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On affinity relating two positive measures and the connection coefficients between polynomials orthogonalized by these measures
We consider two positive, normalized measures dA(x) and dB(x) related by the
relationship dA(x)=(C/(x+D))dB(x) or by dA(x) = (C/(x^2+E))dB(x) and dB(x) is
symmetric. We show that then the polynomial sequences {a_{n}(x)}, {b_{n}(x)}
orthogonal with respect to these measures are related by the relationship
a_{n}(x)=b_{n}(x)+{\kappa}_{n}b_{n-1}(x) or by a_{n}(x) = b_{n}(x) +
{\lambda}_{n}b_{n-2}(x) for some sequences {{\kappa}_{n}} and {{\lambda}_{n}}.
We present several examples illustrating this fact and also present some
attempts for extensions and generalizations. We also give some universal
identities involving polynomials {b_{n}(x)} and the sequence {{\kappa}_{n}}
that have a form of Fourier series expansion of the Radon--Nikodym derivative
of one measure with respect to the other
Comultiplication rules for the double Schur functions and Cauchy identities
The double Schur functions form a distinguished basis of the ring
\Lambda(x||a) which is a multiparameter generalization of the ring of symmetric
functions \Lambda(x). The canonical comultiplication on \Lambda(x) is extended
to \Lambda(x||a) in a natural way so that the double power sums symmetric
functions are primitive elements. We calculate the dual Littlewood-Richardson
coefficients in two different ways thus providing comultiplication rules for
the double Schur functions. We also prove multiparameter analogues of the
Cauchy identity. A new family of Schur type functions plays the role of a dual
object in the identities. We describe some properties of these dual Schur
functions including a combinatorial presentation and an expansion formula in
terms of the ordinary Schur functions. The dual Littlewood-Richardson
coefficients provide a multiplication rule for the dual Schur functions.Comment: 44 pages, some corrections are made in sections 2.3 and 5.
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