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In this paper we present a general class of linear positive operators of discrete type reproducing the third test function of Korovkin theorem. In a certain weighted space it forms an approximation process. A Voronovskaja-type result is established and particular cases are analyzed.

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Abstract

Müntz–Legendre polynomials L n(Λ;x) associated with a sequence Λ={λ k} are obtained by orthogonalizing the system in L 2[0,1] with respect to the Legendre weight. Under very mild conditions on Λ, we establish the endpoint asymptotics close to x=1. The main result is
ea
where and J 0 is the Bessel function of order 0.
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Abstract

Ramanujan suggested an expansion for the nth partial sum of the harmonic series which employs the reciprocal of the nth triangular number. This has been proved in 2006 by Villarino, who speculated that there might also exist a similar expansion for the logarithm of the factorial. This study shows that such an asymptotic expansion indeed exists and provides formulas for its generic coefficient and for the bounds on its errors.

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