A Note on Quantum Coherence

In this paper, we discuss the coherence of the reduced state in system H _A ?H _B under taking different quantum operations acting on subsystem H _B . Firstly, we show that for a pure bipartite state, the coherence of the final subsystem H _A under the sum of two orthonormal rank 1 projections acting on H _B is less than or equal to the sum of the coherence of the state after two orthonormal projections acting on H _B , respectively. Secondly, we obtain that the coherence of reduced state in subsystem H _A under random unitary channel \({\Phi }(\rho )={\sum }_{s}\lambda _{s}U_{s}\rho U_{s}^{\ast }\) acting on H _B , is equal to the coherence of the state after each operation \({\Phi }_{s}(\rho )=\lambda _{s}U_{s}\rho U_{s}^{\ast }\) acting on H _B for every s. In addition, for general quantum operation \({\Phi }(\rho )={\sum }_{s}F_{s}\rho F_{s}^{\ast }\) on H _B , we get the relation

$$ C\left (\left ((I\otimes {\Phi })\rho ^{AB}\right )^{A}\right )\leq \sum \limits _{s}C\left (\left ((I\otimes {\Phi }_{s})\rho ^{AB}\right )^{A}\right ). $$

     

Evolution Law of the Optical Field of Degenerate Parametric Amplifier in Dissipative Channel

We explore the time-evolution law of the optical field of degenerate parametric amplifier (DPA) in dissipative channel. It turns out that its density operator at initial time ρ ₀ = A exp(E ^? a ^?2) exp(a ^? alnλ) exp(E a ²) evolves into \(\rho (t)= \frac {A}{\lambda ^{\prime }}\) \(\exp \left (\frac {E^{\ast }e^{-2\kappa t}a^{\dag 2}}{ \lambda ^{\prime 2}}\right )\exp \left \{a^{\dag }a\ln \frac {[\lambda -(\lambda ^{2}-4|E|^{2})T]e^{-2\kappa t}}{\lambda ^{\prime 2}}\right \} \exp \left (\frac { Ee^{-2\kappa t}a^{2}}{\lambda ^{\prime 2}}\right ),\) where κ is the damping constant of the channel, T = 1 ? e ^?2κt , and \(\lambda ^{\prime }\equiv \sqrt {(1-\lambda T)^{2}-4|E|^{2}T^{2}}.\) We employ the method of integration (or summation) within an ordered (normally ordered or antinormally ordered) of operators to overcome the obstacles in the process of calculation.     

Uncertainty Relation on Generalized Skew Information with aMonotone Pair

In this paper, we first define a generalized (f,g)-skew information \(\left |I_{ \rho }^{(f, g)}\right |(A)\) and two related quantity \(\left |J_{ \rho }^{(f, g)}\right |(A)\) and \(\left |U_{ \rho }^{(f, g)}\right |(A)\) for any non-Hermitian Hilbert-Schmidt operator A and a density operator ρ on a Hilbert space H and discuss some properties of them. And then, we obtain the following uncertainty relation in terms of \(\left |U_{ \rho }^{(f, g)}\right |(A)\):

$$\begin{array}{@{}rcl@{}} \left|U_{ \rho}^{(f, g)}\right|(A)\left|U_{ \rho}^{(f, g)}\right|(B)\geq \beta_{(f, g)}\left|Tr\left( f(\rho)g(\rho)[A, B]^{0}\right)\right|^{2}, \end{array} $$

which is a generalization of a known uncertainty relation in Ko and Yoo (J. Math. Anal. Appl. 383, 208–214, 11).

     

Domain and Range of the Modified Wave Operator for Schrödinger Equations with a Critical Nonlinearity

We study the final problem for the nonlinear Schrödinger equation

$i{\partial }_{t}u+\frac{1}{2}\Delta u=\lambda|u|^{\frac{2}{n}}u,\quad (t,x)\in {\mathbf{R}}\times \mathbf{R}^{n},$

where\(\lambda \in{\bf R},n=1,2,3\). If the final data\(u_{+}\in {\bf H}^{0,\alpha }=\left\{ \phi \in {\bf L}^{2}:\left( 1+\left\vert x\right\vert \right) ^{\alpha }\phi \in {\bf L}^{2}\right\} \) with\(\frac{ n}{2} < \alpha < \min \left( n,2,1+\frac{2}{n}\right) \) and the norm\(\Vert \widehat{u_{+}}\Vert _{{\bf L}^{\infty }}\) is sufficiently small, then we prove the existence of the wave operator in L ². We also construct the modified scattering operator from H ^0,α to H ^0,δ with\(\frac{n}{2} < \delta < \alpha\).

