(M, N) -Coherent pairs of linear functionals and Jacobi matrices Articles uri icon

publication date

  • April 2014

start page

  • 76

end page

  • 83

volume

  • 232

international standard serial number (ISSN)

  • 0096-3003

electronic international standard serial number (EISSN)

  • 1873-5649

abstract

  • A pair of regular linear functionals (U,V) in the linear space of polynomials with complex coefficients is said to be an (M,N)-coherent pair of order m if their corresponding sequences of monic orthogonal polynomials { Pn(x)}n≥0 and {Qn(x)}n≥0 satisfy a structure relationSigmai=0Mai,nPn+m-i(m)(x)=Sigmai= 0Nbi,nQn-i(x),n≥0,where M,N, and m are non-negative integers, {ai,n}n≥0,0≤i≤M, and { bi,n}n≥0,0≤i≤N, are sequences of complex numbers such that aM,n≠0 if n≥M,bN,n≠0 if n≥N, and ai,n=bi,n=0 if i>n. When m=1,(U,V) is called an (M,N)-coherent pair. In this work, we give a matrix interpretation of (M,N)-coherent pairs of linear functionals. Indeed, an algebraic relation between the corresponding monic tridiagonal (Jacobi) matrices associated with such linear functionals is stated. As a particular situation, we analyze the case when one of the linear functionals is classical. Finally, the relation between the Jacobi matrices associated with (M,N)-coherent pairs of linear functionals of order m and the Hessenberg matrix associated with the multiplication operator in terms of the basis of monic polynomials orthogonal with respect to the Sobolev inner product defined by the pair (U,V) is deduced.

keywords

  • coherent pairs; structure relations; regular linear functionals; orthogonal polynomials; classical orthogonal polynomials; sobolev orthogonal polynomials; monic jacobi matrix