Expanding Legendre polynomials

Abstract The expansion of Legendre polynomials <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>P</m:mi> <m:mi mathvariant="normal">ℓ</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>cos</m:mi> <m:mo>⁡</m:mo> <m:mi>θ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {P_{\ell}(\cos\theta)} in cosines of multiples of θ is obtained by comparing the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mn>2</m:mn> <m:mi mathvariant="normal">ℓ</m:mi> </m:msup> </m:math> {2^{\ell}} -pole solution of Laplace equation in complex coordinates with its usual form in spherical coordinates.