     

A new optical field state as an output of diffusion channel when the input being number state

By analyzing theoretical scheme of quantum controlling through photon addition,we propose a new optical field whose density operator asρ=λ(1-λ)l:Ll(-λ2aa/1-λ)e-λaa:(here::denotes normal ordering symbol),which is named Laguerre-polynomialweighted chaotic state.We show that such state is the solution to the master equation d/dtρ=-κ(aaρ+ρaa-aρa-aρa),describing a diffusion channel,with the initial number state|l l|,andλ=1/(1+κt).This new state is characteristic of possessing photon number l+κt at time t,so the photon number by adjusting the diffusion parameterκcan be controlled.This master equation is solved using the summation method within ordered product of operators and the entangled state representation.The physical difference between the diffusion and the amplitude damping is noted.     

Quantum Marginal Inequalities and the Conjectured Entropic Inequalities

A conjecture – the modified super-additivity inequality of relative entropy – was proposed in Zhang et al. (Phys. Lett. A 377:1794–1796, 2013): There exist three unitary operators \(U_{A}\in \mathrm {U}(\mathcal {H}_{A}), U_{B}\in \mathrm {U}(\mathcal {H}_{B})\) , and \(U_{AB}\in \mathrm {U}(\mathcal {H}_{A}\otimes \mathcal {H}_{B})\) such that $$\mathrm{S}\left(U_{AB}\rho_{AB}U^{\dagger}_{AB}||\sigma_{AB}\right)\geqslant \mathrm{S}\left(U_{A}\rho_{A}U^{\dagger}_{A}||\sigma_{A}\right) + \mathrm{S}\left(U_{B}\rho_{B}U^{\dagger}_{B}||\sigma_{B}\right), $$ where the reference state σ is required to be full-ranked. A numerical study on the conjectured inequality is conducted in this note. The results obtained indicate that the modified super-additivity inequality of relative entropy seems to hold for all qubit pairs.     

Rules of correspondence in atomic physics

The Smirnov method of analytic continuation (B.M. Smirnov, Sov. Phys. JETP 20, 345 (1964)) has been justified and developed for atomic physics. It has been shown that the polarizability of alkali atoms α, their van der Waals interaction constant C ₆, and the oscillator strength of the transition to the first P state f ₀₁ are related to the parameter 〈r ²〉 and gap in the spectrum \(\frac{3}{2}\frac{f}{\Delta } \approx \frac{3}{2}\alpha \Delta \approx {\left( {3{C_6}\Delta } \right)^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}}} \approx \left\langle {{r^2}} \right\rangle \). The average square of the coordinate of the valence electron 〈r ²〉 in the first approximation has a hydrogen dependence \({J_1} = \frac{1}{{2{v^2}}}.\) on the filling factor ν, which is defined in terms of the first ionization potential: xxxxxxxxx     

A new method for calculating three-particle interaction in the theory of modulus elasticity

We propose a new method for calculating the potential of multiparticle interaction. Our method considers the energy symmetry for clusters that contain N identical particles with respect to permutation of the number of atoms and free rotation in three-dimensional space. As an example, we calculate moduli of third-order rigidity for copper considering only the three-particle interaction. We analyze nine models of energy dependence on the polynomials that form the integral rational basis of invariants (IRBI) for the group G ₃ = O(3) ? P ₃. In this work, we use only the simplest relation between energy and the invariants forming the IRBI: \(\varepsilon \left( {\left. {i,k,l} \right|j} \right) = \sum\nolimits_{i,k,l} {\left[ { - A_1 r_{ik}^{ - 6} + A_2 r_{ik}^{ - 12} + Q_j I_j^{ - n} } \right]}\), where I _j is the invariant number j (j = 1, 2,..., 9). The results are in good agreement with the experimental values. The best agreement is observed at n = 2, j = 4: \(I_4 = \left( {\vec r_{ik} \vec r_{kl} } \right)\left( {\vec r_{kl} \vec r_{li} } \right) + \left( {\vec r_{kl} \vec r_{li} } \right)\left( {\vec r_{li} \vec r_{ik} } \right) + \left( {\vec r_{li} \vec r_{ik} } \right)\left( {\vec r_{ik} \vec r_{kl} } \right)\).     

