In the answer to Clausen’s identity for associated Legendre polynomials the following result was indicated: $$ \small{\left(P_n^m(\cos\theta)\right)^2=(\sin{\theta})^{2m}\frac{(m+n)!}{(n-m)!}\sum_{k=0}^{n-m}\frac{(-1)^k}{4^{k+m}}\binom{n+m}{k+2m}\binom{n+k+m}{n+m}\binom{2k+2m}{k+m}\sin^{2k}{\theta}}. $$ Can it be further generalized to the product of two different associated Legendre polynomials $P_{n_1}^{m_1}(\cos\theta)P_{n_2}^{m_2}(\cos\theta)$? Such a generalization, if found, can be used in atomic transition form factor calculations.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.