# An integral formula for L 2-eigenfunctions of a fourth-order Bessel-type differential operator

@article{Kobayashi2011AnIF, title={An integral formula for L 2-eigenfunctions of a fourth-order Bessel-type differential operator}, author={Toshiyuki Kobayashi and Jan M{\"o}llers}, journal={Integral Transforms and Special Functions}, year={2011}, volume={22}, pages={521 - 531} }

We find an explicit integral formula for the eigenfunctions of a fourth-order differential operator against the kernel involving two Bessel functions. Our formula establishes the relation between K-types in two different realizations of the minimal representation of the indefinite orthogonal group, namely the L 2-model and the conformal model.

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## References

SHOWING 1-10 OF 15 REFERENCES

### Special functions associated with a certain fourth-order differential equation

- Mathematics
- 2011

We develop a theory of “special functions” associated with a certain fourth-order differential operator $\mathcal{D}_{\mu,\nu}$ on ℝ depending on two parameters μ,ν. For integers μ,ν≥−1 with μ+ν∈2ℕ0,…

### Orthogonal polynomials associated to a certain fourth order differential equation

- Mathematics
- 2011

We introduce orthogonal polynomials $M_{j}^{\mu,\ell}(x)$ as eigenfunctions of a certain self-adjoint fourth order differential operator depending on two parameters μ∈ℂ and ℓ∈ℕ0.These polynomials…

### Integral formula of the unitary inversion operator for the minimal representation of O(p,q)

- Mathematics
- 2007

The indefinite orthogonal group G = O(p,q) has a distinguished infinite dimen- sional unitary representation …, called the minimal representation for p+q even and greater than 6. The Schrodinger…

### Analysis on the minimal representation of O(p;q) { I. Realization via conformal geometry

- Mathematics
- 2001

### The Schrödinger model for the minimal representation of the indefinite orthogonal group (

- Mathematics
- 2011

The indenite orthogonal group G = O(p;q) has a distinguished innite dimensional irreducible unitary representation for p +q even and greater than 4, which is the \smallest" in the sense that the…

### Table of Integrals, Series, and Products

- Mathematics
- 1943

Introduction. Elementary Functions. Indefinite Integrals of Elementary Functions. Definite Integrals of Elementary Functions. Indefinite Integrals of Special Functions. Definite Integrals of Special…

### Analysis on the minimal representation of O(p,q) -- III. ultrahyperbolic equations on R^{p-1,q-1}

- Mathematics
- 2001

### The Schrödinger model for the minimal representation of the indefinite orthogonal group O(p, q)

- Mathematics
- 2011

The authors introduce a generalization of the Fourier transform, denoted by $\mathcal{F}_C$, on the isotropic cone $C$ associated to an indefinite quadratic form of signature $(n_1,n_2)$ on…

### A Table of Integrals

- Mathematics
- 2010

Basic Forms x n dx = 1 n + 1 x n+1 (1) 1 x dx = ln |x| (2) udv = uv − vdu (3) 1 ax + b dx = 1 a ln |ax + b| (4) Integrals of Rational Functions 1 (x + a) 2 dx = −