Mathematics – Classical Analysis and ODEs
Scientific paper
2009-07-15
The Ramanujan Journal 26 (2011), no. 1, 1-34
Mathematics
Classical Analysis and ODEs
Scientific paper
10.1007/s11139-011-9315-0
We develop a theory of "special functions" 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 $L^2(\mathbb{R}_+,x^{\mu+\nu+1}dx)$ 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 "special functions" give $K$-finite vectors.
Hilgert Joachim
Kobayashi Toshiyuki
Mano Gen
Möllers Jan
No associations
LandOfFree
Special functions associated to a certain fourth order differential equation does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Special functions associated to a certain fourth order differential equation, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Special functions associated to a certain fourth order differential equation will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-364731