A note on Hermite polynomials
英文摘要:
In this paper, we consider linear differential equations satisfied by the generating function for Hermite polynomials and derive some new identities involving those polynomials.
机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。
Taekyun Kim, Dae San Kim
In this paper, we consider linear differential equations satisfied by the generating function for Hermite polynomials and derive some new identities involving those polynomials.
机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。
继续探索
In this note, we introduce and investigate the Hermite-based Tangent numbers and polynomials, Hermite-based modifieed degenerate- Tangent polynomials, poly-Tangent polynomials. We give some identities and relations for these polynomials.
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We introduce degenerate Hermite polynomials as a degenerate version of the ordinary Hermite polynomials. Then, among other things, by using the formula about representing one lambda-Sheffer polynomial in terms of other lambda-Sheffer polynomials we represent the degenerate Hermite polynomials in terms of the higher-order degenerate Bernoulli, Euler, and Frobenius-Euler polynomials and vice versa.
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In this paper, we show that for each $n\geq 1$, the generalised Hermite-Laguerre Polynomials $G_{\frac{1}{4}}$ and $G_{\frac{3}{4}}$ are either irreducible or linear polynomial times an irreducible polynomial of degree $n-1$.
In this paper, we consider Hermite and poly-Bernoulli mixed-type polynomials and investigate the properties of those polynomials which are derived from umbral calculus. Finally, we give various identities associated with Stirling numbers, Bernoulli and Frobenius-Euler polynomials of higher order.
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In this paper, we study umbral calculus to have alternative ways of obtaining our results. That is, we derive some interesting identities of the higher-order Bernoulli, Euler and Hermite polynomials arising from umbral calculus to have alternative ways.
Hermite equivalence of polynomials
本文重新研究了Hermite在1850年代提出的整系数一元多项式等价概念,通过不变环和不变理想解释了Hermite等价类,并阐明了其与${\rm GL}_2(\mathbb{Z})$-等价和$\mathbb{Z}$-等价的关系。
A polynomial is real-rooted if all of its roots are real. For every polynomial $f(t) \in {\mathbf R}[t]$, the Hermite-Sylvester theorem associates a quadratic form $Φ_2$ such that $f(t)$ is real-rooted if and only if $Φ_2$ is positive semidefinite. In this note, for every positive integer $m$, an $2m$-adic form $Φ_{2m}$ is constructed such that $f(t)$ is real-rooted if and only if $Φ_{2m}$ is positive semidefinite for some $m$ if and only if $Φ_{2m}(x_1,\ldots, x_n)$ is positive semidefinite for all $m$.
In the present paper, we deal mainly with arithmetic properties of Legendre polynomials by using their orthogonality property. We show that Legendre polynomials are proportional with Bernoulli, Euler, Hermite and Bernstein polynomials.
We study the irreducibility of Wronskian Hermite polynomials labelled by partitions. It is known that these polynomials factor as a power of x times a remainder polynomial. We show that the remainder polynomial is irreducible for the partitions (n, m) with m <= 2, and (n, n) when n + 1 is a square. Our main tools are two theorems that we prove for all partitions. The first result gives a sharp upper bound for the slope of the edges of the Newton polygon for the remainder polynomial. The second result is a Schur-type congruence for Wronskian Hermite polynomials. We also explain how irreducibili…
Using the methods of classical invariant theory a general approach to finding of identities for Bernulli, Euler and Hermite polynomials is proposed.
This paper is now part of the new paper "Series with Hermite polynomials and applications" arXiv:1710.00687.
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