arXiv subjects
Publications and source records attributed to Masatoshi Noumi.
Determinant formulas for the general solutions of the Toda and discrete Toda equations are presented. Application to the $τ$ functions for the Painlevé equations is also discussed.
A series of systems of nonlinear equations with affine Weyl group symmetry of type $A^{(1)}_l$ is studied. This series gives a generalization of Painlevé equations $P_{IV}$ and $P_{V}$ to higher orders.
A new class of representations of affine Weyl groups on rational functions are constructed, in order to formulate discrete dynamical systems associated with affine root systems. As an application, some examples of difference and differential systems of Painleve type are discussed.
Raising operators of row type are constructed by means of an interpolation method. These are a dual version of the raising operators of column type by A.N.Kirillov and M.Noumi. An extension of the q-binomial coefficients is introduced in relation to the raising operators.
We propose a new representation of the fourth Painlevé equation in which the $A^{(1)}_2$-symmetries become clearly visible. By means of this representation, we clarify the internal relation between the fourth Painlevé equation and the modified KP hierarchy. We obtain in particular a complete description of the rational solutions of the fourth Painlevé equation in terms of Schur functions. This implies that the so-called Okamoto polynomials, which arise from the $τ$-functions for rational solutions, are in fact expressible by the 3-reduced Schur functions.
We introduce certain raising and lowering operators for Macdonald polynomials (of type $A_{n-1}$) by means of Dunkl operators. The raising operators we discuss are a natural $q$-analogue of raising operators for Jack polynomials introduced by L.Lapointe and L.Vinet. As an application we prove the integrality of double Kostka coefficients. Double analog of the multinomial coefficients are introduced.
We study certain $q$-difference raising operators for Macdonald polynomials (of type $A_{n-1}$) which are originated from the $q$-difference-reflection operators introduced in our previous paper. These operators can be regarded as a $q$-difference version of the raising operators for Jack polynomials introduced by L.Lapointe and L.Vinet. As an application of our $q$-difference raising operators we give an elementary proof of the integrality of the double Kostka coefficients which had been conjectured I.G. Macdonald. We also determine their quasi-classical limits, which give rise to (differental) raising operators for Jack polynomials.
Quantum analogues of the homogeneous spaces $\GL(n)/\SO(n)$ and $\GL(2n)/\Sp(2n)$ are introduced. The zonal spherical functions on these quantum homogeneous spaces are represented by Macdonald's symmetric polynomials $P_{\ld}=P_{\ld}(x_1,\cdots,x_n;q,t)$ with $t=q^{1 \over 2}$ or $t=q^2$.
A class of quantum analogues of compact symmetric spaces of classical type is introduced by means of constant solutions to the reflection equations. Their zonal spherical functions are discussed in connection with $q$-orthogonal polynomials.