right)(a)$
$\displaystyle=\sum_{k=0}^{N}\left(D_{q}^{N-k}g(0,0)H_{q}^{k}D_{q}^{k}\left(f\right)(a+1)-D_{q}^{N-k}g(0,1)H_{q}^{k}D_{q}^{k}\left(f\right)(a)\right)-D_{q}\left(f\right)(a)$
$\displaystyle=D_{q}\left(f\right)(a)-\sum_{k=0}^{N}D_{q}^{N-k}g(0,x)H_{q}^{k}D_{q}^{k}\left(f\right)(a+1-x)|_{x=0}^{x=1}$

By taking $q-$derivative of the summation, we may rewrite this partial sum by
integral presentation, Moreover this relation give a boundary for $R_{N}(x).$

###### Proposition 9

Let $f\in L_{q}^{2}\left[0,1\right]$ and be estimated by truncated
$q-$Bernoulli series $\sum_{n=0}^{N}C_{n}\mathit{\beta}_{n,q}(x).$ Then
$C_{n}$ coefficients for $n=0,1,...,N$ can be calculated as an integral form
$C_{n}=\frac{1}{\left[n\right]_{q}!}\int\limits_{0}^{1}D_{q}^{(n)}f(x)d_{q}x$.
In addition we can write $f(x)$ in the following form

$f(x)=\int_{0}^{1}f(x)d_{q}x-\frac{h}{\left[2\right]_{q}}f(x)+\sum_{n=1}^{N}\left(\frac{1}{\left[n\right]_{q}!}\int\limits_{0}^{1}D_{q}^{(n)}f(x)d_{q}x\right)\mathit{\beta}_{n,q}(x)+R_{q,n}(x)$

Moreover $R_{q,n}(x)$ as a reminder part is bounded by
$\frac{2^{n}}{\left[N\right]_{q}!}Sup_{x\in\left[0,1\right]}|\mathit{\beta}_{n,q}(x)|Sup_{x\in\left[0,1\right]}|D_{q}^{(n)}f(x)|$

Proof. In fact we approximate $f(x)$ as a linear combinations of $q-$Bernoulli
polynomials and we assume that
$f(x)\simeq\sum_{n=0}^{N}C_{n}\mathit{\beta}_{n,q}(x),$ so take the Jackson
integral from both sides lead us to

$\int_{0}^{1}f(x)d_{q}x\simeq\sum_{n=0}^{N}C_{n}\int_{0}^{1}\mathit{\beta}_{n,q}(x)d_{q}x=C_{0}\int_{0}^{1}\mathit{\beta}_{0,q}(x)d_{q}x+C_{1}\int_{0}^{1}\mathit{\beta}_{1,q}(x)d_{q}x+...+C_{N}\int_{0}^{1}\mathit{\beta}_{N,q}(x)d_{q}x$

In the aid of Lemma 1 part (d) all the terms at the right side except the
first one has to be zero. we calculate $\mathit{\beta}_{0,q}(x)=1,$ so
$C_{0}=\frac{1}{\left[0\right]_{q}!}\int\limits_{0}^{1}f(x)d_{q}x.$Using part
(a) of that lemma for $q-$derivative of $q-$Bernoulli polynomial gather by
taking $q$-derivative of $f(x)$ leads to

$\int_{0}^{1}D_{q}\left(f(x)\right)d_{q}x\simeq\sum_{n=0}^{N}C_{n}\left[n\right]_{q}!\int_{0}^{1}\mathit{\beta}_{n-1,q}(x)d_{q}x=C_{1}\int_{0}^{1}\mathit{\beta}_{0,q}(x)d_{q}x=C_{1}$

Repeating this procedure $n$-times, yields the form of $C_{n}$ as we mention
it at theorem. We mention that if $x\in\left[0,1\right]$ and $0<|q|<1$, then
$H_{q}^{n}(x)$ for $h=1$ is bounded by $2^{n}.$In addition, the reminder part
can be presented by integral forms and this boundary can be found easily.

###  3.1 Further works

In this paper, we introduced a proper tool to approximate a given capable
function by combinations of q-Bernoulli polynomials. In spite of a lot of
investigations on q-Bernoulli polynomials, study of approximation properties
of the q-Bernoulli polynomials are not studied. We may apply these results to
solve q-difference equation, $q$-analogue of the things that is done in [20]
or [21]. The techniques of H-operator can be applied in operator theory.