Recent results on kaon decays by the NA48/2 experiment at CERN

The NA48/2 experiment reports the first observation of the rare decay K^± → π^±π⁰e⁺e^?, based on about 2000 candidates from 2003 data. The preliminary branching ratio in the full kinematic region is \(\mathcal {B}(K^{\pm } \to \pi ^{\pm }\pi ^{0}e^{+}e^{-})=(4.06\pm 0.17)\cdot 10^{-6}\). A sample of 4.687 × 10⁶\(K^{\pm }\to \pi ^{\pm }{\pi ^{0}_{D}}\) events collected in 2003/4 is analyzed to search for the dark photon (\(A^{\prime }\)) via the decay chain K^± → π^±π⁰, \(\pi ^{0}\to \gamma A^{\prime }\), \(A^{\prime }\to e^{+}e^{-}\). No signal is observed, limits in the plane mixing parameter ε² versus its mass \(m_{A^{\prime }}\) are reported.     

On Blow-up Profile of Ground States of Boson Stars with External Potential

We study minimizers of the pseudo-relativistic Hartree functional \({\mathcal {E}}_{a}(u):=\Vert (-\varDelta +m^{2})^{1/4}u\Vert _{L^{2}}^{2}+\int _{{\mathbb {R}}^{3}}V(x)|u(x)|^{2}\mathrm{d}x-\frac{a}{2}\int _{{\mathbb {R}}^{3}}(\left| \cdot \right| ^{-1}\star |u|^{2})(x)|u(x)|^{2}\mathrm{d}x\) under the mass constraint \(\int _{{\mathbb {R}}^3}|u(x)|^2\mathrm{d}x=1\). Here \(m>0\) is the mass of particles and \(V\ge 0\) is an external potential. We prove that minimizers exist if and only if a satisfies \(0\le a<a^{*}\), and there is no minimizer if \(a\ge a^*\), where \(a^*\) is called the Chandrasekhar limit. When a approaches \(a^*\) from below, the blow-up behavior of minimizers is derived under some general external potentials V. Here we consider three cases of V: trapping potential, i.e. \(V\in L_{\mathrm{loc}}^{\infty }({\mathbb {R}}^3)\) satisfies \(\lim _{|x|\rightarrow \infty }V(x)=\infty \); periodic potential, i.e. \(V\in C({\mathbb {R}}^3)\) satisfies \(V(x+z)=V(x)\) for all \(z\in \mathbb {Z}^3\); and ring-shaped potential, e.g. \( V(x)=||x|-1|^p\) for some \(p>0\).     

A remark on tensor representations

The identity $$\sum\limits_{v = 0} {\left( {\begin{array}{*{20}c} {n + 1} \\ v \\ \end{array} } \right)\left[ {\left( {\begin{array}{*{20}c} {n - v} \\ v \\ \end{array} } \right) - \left( {\begin{array}{*{20}c} {n - v} \\ {v - 1} \\ \end{array} } \right)} \right] = ( - 1)^n } $$ is proved and, by means of it, the coefficients of the decomposition ofD ₁ ⁿ into irreducible representations are found. It holds: ifD ₁ ⁿ \(\mathop {\sum ^n }\limits_{m = 0} A_{nm} D_m \) , then $$A_{nm} = \mathop \sum \limits_{\lambda = 0} \left( {\begin{array}{*{20}c} n \\ \lambda \\ \end{array} } \right)\left[ {\left( {\begin{array}{*{20}c} \lambda \\ {n - m - \lambda } \\ \end{array} } \right) - \left( {\begin{array}{*{20}c} \lambda \\ {n - m - \lambda - 1} \\ \end{array} } \right)} \right].$$      

Gleichzeitige Messung von Hyperfeinstruktur,Starkeffekt und Zeemaneffekt des133Cs19F mit einer Molekülstrahl-Resonanzapparatur

