Related papers: Baker-Akhiezer functions and generalised Macdonald…
We develop further quaternionic analysis introducing left and right doubly regular functions. We derive Cauchy-Fueter type formulas for these doubly regular functions that can be regarded as another counterpart of Cauchy's integral formula…
A brief non-technical review of the recent study of classical integrable structures in quantum integrable systems is given. It is explained how to identify the standard objects of quantum integrable systems (transfer matrices, Baxter's…
Any generalization of the method of Godement-Jacquet on principal L-functions for GL(n) to other groups as perceived by Braverman-Kazhdan and Ngo requires a Fourier transform on a space of Schwartz functions. In the case of standard…
We consider the generalised Calogero-Moser-Sutherland quantum integrable system associated to the configuration of vectors $AG_2$, which is a union of the root systems $A_2$ and $G_2$. We establish the existence of and construct a suitably…
Some new formulas for the KP hierarchy are derived from the differential Fay identity. They proved to be useful for the $k$-constrained hierarchies providing a series of determinant identities for them. A differential equation is introduced…
We generalize module weak-* Haagerup tensor products to obtain complete quotients of normal Haagerup tensor product included in canonical Hilbert spaces associated to completely positive normal (covariance) maps $\eta$ on a finite von…
We construct differential algebras in which spaces of (one-dimensional) periodic ultradistributions are embedded. By proving a Schwartz impossibility type result, we show that our embeddings are optimal in the sense of being consistent with…
We consider Baxter Q-operators for various versions of quantum affine Toda chain. The interpretation of eigenvalues of the finite Toda chain Baxter operators as local Archimedean L-functions proposed recently is generalized to the case of…
We study a natural measurable selection problem for which the standard uniformisation theorems do not seem to apply directly, yet a Borel selector exists. More precisely, we consider families of finite dimensional functions that admit…
We continue the study of the Hrushovski-Kazhdan integration theory and consider exponential integrals. The Grothendieck ring is enlarged via a tautological additive character and hence can receive such integrals. We then define the Fourier…
In earlier joint work with Ruijsenaars, we constructed and studied symmetric joint eigenfunctions $J_N$ for quantum Hamiltonians of the hyperbolic relativistic $N$-particle Calogero--Moser system. For generic coupling values, they are…
In this work we consider the $\eta$-invariant for pseudodifferential operators of tensor product type, also called bisingular pseudodifferential operators. We study complex powers of classical bisingular operators. We prove the trace…
We introduce the notion of discrete Baker-Akhiezer (DBA) modules, which are modules over the ring of difference operators, as a certain discretization of Baker-Akhiezer modules which are modules over the ring of differential operators. We…
Hypergeometric functions and their generalizations play an important r\^{o}les in diverse applications. Many authors have been established generalizations of hypergeometric functions by a number ways. In this paper, we aim at establishing…
In the first part of this paper, we express the generalized Bessel function associated with dihedral systems and a constant multiplicity function as a infinite series of confluent Horn functions. The key ingredient leading to this…
The Grothendieck-Serre formula for the difference between the Hilbert function and Hilbert polynomial of a graded algebra is generalized for bigraded standard algebras. This is used to get a similar formula for the difference between the…
A unifying scheme of classical special functions of hypergeometric type obeying orthogonality or biorthogonality relations is described. It expands the Askey scheme of classical orthogonal polynomials and its $q$-analogue based on the…
The Nekrasov-Okounkov formula gives an expression for the Fourier coefficients of the Euler functions as a sum of hook length products. This formula can be deduced from a specialization in a renormalization of the affine type $A$ Weyl…
Using a different approach, we derive integral representations for the Riemann zeta function and its generalizations (the Hurwitz zeta, $\zeta(-k,b)$, the polylogarithm, $\mathrm{Li}_{-k}(e^m)$, and the Lerch transcendent,…
We construct a family of BC_n-symmetric biorthogonal abelian functions generalizing Koornwinder's orthogonal polynomials, and prove a number of their properties, most notably analogues of Macdonald's conjectures. The construction is based…