Pablo Román

Matrix-Valued Orthogonal Polynomials Related to (SU(2)×SU(2), diag)

The matrix-valued spherical functions for the pair (K x K, K), K=SU(2), are studied. By restriction to the subgroup A the matrix-valued spherical functions are diagonal. For suitable set of representations we take these diagonals into a matrix-valued function, which are the full spherical functions. Their orthogonality is a consequence of the Schur orthogonality relations. From the full spherical functions we obtain matrix-valued orthogonal polynomials of arbitrary size, and they satisfy a three-term recurrence relation which follows by considering tensor product decompositions. An explicit expression for the weight and the complete block-diagonalization of the matrix-valued orthogonal polynomials is obtained. From the explicit expression we obtain right-hand sided differential operators of first and second order for which the matrix-valued orthogonal polynomials are eigenfunctions. We study the low-dimensional cases explicitly, and for these cases additional results, such as the Rodrigues formula and being eigenfunctions to first order differential-difference and second order differential operators, are obtained.

Matrix-valued Orthogonal Polynomials Related to (SU(2)

In a previous paper we have introduced matrix-valued analogues of the Chebyshev polynomials by studying matrix-valued spherical functions on SU(2)\times SU(2). In particular the matrix-size of the polynomials is arbitrarily large. The matrix-valued orthogonal polynomials and the corresponding weight function are studied. In particular, we calculate the LDU-decomposition of the weight where the matrix entries of L are given in terms of Gegenbauer polynomials. The monic matrix-valued orthogonal polynomials P_n are expressed in terms of Tiraos matrix-valued hypergeometric function using the matrix-valued differential operator of first and second order to which the P_ns are eigenfunctions. From this result we obtain an explicit formula for coefficients in the three-term recurrence relation satisfied by the polynomials P_n. These differential operators are also crucial in expressing the matrix entries of P_nL as a product of a Racah and a Gegenbauer polynomial. We also present a group theoretic derivation of the matrix-valued differential operators by considering the Casimir operators corresponding to SU(2)\times SU(2).

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We study the asymptotic zero distribution of type II multiple orthogonal polynomials associated with two Macdonald functions (modified Bessel functions of the second kind). On the basis of the four-term recurrence relation, it is shown that, after proper scaling, the sequence of normalized zero counting measures converges weakly to the first component of a vector of two measures which satisfies a vector equilibrium problem with two external fields. We also give the explicit formula for the equilibrium vector in terms of solutions of an algebraic equation.

SU(2), diag), II

We present a method to obtain infinitely many examples of pairs (W;D) consis- ting of a matrix weight W in one variable and a symmetric second-order differential opera- torD. The method is based on a uniform construction of matrix valued polynomials starting from compact Gelfand pairs (G;K) of rank one and a suitable irreducible K-representation. The heart of the construction is the existence of a suitable base change 0. We analyze the base change and derive several properties. The most important one is that 0 satisfies a first-order differential equation which enables us to compute the radial part of the Casimir operator of the group G as soon as we have an explicit expression for 0. The weight W is also determined by 0. We provide an algorithm to calculate 0 explicitly. For the pair (USp(2n); USp(2n 2) USp(2)) we have implemented the algorithm in GAP so that individual pairs (W;D) can be calculated explicitly. Finally we classify the Gelfand pairs (G;K) and theK-representations that yield pairs (W;D) of size 2 2 and we provide explicit expressions for most of these cases.

The asymptotic zero distribution of multiple orthogonal polynomials associated with Macdonald functions

Burchnalls method to invert the Feldheim-Watson linearization formula for the Hermite polynomials is extended to all polynomial families in the Askey-scheme and its

Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One

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Generalized Burchnall-Type Identities for Orthogonal Polynomials and Expansions

-analogue. The resulting expansion formulas are made explicit for several families corresponding to measures with infinite support, including the Wilson and Askey-Wilson polynomials. An integrated version gives the possibility to give alternate expression for orthogonal polynomials with respect to a modified weight. This gives expansions for polynomials, such as Hermite, Laguerre, Meixner, Charlier, Meixner-Pollaczek and big

Neuronal (Bi)Polarity as a Self-Organized Process Enhanced by Growing Membrane

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Non-intersecting squared Bessel paths with one positive starting and ending point

-Jacobi polynomials and big

Matrix-valued Gegenbauer polynomials

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