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The place d three 3)2 ( L x - -- i3x j3x j i i>j hence the Gegenbauer polynomials play a job within the concept of hyper round harmonics which is similar to the position performed through Legendre polynomials within the commonly used thought of third-dimensional round harmonics; and whilst d = three, the Gegenbauer polynomials lessen to Legendre polynomials. The commonplace sum rule, in 'lrlhich a sum of round harmonics is expressed as a Legendre polynomial, additionally has a d-dimensional generalization, within which a sum of hyper round harmonics is expressed as a Gegenbauer polynomial (equation (3-27»: The hyper round harmonics which seem during this sum rule are eigenfunctions of the generalized angular monentum 2 operator A , selected in any such means as to fulfil the orthonormality relation: VIe are all conversant in the truth that a airplane wave could be multiplied by way of round Bessel services and both Legendre polynomials or round harmonics in a three-dimensional area. equally, one reveals d-dimensional aircraft wave will be extended by way of HYPERSPHERICAL HARMONICS xii "hyperspherical Bessel services" and both Gegenbauer polynomials in any other case hyperspherical harmonics (equations ( four - 27) and ( four - 30) ) : 00 ik·x e = (d-4)!!A~oiA(d+2A-2)j~(kr)C~(~k'~) 00 (d-2)!!I(0) 2: iAj~(kr) 2:Y~ (["2k)Y (["2) A A=O ). l). l)J the place I(O) is the whole reliable attitude. This enlargement of a d-dimensional aircraft wave turns out to be useful after we desire to calculate Fourier transforms in a d-dimensional house.

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