The combinatorics of associated Hermite polynomials

多重变量广义勒规则-Genocchi多项式（阶a）

The Multivariable Generalized Hermite-Type-Genocchi Polynomials of Order a

本文提出一种新的多项Genocchi多项式类，通过整合有趣多项式构建，研究其性质，包括显式表示、加法公式和多项式展开，并建立与广义第一类Stirling数的关系。

A curious $q$-analogue of Hermite polynomials

Two well-known $q$-Hermite polynomials are the continuous and discrete $q$-Hermite polynomials. In this paper we consider a new family of $q$-Hermite polynomials and prove several curious properties about these polynomials. One striking property is the connection with $q$-Fibonacci and $q$-Lucas polynomials. The latter relation yields a generalization of the Touchard-Riordan formula.

Discrete q-Hermite polynomials: An elementary approach

We present a simple approach to discrete q-Hermite polynomials with special emphasis on analogies with the classical case.

Continuous q-Hermite polynomials: An elementary approach

This overview article gives an elementary approach to continuous q-Hermite polynomials. We stress their relation to Fibonacci, Lucas and Chebyshev polynomials and to some q-analogues of these polynomials.

A new characterization of ultraspherical, Hermite, and Chebyshev polynomials of the first kind

We show that the only polynomial sets with a generating function of the form F (xt -- R(t)) and satisfying a three-term recursion relation are the monomial set and the rescaled ultraspherical, Hermite, and Chebyshev polynomials of the first kind.

Coefficients of Wronskian Hermite polynomials

We study Wronskians of Hermite polynomials labelled by partitions and use the combinatorial concepts of cores and quotients to derive explicit expressions for their coefficients. These coefficients can be expressed in terms of the characters of irreducible representations of the symmetric group, and also in terms of hook lengths. Further, we derive the asymptotic behaviour of the Wronskian Hermite polynomials when the length of the core tends to infinity, while fixing the quotient. Via this combinatorial setting, we obtain in a natural way the generalization of the correspondence between Hermi…

On q-Hermite polynomials and their relationship with some other families of orthogonal polynomials

We review properties of the $q-$Hermite polynomials and indicate their links with the Chebyshev, Rogers--Szegö, Al-Salam--Chihara, continuous $q-$% utraspherical polynomials. In particular we recall the connection coefficients between these families of polynomials. We also present some useful and important finite and infinite expansions involving polynomials of these families including symmetric and non-symmetric kernels. In the paper we collect scattered throughout literature useful but not widely known facts concerning these polynomials. It is based on 43 positions of predominantly recent li…

Asymptotic properties of biorthogonal polynomials systems related to Hermite and Laguerre polynomials

In this paper, the structures to a family of biorthogonal polynomials that approximate to the Hermite and Generalized Laguerre polynomials are discussed respectively. Therefore, the asymptotic relation between several orthogonal polynomials and combinatorial polynomials are derived from the systems, which in turn verify the Askey scheme of hypergeometric orthogonal polynomials. As the applications of these properties, the asymptotic representations of the generalized Buchholz, Laguerre, Ultraspherical(Gegenbauer), Bernoulli, Euler, Meixner and Meixner-Pllaczekare polynomials are derived from t…

Covariants of binary forms and new identities for Bernoulli, Euler and Hermite polynomials

Using the methods of classical invariant theory a general approach to finding of identities for Bernulli, Euler and Hermite polynomials is proposed.

Recurrence Relations for Wronskian Hermite Polynomials

We consider polynomials that are defined as Wronskians of certain sets of Hermite polynomials. Our main result is a recurrence relation for these polynomials in terms of those of one or two degrees smaller, which generalizes the well-known three term recurrence relation for Hermite polynomials. The polynomials are defined using partitions of natural numbers, and the coefficients in the recurrence relation can be expressed in terms of the number of standard Young tableaux of these partitions. Using the recurrence relation, we provide another recurrence relation and show that the average of the …

A characterization of ultraspherical, Hermite, and Chebyshev polynomials of the first kind

We show that the only orthogonal polynomials with a generating function of the form $F(x z - αz^2)$ are the ultraspherical, Hermite, and Chebyshev polynomials of the first kind. For special $F$ for which this is the case, we then finish the classification of orthogonal polynomials with more general generating functions $F(x w(z) - R(z))$.

The Real-rootedness of Eulerian Polynomials via the Hermite--Biehler Theorem

Based on the Hermite--Biehler theorem, we simultaneously prove the real-rootedness of Eulerian polynomials of type $D$ and the real-rootedness of affine Eulerian polynomials of type $B$, which were first obtained by Savage and Visontai by using the theory of $\mathbf{s}$-Eulerian polynomials. We also confirm Hyatt's conjectures on the interlacing property of half Eulerian polynomials. Borcea and Brändén's work on the characterization of linear operators preserving Hurwitz stability is critical to this approach.

On the inequalities in Hermite's theorem for a real polynomial to have real zeros

We prove expressions for the inequalities in Hermite's theorem which are conditions for a real polynomial to have real zeros. These expressions generalize the discriminant of a quadratic polynomial and the expression of J. Marík for a cubic polynomial. We show that the $(k+1)$-th minor of the Hermite matrix associated a polynomial $p(x)$ is equal to the $k$-th minor of another matrix we call $E(n)$ times $n^{k-1}$ and a simple integer. To prove this equivalence, we prove generalizations of the discriminant of a polynomial and analyze certain labeled directed graphs. To define this matrix $E(n)…

关于Hermite多项式Carlitz-Mehler公式的研究

On the Carlitz-Mehler formula for Hermite polynomials

本文给出Hermite多项式Carlitz-Mehler公式的进一步推广的直接证明，扩展了Carlitz和Srivastava等人的工作。

Dynamic ASEP, duality and continuous $q^{-1}$-Hermite polynomials

We demonstrate a Markov duality between the dynamic ASEP and the standard ASEP. We then apply this to step initial data, as well as a half-stationary initial data (which we introduce). While investigating the duality for half-stationary initial data, we uncover and utilize connections to continuous $q^{-1}$-Hermite polynomials. Finally, we introduce a family of stationary initial data which are related to the indeterminate moment problem associated with these $q^{-1}$-Hermite polynomials.

Ordered Bell numbers, Hermite polynomials, Skew Young Tableaux, and Borel orbits

We give three interpretations of the number $b$ of orbits of the Borel subgroup of upper triangular matrices on the variety $\ms{X}$ of complete quadrics. First, we show that $b$ is equal to the number of standard Young tableaux on skew-diagrams. Then, we relate $b$ to certain values of a modified Hermite polynomial. Third, we relate $b$ to a certain cell decomposition on $\ms{X}$ previously studied by De Concini, Springer, and Strickland. Using these, we give asymptotic estimates for $b$ as the dimension of the quadrics increases.