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A summation formula over the zeros of a combination of the associated Legendre functions with a physical application. (English) Zbl 1185.33012

Let \(P^{\mu}_{\nu-1/2}(u)\) and \(Q^{\mu}_{\nu-1/2}(u)\) be the associated Legendre functions of the first and second kind, respectively. Furthermore, let \(z=z_k,\) \(k=1,2,\dots\) be zeros of the function
\[ X^{\mu}_{iz}(u,v)=\frac{P^{\mu}_{iz-1/2}(u)P^{-\mu}_{iz-1/2}(v)-P^{-\mu}_{iz-1/2}(u)P^{\mu}_{iz-1/2}(v)}{\sin(\mu\pi)} \]
in the right half of the complex plane. The author finds a representation for the sum of the series
\[ \left.\sum_{k=1}^{\infty}\frac{h(z)}{\partial_zX^{\mu}_{iz}(u,v)}\frac{Q^{\mu}_{iz-1/2}(v)}{Q^{\mu}_{iz-1/2}(u)}\right|_{z=z_k}, \]
where \(h(z)\) is a meromorphic function in the right half-plane satisfying a constraint for its growth at \(\infty\). As physical applications of that summation formula the author evaluates the Wightman function for a scalar field with a general curvature coupling parameter in the region between concentric spherical shells on a background of constant negative curvature space in the form of the sum of two integrals. The first one corresponds to the Wightman function for the geometry of a single spherical shell and the second one is induced by the presence of the second shell. The boundary-induced part in the vacuum expectation value of the field squared is investigated. For points away from the boundary the corresponding renormalization procedure is reduced to that for the boundary-free part.

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33C90 Applications of hypergeometric functions
81T99 Quantum field theory; related classical field theories
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