Commuting differential operators and zonal spherical functions

1987 ◽  
pp. 189-200 ◽  
Author(s):  
I. G. Macdonald
Keyword(s):  
Differential Operators ◽  
Spherical Functions ◽  
Commuting Differential Operators

Commuting differential operators

2010 ◽  
Vol 162 (3) ◽  
pp. 276-285 ◽  
Author(s):  
A. B. Shabat ◽  
Z. S. Elkanova
Keyword(s):  
Differential Operators ◽  
Commuting Differential Operators

SPHERICAL FUNCTIONS, THE COMPLEX HYPERBOLIC PLANE AND THE HYPERGEOMETRIC OPERATOR

2006 ◽  
Vol 17 (10) ◽  
pp. 1151-1173 ◽  
Author(s):  
P. ROMÁN ◽  
J. TIRAO
Keyword(s):  
Hypergeometric Function ◽  
Hyperbolic Plane ◽  
Casimir Operator ◽  
Differential Operators ◽  
Real Variable ◽  
Spherical Functions ◽  
Matrix Coefficients ◽  
Vector Valued Function ◽  
Vector Valued ◽  
Complex Hyperbolic Plane

In this paper, we determine all irreducible spherical functions Φ of any K-type associated to the dual Hermitian symmetric pairs (G, K) = ( SU (3), U (2)) and ( SU (2,1), U (2)). This is accomplished by associating to Φ a vector valued function H = H(u) of a real variable u, analytic at u = 0, which is a simultaneous eigenfunction of two second order differential operators with matrix coefficients. One of them comes from the Casimir operator of G and we prove that it is conjugated to a hypergeometric operator, allowing us to express the function H in terms of a matrix valued hypergeometric function. For the compact pair ( SU (3), U (2)), this project was started in [4].

Burchnall–Chaundy theory, Ore extensions and σ-differential operators

2014 ◽  
Vol 13 (07) ◽  
pp. 1450049
Author(s):  
Daniel Larsson
Keyword(s):  
Algebraic Curve ◽  
Differential Operators ◽  
Classical Theorem ◽  
Precise Statement ◽  
Commuting Differential Operators ◽  
Ore Extensions

A classical theorem of J. L. Burchnall and T. W. Chaundy shows that two commuting differential operators P and Q give rise, via a differential resultant, to a complex algebraic curve with equation F (x, y) = 0, such that formally inserting P and Q for x and y in F (x, y) , gives identically zero. In addition, the points on this curve have coordinates which are exactly the eigenvalues associated with the operators P and Q (see the Introduction for a more precise statement). In this paper, we prove a generalization of this result using resultants in Ore extensions.

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