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Special functions associated to a certain fourth order differential equation

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Special functions associated to a certain fourth order differential equation. / Hilgert, Joachim; Kobayashi, Toshiyuki; Mano, Gen; Möllers, Jan.

I: Ramanujan Journal, Bind 26, Nr. 1, 2011, s. 1-34.

Publikation: Bidrag til tidsskrift/Konferencebidrag i tidsskrift /Bidrag til avisTidsskriftartikelForskningpeer review

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Hilgert, J, Kobayashi, T, Mano, G & Möllers, J 2011, 'Special functions associated to a certain fourth order differential equation', Ramanujan Journal, bind 26, nr. 1, s. 1-34. https://doi.org/10.1007/s11139-011-9315-0

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Hilgert, Joachim ; Kobayashi, Toshiyuki ; Mano, Gen ; Möllers, Jan. / Special functions associated to a certain fourth order differential equation. I: Ramanujan Journal. 2011 ; Bind 26, Nr. 1. s. 1-34.

Bibtex

@article{4b8cbe0a4c504382914fa34bef361196,
title = "Special functions associated to a certain fourth order differential equation",
abstract = "We develop a theory of \lq special functions\rq\ associated to a certain fourth order differential operator $\mathcal{D}_{\mu,\nu}$ on $\mathbb{R}$ depending on two parameters $\mu,\nu$. For integers $\mu,\nu\geq-1$ with $\mu+\nu\in2\mathbb{N}_0$ this operator extends to a self-adjoint operator on $L^2(\mathbb{R}_+,x^{\mu+\nu+1}\td x)$ with discrete spectrum. We find a closed formula for the generating functions of the eigenfunctions, from which we derive basic properties of the eigenfunctions such as orthogonality, completeness, $L^2$-norms, integral representations and various recurrence relations.This fourth order differential operator $\mathcal{D}_{\mu,\nu}$ arises as the radial part of the Casimir action in the Schr\{"}odinger model of the minimal representation of the group $O(p,q)$, and our \lq special functions\rq\ give $K$-finite vectors.",
author = "Joachim Hilgert and Toshiyuki Kobayashi and Gen Mano and Jan M{\"o}llers",
year = "2011",
doi = "10.1007/s11139-011-9315-0",
language = "English",
volume = "26",
pages = "1--34",
journal = "Ramanujan Journal",
issn = "1382-4090",
publisher = "Springer New York LLC",
number = "1",

}

RIS

TY - JOUR

T1 - Special functions associated to a certain fourth order differential equation

AU - Hilgert, Joachim

AU - Kobayashi, Toshiyuki

AU - Mano, Gen

AU - Möllers, Jan

PY - 2011

Y1 - 2011

N2 - We develop a theory of \lq special functions\rq\ associated to a certain fourth order differential operator $\mathcal{D}_{\mu,\nu}$ on $\mathbb{R}$ depending on two parameters $\mu,\nu$. For integers $\mu,\nu\geq-1$ with $\mu+\nu\in2\mathbb{N}_0$ this operator extends to a self-adjoint operator on $L^2(\mathbb{R}_+,x^{\mu+\nu+1}\td x)$ with discrete spectrum. We find a closed formula for the generating functions of the eigenfunctions, from which we derive basic properties of the eigenfunctions such as orthogonality, completeness, $L^2$-norms, integral representations and various recurrence relations.This fourth order differential operator $\mathcal{D}_{\mu,\nu}$ arises as the radial part of the Casimir action in the Schr\"odinger model of the minimal representation of the group $O(p,q)$, and our \lq special functions\rq\ give $K$-finite vectors.

AB - We develop a theory of \lq special functions\rq\ associated to a certain fourth order differential operator $\mathcal{D}_{\mu,\nu}$ on $\mathbb{R}$ depending on two parameters $\mu,\nu$. For integers $\mu,\nu\geq-1$ with $\mu+\nu\in2\mathbb{N}_0$ this operator extends to a self-adjoint operator on $L^2(\mathbb{R}_+,x^{\mu+\nu+1}\td x)$ with discrete spectrum. We find a closed formula for the generating functions of the eigenfunctions, from which we derive basic properties of the eigenfunctions such as orthogonality, completeness, $L^2$-norms, integral representations and various recurrence relations.This fourth order differential operator $\mathcal{D}_{\mu,\nu}$ arises as the radial part of the Casimir action in the Schr\"odinger model of the minimal representation of the group $O(p,q)$, and our \lq special functions\rq\ give $K$-finite vectors.

U2 - 10.1007/s11139-011-9315-0

DO - 10.1007/s11139-011-9315-0

M3 - Journal article

VL - 26

SP - 1

EP - 34

JO - Ramanujan Journal

JF - Ramanujan Journal

SN - 1382-4090

IS - 1

ER -