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On the derivative of the Legendre function of the first kind with respect to its degree. (English) Zbl 1112.33009

Summary: We study the derivative of the Legendre function of the first kind, P ν (z), with respect to its degree ν. At first, we provide two contour integral representations for P ν (z)/ν. Then, we proceed to investigate the case of [P ν (z)/ν] ν=n , with n being an integer; this case is met in some physical and engineering problems. Since it holds that [P ν ' (z)/ν ' ] ν ' =-ν-1 =-[P ν ' (z)/ν ' ] ν ' =ν , we focus on the sub-case of n being a non-negative integer. We show that

P ν (z) ν ν=n =P n (z)lnz+1 2+R n (z)(n)

where R n (z) is a polynomial in z of degree n. We present alternative derivations of several known explicit expressions for R n (z) and also add some new. A generating function for R n (z) is also constructed. Properties of the polynomials V n (z)=[R n (z)+(-1) n R n (-z)]/2 and W n-1 (z)=-[R n (z)-(-1) n R n (-z)]/2 are also investigated. It is found that W n-1 (z) is the Christoffel polynomial, well known from the theory of the Legendre function of the second kind, Q n (z). As examples of applications of the results obtained, we present non-standard derivations of some representations of Q n (z), sum to closed forms some Legendre series, evaluate some definite integrals involving Legendre polynomials and also derive an explicit representation of the indefinite integral of the Legendre polynomial squared.


MSC:
33C45Orthogonal polynomials and functions of hypergeometric type