Saharian, A. A. A summation formula over the zeros of a combination of the associated Legendre functions with a physical application. (English) Zbl 1185.33012 J. Phys. A, Math. Theor. 42, No. 46, Article ID 465210, 15 p. (2009). 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. Reviewer: Alexei Lukashov (Istanbul) Cited in 1 Document 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 Keywords:Legendre function; Wightman function; summation formula PDFBibTeX XMLCite \textit{A. A. Saharian}, J. Phys. A, Math. Theor. 42, No. 46, Article ID 465210, 15 p. (2009; Zbl 1185.33012) Full Text: DOI arXiv