An electric molecular beam resonance spectrometer has been used to measure simultaneously the Zeeman- and Stark-effect splitting of the hyperfine structure of¹³³Cs¹⁹F. Electric four pole lenses served as focusing and refocusing fields of the spectrometer. A homogenous magnetic field (Zeeman field) was superimposed to the electric field (Stark field) in the transition region of the apparatus. Electrically induced (Δ m _J =±1)-transitions have been measured in theJ=1 rotational state, υ=0, 1 vibrational state. The obtained quantities are: The electric dipolmomentμ _el of the molecule for υ=0, 1; the rotational magnetic dipolmomentμ _J for υ=0, 1; the anisotropy of the magnetic shielding (σ _⊥-σ‖) by the electrons of both nuclei as well as the anisotropy of the molecular susceptibility (ξ _⊥-ξ‖), the spin rotational interaction constantsc _Cs andc _F, the scalar and the tensor part of the nuclear dipol-dipol interaction, the quadrupol interactioneqQ for υ=0, 1. The numerical values are:

$$\begin{gathered} \mu _{el} \left( {\upsilon = 0} \right) = 73878\left( 3 \right)deb \hfill \\ \mu _{el} \left( {\upsilon = 1} \right) - \mu _{el} \left( {\upsilon = 0} \right) = 0.07229\left( {12} \right)deb \hfill \\ \mu _J /J\left( {\upsilon = 0} \right) = - 34.966\left( {13} \right) \cdot 10^{ - 6} \mu _B \hfill \\ \mu _J /J\left( {\upsilon = 1} \right) = - 34.823\left( {26} \right) \cdot 10^{ - 6} \mu _B \hfill \\ \left( {\sigma _ \bot - \sigma _\parallel } \right)_{Cs} = - 1.71\left( {21} \right) \cdot 10^{ - 4} \hfill \\ \left( {\sigma _ \bot - \sigma _\parallel } \right)_F = - 5.016\left( {15} \right) \cdot 10^{ - 4} \hfill \\ \left( {\xi _ \bot - \xi _\parallel } \right) = 14.7\left( {60} \right) \cdot 10^{ - 30} erg/Gau\beta ^2 \hfill \\ c_{cs} /h = 0.638\left( {20} \right)kHz \hfill \\ c_F /h = 14.94\left( 6 \right)kHz \hfill \\ d_T /h = 0.94\left( 4 \right)kHz \hfill \\ \left| {d_s /h} \right|< 5kHz \hfill \\ eqQ/h\left( {\upsilon = 0} \right) = 1238.3\left( 6 \right) kHz \hfill \\ eqQ/h\left( {\upsilon = 1} \right) = 1224\left( 5 \right) kHz \hfill \\ \end{gathered} $$     

The Andreev conductance in superconductor–insulator–normal metal structures

The Andreev subgap conductance at 0.08–0.2 K in thin-film superconductor (aluminum)–insulator–normal metal (copper, hafnium, or aluminum with iron-sublayer-suppressed superconductivity) structures is studied. The measurements are performed in a magnetic field oriented either along the normal or in the plane of the structure. The dc current–voltage (I–U) characteristics of samples are described using a sum of the Andreev subgap current dominating in the absence of the field at bias voltages U < (0.2–0.4)Δ_c/e (where Δ_c is the energy gap of the superconductor) and the single-carrier tunneling current that predominates at large voltages. To within the measurement accuracy of 1–2%, the Andreev current corresponds to the formula \({I_n} + {I_s} = {K_n}\tanh \left( {{{eU} \mathord{\left/ {\vphantom {{eU} {2k{T_{eff}}}}} \right. \kern-\nulldelimiterspace} {2k{T_{eff}}}}} \right) + {K_s}{{\left( {{{eU} \mathord{\left/ {\vphantom {{eU} {{\Delta _c}}}} \right. \kern-\nulldelimiterspace} {{\Delta _c}}}} \right)} \mathord{\left/ {\vphantom {{\left( {{{eU} \mathord{\left/ {\vphantom {{eU} {{\Delta _c}}}} \right. \kern-\nulldelimiterspace} {{\Delta _c}}}} \right)} {\sqrt {1 - {{eU} \mathord{\left/ {\vphantom {{eU} {{\Delta _c}}}} \right. \kern-\nulldelimiterspace} {{\Delta _c}}}} }}} \right. \kern-\nulldelimiterspace} {\sqrt {1 - {{eU} \mathord{\left/ {\vphantom {{eU} {{\Delta _c}}}} \right. \kern-\nulldelimiterspace} {{\Delta _c}}}} }}\) following from a theory that takes into account mesoscopic phenomena with properly selected effective temperature T _eff and the temperature- and fieldindependent parameters K _n and K _s (characterizing the diffusion of electrons in the normal metal and superconductor, respectively). The experimental value of K _n agrees in order of magnitude with the theoretical prediction, while K _s is several dozen times larger than the theoretical value. The values of T _eff in the absence of the field for the structures with copper and hafnium are close to the sample temperature, while the value for aluminum with an iron sublayer is several times greater than this temperature. For the structure with copper at T = 0.08–0.1 K in the magnetic field B_|| = 200–300 G oriented in the plane of the sample, the effective temperature T _eff increases to 0.4 K, while that in the perpendicular (normal) field B _⊥ ≈ 30 G increases to 0.17 K. In large fields, the Andreev conductance cannot be reliably recognized against the background of single- carrier tunneling current. In the structures with hafnium and in those with aluminum on an iron sublayer, the influence of the magnetic field is not observed.     

