Geodesic
Some Continuous Analogs of the Expansion in Jacobi Polynomials and Vector-Valued Orthogonal Bases
Funkcionalʹnyj analiz i ego priloženiâ, Tome 39 (2005) no. 2, pp. 31-46

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We obtain the spectral decomposition of the hypergeometric differential operator on the contour $\operatorname{Re}z=1/2$. (The multiplicity of the spectrum of this operator is $2$.) As a result, we obtain a new integral transform different from the Jacobi (or Olevskii) transform. We also construct an ${}_3F_2$-orthogonal basis in a space of functions ranging in $\mathbb{C}^2$. The basis lies in the analytic continuation of continuous dual Hahn polynomials with respect to the index $n$ of a polynomial.
Keywords: hypergeometric differential operator
Mots-clés : spectral decomposition, Jacobi transform, Hahn polynomial. 
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     author = {Yu. A. Neretin},
     title = {Some {Continuous} {Analogs} of the {Expansion} in {Jacobi} {Polynomials} and {Vector-Valued} {Orthogonal} {Bases}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {31--46},
     publisher = {mathdoc},
     volume = {39},
     number = {2},
     year = {2005},
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     url = {http://geodesic.mathdoc.fr/item/FAA_2005_39_2_a2/}
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Yu. A. Neretin. Some Continuous Analogs of the Expansion in Jacobi Polynomials and Vector-Valued Orthogonal Bases. Funkcionalʹnyj analiz i ego priloženiâ, Tome 39 (2005) no. 2, pp. 31-46. http://geodesic.mathdoc.fr/item/FAA_2005_39_2_a2/