On the eigenfunctions for the multi-species q-Boson system

In a previous paper a multi-species version of the q-Boson stochastic particle system is introduced and the eigenfunctions of its backward generator are constructed by using a representation of the Hecke algebra. In this article we prove a formula which expresses the eigenfunctions by means of the q-deformed bosonic operators, which are constructed from the L-operator of higher rank found in the recent work by Garbali, de Gier and Wheeler. The L-operator is obtained from the universal R-matrix of the quantum affine algebra of type A_{r}^{(1)} by the use of the q-oscillator representation. Thus our formula may be regarded as a bridge between two approaches to studying integrable stochastic systems by means of the quantum affine algebra and the affine Hecke algebra.Comment: We added an explanation of the relation between our result and the previous results due to Borodin, and Motegi and Sakai in Section 5.

Differential equations compatible with boundary rational qKZ equation

We give differential equations compatible with the rational qKZ equation with boundary reflection. The total system contains the trigonometric degeneration of the bispectral qKZ equation of type (C_{n}^{\vee}, C_{n}) which in the case of type GL_{n} was studied by van Meer and Stokman. We construct an integral formula for solutions to our compatible system in a special case.Comment: 25 pages, no figure; Section 4.2, 4.4 and 4.6 are revised

Algebraic construction of multi-species q-Boson system

We construct a stochastic particle system which is a multi-species version of the q-Boson system due to Sasamoto and Wadati. Its transition rate matrix is obtained from a representation of a deformation of the affine Hecke algebra of type GL.Comment: 18 page

On solutions of the q-hypergeometric equation with q^{N}=1

We consider the q-hypergeometric equation with q^{N}=1 and $\alpha, \beta, \gamma \in {\Bbb Z}$. We solve this equation on the space of functions given by a power series multiplied by a power of the logarithmic function. We prove that the subspace of solutions is two-dimensional over the field of quasi-constants. We get a basis for this space explicitly. In terms of this basis, we represent the q-hypergeometric function of the Barnes type constructed by Nishizawa and Ueno. Then we see that this function has logarithmic singularity at the origin. This is a difference between the q-hypergeometric functions with 0<|q|<1 and at |q|=1.Comment: 9 page

The q-twisted cohomology and the q-hypergeometric function at |q|=1

We construct the q-twisted cohomology associated with the q-multiplicative function of Jordan-Pochhammer type at |q|=1. In this framework, we prove the Heine's relations and a connection formula for the q-hypergeometric function of the Barnes type. We also prove an orthogonality relation of the q-little Jacobi polynomials at |q|=1.Comment: 16 page

Emission of diffuse bands of sodium behind shock fronts

N/

On Form Factors of SU(2) Invariant Thirring Model

Integral formulae for form factors of a large family of charged local operators in SU(2) invariant Thirring model are given extending Smirnov's construction of form factors of chargeless local operators in the sine-Gordon model. New abelian symmetry acting on this family of local operators is found. It creates Lukyanov's operators which are not in the above family of local operators in general.Comment: 27 pages, 6 figure

Determinant Formula for the Solutions of the Quantum Knizhnik-Zamolodchikov Equation with |q|=1

The fundamental matrix solution of the quantum Knizhnik-Zamolodchikov equation associated with quantum affine sl2 algebra is constructed for |q|=1. The formula for its determinant is given in terms of the double sine function.Comment: 17 pages, submitted to Contemporary Math. Proceedings for a North-Carolina meetin

A restricted sum formula for a q-analogue of multiple zeta values

We prove a new linear relation for a q-analogue of multiple zeta values. It is a q-extension of the restricted sum formula obtained by Eie, Liaw and Ong for multiple zeta values.Comment: 12 pages, no figur

A deformation of affine Hecke algebra and integrable stochastic particle system

We introduce a deformation of the affine Hecke algebra of type GL which describes the commutation relations of the divided difference operators found by Lascoux and Schutzenberger and the multiplication operators. Making use of its representation we construct an integrable stochastic particle system. It is a generalization of the q-Boson system due to Sasamoto and Wadati. We also construct eigenfunctions of its generator using the propagation operator. As a result we get the same eigenfunctions for the (q, \mu, \nu)-Boson process obtained by Povolotsky.Comment: 18 pages, no fugur