GENERALIZATIONS OF LAGUERRE-POLYNOMIALS

It is shown that the polynomials {Lnα,M(x)}n = 0∞ defined by Lnα,M(x)= ∑ k=0 N+1Ak·DkLn(α)(x ) for certain real coefficients {Ak}k = 0N + 1 are orthogonal with respect to the inner product 〈f,g〉= 1 Γ(α+1)·∫0∞x αe-x·f(x)g(x)dx+ ∑ v=0 NMv·f(v)(0)g(v)(0), where α > - 1, N ε{lunate} N and Mv ≥ 0 for all v ε{lunate} {0, 1, 2, ..., N}. For these new polynomials {n,M(X )}N = 0∞ an orthogonality relation and a second order differential equation are derived. Further we obtain a representation as a N + 2FN + 2 hypergeometric series and a (2N + 3)-terms recurrence relation, which gives rise to a Christoffel-Darboux type formula. © 1990.