When do Linear Combinations of Orthogonal Polynomials Yield New Sequences of Orthogonal Polynomials? Articles uri icon

publication date

  • January 2010

start page

  • 1446

end page

  • 1452

issue

  • 6

volume

  • 233

international standard serial number (ISSN)

  • 0377-0427

electronic international standard serial number (EISSN)

  • 1879-1778

abstract

  • Given {P"n}"n">="0 a sequence of monic orthogonal polynomials, we analyze their linear combinations with constant coefficients and fixed length, i.e.,
    Q"n(x)=P"n(x)+a"1P"n"-"1(x)+...+a"kP"n"-"k,a"k<>0,n>k.
    Necessary and sufficient conditions are given for the orthogonality of
    the sequence {Q"n}"n">="0. An interesting interpretation in terms of
    the Jacobi matrices associated with {P"n}"n">="0 and {Q"n}"n">="0
    is shown. Moreover, in the case k=2, we characterize the families
    {P"n}"n">="0 such that the corresponding polynomials {Q"n}"n">="0
    are also orthogonal.