Evaluate more general integrals involving universal associated Legendre polynomials via Taylor’s theorem. (English) Zbl 1375.33019
Summary: A few important integrals involving the product of two universal associated Legendre polynomials \(P_{l'}^{m'}(x)\), \(P_{k'}^{n'}(x)\) and \(x^{2a}(1-x^2)^{-p-1}\), \(x^b(1\pm x)^{-p-1}\) and \(x^c(1-x^2)^{-p-1}(1\pm x)\) are evaluated using the operator form of Taylor’s theorem and an integral over a single universal associated Legendre polynomial. These integrals are more general since the quantum numbers are unequal, i.e. \(l'\neq k'\) and \(m'\neq n'\). Their selection rules are also given. We also verify the correctness of those integral formulas numerically.
MSC:
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
| 42C10 | Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) |
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