Spherical Model on a Cayley Tree: Large Deviations

We study the spherical model of a ferromagnet on a Cayley tree and show that in the case of empty boundary conditions a ferromagnetic phase transition takes place at the critical temperature \(T_\mathrm{c} =\frac{6\sqrt{2}}{5}J\), where J is the interaction strength. For any temperature the equilibrium magnetization, \(m_n\), tends to zero in the thermodynamic limit, and the true order parameter is the renormalized magnetization \(r_n=n^{3/2}m_n\), where n is the number of generations in the Cayley tree. Below \(T_\mathrm{c}\), the equilibrium values of the order parameter are given by \(\pm \rho ^*\), where

$$\begin{aligned} \rho ^*=\frac{2\pi }{(\sqrt{2}-1)^2}\sqrt{1-\frac{T}{T_\mathrm{c}}}. \end{aligned}$$

One more notable temperature in the model is the penetration temperature

$$\begin{aligned} T_\mathrm{p}=\frac{J}{W_\mathrm{Cayley}(3/2)}\left( 1-\frac{1}{\sqrt{2}}\left( \frac{h}{2J}\right) ^2\right) . \end{aligned}$$

Below \(T_\mathrm{p}\) the influence of homogeneous boundary field of magnitude h penetrates throughout the tree. The main new technical result of the paper is a complete set of orthonormal eigenvectors for the discrete Laplace operator on a Cayley tree.

     

Time Delay for the Dirac Equation

We consider time delay for the Dirac equation. A new method to calculate the asymptotics of the expectation values of the operator \({\int\limits_{0} ^{\infty}{\rm e}^{iH_{0}t}\zeta(\frac{\vert x\vert }{R}) {\rm e}^{-iH_{0}t}{\rm d}t}\), as \({R \rightarrow \infty}\), is presented. Here, H₀ is the free Dirac operator and \({\zeta\left(t\right)}\) is such that \({\zeta\left(t\right) = 1}\) for \({0 \leq t \leq 1}\) and \({\zeta\left(t\right) = 0}\) for \({t > 1}\). This approach allows us to obtain the time delay operator \({\delta \mathcal{T}\left(f\right)}\) for initial states f in \({\mathcal{H} _{2}^{3/2+\varepsilon}(\mathbb{R}^{3};\mathbb{C}^{4})}\), \({\varepsilon > 0}\), the Sobolev space of order \({3/2+\varepsilon}\) and weight 2. The relation between the time delay operator \({\delta\mathcal{T}\left(f\right)}\) and the Eisenbud–Wigner time delay operator is given. In addition, the relation between the averaged time delay and the spectral shift function is presented.     

