Can some body suggest how to relate Gegenbauer Polynomial G_n{(\zeta)} and Associated Legendre polynomial P^{1}_{(\zeta)}?
The normalized Gegenbauer polynomials are generalizations of the Legendre ones in higher dimensions. For example, if you get the Wolfram Mathworld formulas for the Gegenbauer polynomials you'll get the Legendre ones for \lambda=1/2
You can derive a connection formula between those two families of polynomials if you proceed as Koepf W. and Schmersau D did in "Representation of orthogonal polynomial". In that case, you simply need to know the differential equation satisfied by each family.
Dear B. R. Jaiswal,
These two links will help you to find such relation.
Best Wishes
http://mathworld.wolfram.com/GegenbauerPolynomial.html
dlmf.nist.gov/15.9#iii