NDLI: Spherical functions associated with the three-dimensional sphere

Content Provider	Springer Nature Link
Author	Pacharoni, Inés Tirao, Juan Zurrián, Ignacio
Copyright Year	2013
Abstract	In this paper, we determine all irreducible spherical functions $$\Phi $$ of any $$K $$ -type associated with the pair $$(G,K)=(\mathrm{SO }(4),\mathrm{SO }(3))$$ . This is accomplished by associating with $$\Phi $$ a vector-valued function $$H=H(u)$$ of a real variable $$u$$ , which is analytic at $$u=0$$ and whose components are solutions of two coupled systems of ordinary differential equations. By an appropriate conjugation involving Hahn polynomials, we uncouple one of the systems. Then, this is taken to an uncoupled system of hypergeometric equations, leading to a vector-valued solution $$P=P(u)$$ , whose entries are Gegenbauer’s polynomials. Afterward, we identify those simultaneous solutions and use the representation theory of $$\mathrm{SO }(4)$$ to characterize all irreducible spherical functions. The functions $$P=P(u)$$ corresponding to the irreducible spherical functions of a fixed $$K$$ -type $$\pi _\ell $$ are appropriately packaged into a sequence of matrix-valued polynomials $$(P_w)_{w\ge 0}$$ of size $$(\ell +1)\times (\ell +1)$$ . Finally, we prove that $$\widetilde{P}_w={P_0}^{-1}P_w$$ is a sequence of matrix orthogonal polynomials with respect to a weight matrix $$W$$ . Moreover, we show that $$W$$ admits a second-order symmetric hypergeometric operator $$\widetilde{D}$$ and a first-order symmetric differential operator $$\widetilde{E}$$ .
Ending Page	1778
Page Count	52
Starting Page	1727
File Format	PDF
ISSN	03733114
e-ISSN	16181891
Journal	Annali di Matematica Pura ed Applicata
Issue Number	6
Volume Number	193
Language	English
Publisher	Springer Berlin Heidelberg
Publisher Date	2013-07-10
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) Three-dimensional sphere Representations of Lie and linear algebraic groups over real fields: analytic methods Matrix orthogonal polynomials Matrix-valued spherical functions Mathematics The matrix hypergeometric operator Other special orthogonal polynomials and functions
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics

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