This paper deals with the classical problem of how to evaluate the joint rank statistics distributions for two independent i.i.d. samples from a common continuous distribution. It is pointed that ...these distributions rely on an underlying polynomial structure of negative binomial type. That property is exploited to obtain, in a systematic and unified way, closed forms and simple recursions, some well established, for computing the joint tail and rectangular probabilities of interest.
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We introduce a class of Schur type functions associated with polynomial sequences of binomial type. This can be regarded as a generalization of the ordinary Schur functions and the factorial Schur ...functions. This generalization satisfies some interesting expansion formulas, in which there is a curious duality. Moreover, this class includes examples which are useful to describe the eigenvalues of Capelli type central elements of the universal enveloping algebras of classical Lie algebras.
In this paper, we study the Jordan canonical form of the generalized Pascal functional matrix associated with a sequence of binomial type, and demonstrate that the transition matrix between the ...generalized Pascal functional matrix and its Jordan canonical form is the iteration matrix associated with the binomial sequence. In addition, some combinatorial identities are derived from the corresponding matrix factorization.
The basic results of spectral theory are obtained using the sequence of powers of a bounded linear operator
T,
T
2,…,
T
n
,…. In this paper, we replace the powers
T
n
by certain polynomials
p
n
(
T), ...and make use of special properties of the polynomial sequence
{
p
n}
n⩾0
to derive some new results concerning operators. For example, using an arbitrary polynomial sequence
{
p
n}
n⩾0
, we obtain “binomial” spectral radii and semidistances, which reduce, in the case of the sequence of powers, to the usual spectral radius and semidistance.
We give a matrix generalization of the family of exponential polynomials in one variable φk(x). Our generalization consists of a matrix of polynomials Φk(X)=(Φ(k)i, j(X))ni, j=1 depending on a matrix ...of variables X=(xi, j)ni, j=1. We prove some identities of the matrix exponential polynomials which generalize classical identities of the ordinary exponential polynomials. We also introduce matrix generalizations of the decreasing factorial (x)k=x(x−1)(x−2)…(x−k+1), the increasing factorial (x)(k)=x(x+1)(x+2)…(x+k−1), and the Laguerre polynomials. These polynomials have interesting combinatorial interpretations in terms of different kinds of walks on directed graphs.
Generalized Rook Polynomials Goldman, Jay; Haglund, James
Journal of combinatorial theory. Series A,
07/2000, Letnik:
91, Številka:
1-2
Journal Article
Recenzirano
Odprti dostop
Generalizing the notion of placing rooks on a Ferrers board leads to a new class of combinatorial models and a new class of rook polynomials. Connections are established with absolute Stirling ...numbers and permutations, Bessel polynomials, matchings, multiset permutations, hypergeometric functions, Abel polynomials and forests, and polynomial sequences of binomial type. Factorization and reciprocity theorems are proved and a q-analogue is given.
We apply recent constructions of free Baxter algebras to the study of the umbral calculus. We give a characterization of the umbral calculus in terms of the Baxter algebra. This characterization ...leads to a natural generalization of the umbral calculus that includes the classical umbral calculus in a family of λ-umbral calculi parameterized by λ in the base ring.
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