Starting from the second-order diierence hypergeometric equation satissed by the set of discrete orthogonal polynomials fp n g, we nd the analytical expressions of the expansion coeecients of any polynomial r m (x) and of the product r m (x)q j (x) in series of the set fp n g. These coeecients are given in terms of the polynomial coeecients of the second-order diierence equations satissed by th...

We present a new family of hook-length formulas for the number standard increasing tableaux which arise in study factorial Grothendieck polynomials. In case straight shapes our generalize classical formula and Stanley's formula. For skew shapes, Naruse its $q$-analogues, were studied previous papers series.

An algebraically motivated generalization of Gosper’s algorithm to indefinite bibasic hypergeometric summation is presented. In particular, it is shown how Paule’s concept of greatest factorial factorization of polynomials can be extended to the bibasic case. It turns out that most of the bibasic hypergeometric summation identities from literature can be proved and even found this way. A Mathem...

Measurement bias refers to systematic differences across subpopulations in the relation between observed test scores and the latent variant underlying the test scores. Comparisons of subpopulations with the same score on the latent variable can be expected to have the same observed test score. Measurement invariance is therefore one of the key issues in psychological testing. It has been establ...

in this paper we demonstrate the existence of a set of polynomials pi , 1 i  n , which arepositive semi-definite on an interval [a , b] and satisfy, partially, the conditions of polynomials found in thelagrange interpolation process. in other words, if a  a1  an  b is a given finite sequence of realnumbers, then pi (a j )  ij (ij is the kronecker delta symbol ) ; moreover, the sum of ...

Ehrenborg gave a combinatorial proof of Radoux’s theorem which states that the determinant of the (n + 1)× (n + 1) dimensional Hankel matrix of exponential polynomials is x ∏n i=0 i!. This proof also shows the result that the (n + 1) × (n + 1) Hankel matrix of factorial numbers is ∏n k=1(k!) . We observe that two polynomial generalizations of factorial numbers also have interesting determinant ...

In 1977, Strassen invented a famous baby-step/giant-step algorithm that computes the factorial N ! in arithmetic complexity quasi-linear . 1988, Chudnovsky brothers generalized Strassen's to computation of -th term any holonomic sequence essentially same complexity. We design q -analogues these algorithms. first extend -factorial , then Chudnovskys' -holonomic sequence. Both algorithms work ; s...

By a deformation of the integers we mean a sequence x = {xn, n ∈ N} of polynomials in one or more variables and with integral coefficients, having the property that there exists some value q0 of the variables such that ∀n ∈ N, xn(q0) = n. The quantum integers xn = ∑n−1 l=0 q l are a typical example of a deformation of the integers. Another example is given by the version of the Chebyshev polyno...