##  4
conflict of interest

The authors declare that there is no conflict of interest regarding the
publication of this paper

## References

  * [1] kac, V. Cheung, P. (2000). Quantum Calculus. USA: New York.
  * [2] Carlitz, L. (1948). Duke Math. Journal. $q$-Bernoulli numbers and polynomials. (15) pp. 987–1000.
  * [3] Mahmudov, N, I. and Momenzadeh, M. (2014). Abstract and Applied Analysis. On a Class of q-Bernoulli, q-Euler, and q-Genocchi Polynomials. (2014) 10 pages.
  * [4] Van Assche, W. (2001). The Ramanujan Journal. Little $q-$Legendre polynomials and irrationality of certain Lambert (5). pp. 295-310.
  * [5] Cie´sli´nski, L. (2011). Applied Mathematics Letters. Improved q-exponential and q-trigonometric functions.(24). Issue 12. pp. 2110–2114.
  * [6] Ernst, T. (2000). the history of q-calculus and a new method. USA:Uppsala.
  * [7] Tohidi, E. Klicman, A. (2013). Hindawi Journal of applied analysis. A Collocation Method Based on the Bernoulli Operational Matrix for Solving Nonlinear BVPs Which Arise from the Problems in Calculus of Variatio. (2013) pp. 1-10.
  * [8] Evans, D, J. Raslan, K, R. (2005). Int. J. Comput. Math. The adomain decomposition method for solving differential equation (82) pp. 49-54.
  * [9] Jackson, F, H. (1908). Trans. Roy. Soc. Edinb. On q-functions and a certain difference operator. (46). pp. 64–72.
  * [10] Carmichael, R, D. (1913). Am. J. Math. On the theory of linear difference equations. (2)35. pp. 163–182.
  * [11] Chen, B, Q. Chen, X, Z. Li, S. (2010). Acta Math. Sin. (Engl. Ser.). Properties on solutions of some q-difference equations. (10)26. pp. 1877–1886 .
  * [12] Dobrogowska, A. Odzijewicz, A. (2006). J. Comput. Appl. Math. Second order q-difference equations solvable by factorization method. (1)193. pp. 319–346 .
  * [13] Abdel-Gawad, H, I. Aldailami, A, A. (2010). Appl. Math. Model. On q-dynamic equations modelling and complexity. (3)34. pp. 697–709 .
  * [14] Algin, A. (2011). Int. J. Theor. Phys. A comparative study on q-deformed fermion oscillators. (5)50. pp. 1554–1568.
  * [15] Dobrogowska, A. Odzijewicz, A. (2007). J. Phys. A Math. Theor. Solutions of the q-deformed Schr¨odinger equation for special potentials. (9)40. pp. 2023–2036 .
  * [16] Mabrouk, H. (2006). Fract.Calc.Appl.Anal. q-heat operatorand q-Poisson’s operator. (3)9. pp. 265–286 .
  * [17] A. Nemri, A. Fitouhi, (2010). Journal of Matematiche. Polynomial expansions for solution of wave equation in quantum calculus. (1)65. pp. 73–82 .
  * [18] Annaby, M, H. Mansour, Z. (2012). $q$-fractional calculus and equations, USA: New York.
  * [19] Rudin, W. (1964). Principles of mathematical analysis.USA: New York.
  * [20] Tohidi, E. zak, M. (2016). Mediterranean journal of mathematics. A new matrix approach for solving second-order linear matrix partial differential equations. (13). pp 1353-1376.
  * [21] Zogheib, B. Tohidi, E. (2016). Applied mathematics and computation. A new matrix method for solving two-dimentional time dependent diffusion equation with Dirichlet boundary conditions. (291). pp 1-12.
  * [22] Chan, O. Manna, D. (2010). Ams clas. A new q-analogue for Bernoulli numbers. (11).68. pp. 1-6.
  * [23] Carmichael, R. (1855). Longman, brown, green, and longmans. A treatise on the calculus of operations. pp 92-103.
  * [24] Ricci, P. Tavkelidze, I. (2009). Journal of mathematical sciences. An introduction to opertional techniques and special polynomials. (157)1. pp. 1-7.
  * [25] Ricci, P. Tavkelidze, I. (2009). Journal of mathematical sciences. Operational methods and solutions of boundary-value problems with periodic data. (157)1. pp. 1-8.
  * [26] Davis, H. (1936). The thory of linear operators. USA: Principia.
  * [27] Boole, G. (1880). A tretise on the calculus of finite differences. USA: Philadelphia.
  * [28] Goldman, R. Simeonov, P. Simsek, Y. (2014). SIAM Journal of Discrete Mathematics. Generating functions for the $q-$Bernstein bases. (28). pp. 1009-1025.
  * [29] Hegazi, A. Mansour, M. (2006). Journal of nonlinear mathematical physics. A note on $q$-Bernoulli numbers and polynomials. (13)1. pp. 9-18.