Constraining the electric charges of some astronomical bodies in Reissner–Nordström spacetimes and generic <Emphasis Type="Italic">r</Emphasis><Superscript>−2</Superscript>-type power-law potentials from orbital motions

We put independent model dynamical constraints on the net electric charge Q of some astronomical and astrophysical objects by assuming that their exterior spacetimes are described by the Reissner-Nordström, metric, which induces an additional potential \({U_{\rm RN} \propto Q^2 r^{-2}}\). From the current bounds \({\Delta \dot \varpi}\) on any anomalies in the secular perihelion rate \({\dot \varpi}\) of Mercury and the Earth–mercury ranging Δρ, we have \({\left|Q_{\odot}\right| \lesssim 1-0.4 \times 10^{18}\ {\rm C}}\). Such constraints are ~60–200 times tighter than those recently inferred in literature. For the Earth, the perigee precession of the Moon, determined with the Lunar Laser Ranging technique, and the intersatellite ranging Δρ for the GRACE mission yield \({\left|Q_{\oplus} \right| \lesssim 5-0.4 \times 10^{14}\ {\rm C}}\). The periastron rate of the double pulsar PSR J0737-3039A/B system allows to infer \({\left|Q_{\rm NS} \right| \lesssim 5\times 10^{19}\ {\rm C}}\). According to the perinigricon precession of the main sequence S2 star in Sgr A^*, the electric charge carried by the compact object hosted in the Galactic Center may be as large as \({\left|Q_{\bullet} \right| \lesssim 4\times 10^{27} \ {\rm C}}\). Our results extend to other hypothetical power-law interactions inducing extra-potentials \({U_{\rm pert} = \Psi r^{-2}}\) as well. It turns out that the terrestrial GRACE mission yields the tightest constraint on the parameter \({\Psi}\), assumed as a universal constant, amounting to \({|\Psi| \lesssim 5\times 10^{9}\ {\rm m^4\ s^{-2}}}\).     

Spherically Symmetric Solutions of Light Galileon

We have been studied the model of light Galileon with translational shift symmetry ? → ? + c. The matter Lagrangian is presented in the form \(\mathcal {L}_{\phi }= -\eta (\partial \phi )^{2}+\beta G^{\mu \nu }\partial _{\mu }\phi \partial _{\nu }\phi \). We have been addressed two issues: the first is that, we have been proven that, this type of Galileons belong to the modified matter-curvature models of gravity in type of \(f(R,R^{\mu \nu }T_{\mu \nu }^{m})\). Secondly, we have been investigated exact solution for spherically symmetric geometries in this model. We have been found an exact solution with singularity at r = 0 in null coordinates. We have been proven that the solution has also a non-divergence current vector norm. This solution can be considered as an special solution which has been investigated in literature before, in which the Galileon’s field is non-static (time dependence). Our scalar-shift symmetrized Galileon has the simple form of ? = t, which it is remembered by us dilaton field.     

Unifying the theory of integration within normal-, Weyl- and antinormal-ordering of operators and the s-ordered operator expansion formula of density operators

By introducing the s-parameterized generalized Wigner operator into phase-space quantum mechanics we invent the technique of integration within s-ordered product of operators(which considers normally ordered,antinormally ordered and Weyl ordered product of operators as its special cases).The s-ordered operator expansion(denoted by...) formula of density operators is derived,which is ρ = 2 1 s ∫ d2βπβ|ρ |β exp { 2 s 1(s|β|2 β a + βa a a) }.The s-parameterized quantization scheme is thus completely established.     

Higher recurrences for Apostol-Bernoulli-Euler numbers

We present explicit formulas for sums of products of Apostol-Bernoulli and Apostol-Euler numbers of the form

$\sum\limits_{_{m_1 , \cdots ,m_N \geqslant n}^{m_1 + \cdots + m_N = n} } {\left( {_{m_1 , \cdots m_N }^n } \right)B_{m_1 } (q) \cdots B_{m_N } (q),} \sum\limits_{_{m_1 , \cdots ,m_N \geqslant n}^{m_1 + \cdots + m_N = n} } {\left( {_{m_1 , \cdots m_N }^n } \right)E_{m_1 } (q) \cdots E_{m_N } (q),}$

where N and n are positive integers, B _m (q) _n stand for the Apostol-Bernoulli numbers, E _m (q) for the Apostol-Euler numbers, and \(\left( {\begin{array}{*{20}c} n \\ {m_1 , \cdots ,m_N } \\ \end{array} } \right) = \frac{{n!}}{{m_1 ! \cdots m_N !}}.\) Our formulas involve Stirling numbers of the first kind. We also derive results for Apostol-Bernoulli and Apostol-Euler polynomials. As an application, for q = 1 we recover results of Dilcher, and our paper can be regarded as a q-extension of that of Dilcher.