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Spherical Functions Associated With the Three Dimensional Sphere

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Inés Pacharoni, Juan Tirao, Ignacio Zurrián

Classical Analysis and ODEs · Submitted 19 Mar 2012

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim In this paper, we determine all irreducible spherical functions \Phi of any K -type associated to the pair (G,K)=(\SO(4),\SO(3)).

how it is stated · asserted 1.00 read the record →

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arXiv ID1203.4275 (version 2)
Pith IDGROTBI7D, Pith's permanent record of this paper and its review
CitationIgnacio Zurri\'an, In\'es Pacharoni, Juan Tirao. "Spherical Functions Associated With the Three Dimensional Sphere." arXiv:1203.4275 (2012). https://pith.science/paper/GROTBI7D
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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,435 input tokens, $0.0003 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.05 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.04 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

In this paper, we determine all irreducible spherical functions \Phi of any K -type associated to the pair (G,K)=(\SO(4),\SO(3)). This is accomplished by associating to \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 \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\ge0} of size (\ell+1)\times(\ell+1). Finally we proved 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 showed that W admits a second order symmetric hypergeometric operator \widetilde D and a first order symmetric differential operator \widetilde E.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 In this paper, we determine all irreducible spherical functions \Phi of any K -type associated to the pair (G,K)=(\SO(4),\SO(3)). contribution 0.83 0.97 0.85 asserted 1.00
s1 This is accomplished by associating to \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. method 0.99 0.95 0.04 asserted 1.00
s2 By an appropriate conjugation involving Hahn polynomials we uncouple one of the systems. method 0.99 0.94 0.02 asserted 1.00
s3 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. method 0.97 0.94 0.06 asserted 1.00
s4 Afterward, we identify those simultaneous solutions and use the representation theory of \SO(4) to characterize all irreducible spherical functions. method 0.66 0.96 0.42 asserted 1.00
s5 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\ge0} of size (\ell+1)\times(\ell+1). contribution 0.44 0.93 0.10 asserted 1.00
s6 Finally we proved that \widetilde P_w={P_0}^{-1}P_w is a sequence of matrix orthogonal polynomials with respect to a weight matrix W. result 0.94 0.97 0.27 asserted 1.00
s7 Moreover we showed that W admits a second order symmetric hypergeometric operator \widetilde D and a first order symmetric differential operator \widetilde E. result 0.96 0.96 0.16 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

In this paper, we determine all irreducible spherical functions \Phi of any K -type associated to the pair (G,K)=(\SO(4),\SO(3)). This is accomplished by associating to \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 \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\ge0} of size (\ell+1)\times(\ell+1). Finally we proved 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 showed that W admits a second order symmetric hypergeometric operator \widetilde D and a first order symmetric differential operator \widetilde E.

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