# Publication Citation: On a generalization of the generating function for Gegenbauer polynomials

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Author(s): Howard S. Cohl; On a generalization of the generating function for Gegenbauer polynomials January 28, 2013 We derive a Gegenbauer polynomial expansion for complex powers of the distance between two points in $d$-dimensional Euclidean space. The argument of the Gegenbauer polynomial in the expansion is given by the cosine of the separation angle between the two points as measured from the origin. The order of the Gegenbauer polynomial is given by $d/2-1$, which is ideal for utilization of the addition theorem for hyperspherical harmonics. The coefficients of the expansion are given in terms of an associated Legendre function of the second kind with argument $(r^2+{r^\prime}^2)/(2rr^\prime)>1$, where $r,r^\prime$ represent the Euclidean norm of the vectors representing the distances to the two points as measured from the origin. We extend this result by proving a generalization of the generating function for Gegenbauer polynomials. Integral Transforms and Special Functions Euclidean space; Polyharmonic equation; Fundamental solution; Gegenbauer polynomials; associated Legendre functions Modeling http://dx.doi.org/10.1080/10652469.2012.761613  (Note: May link to a non-U.S. Government webpage) Click here to retrieve PDF version of paper (134KB)