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Some sums of Legendre and Jacobi polynomials

Jan Gustavsson (2001)

Mathematica Bohemica

We prove identities involving sums of Legendre and Jacobi polynomials. The identities are related to Green’s functions for powers of the invariant Laplacian and to the Minakshisundaram-Pleijel zeta function.

Spectral properties of some regular boundary value problems for fourth order differential operators

Nazim Kerimov, Ufuk Kaya (2013)

Open Mathematics

In this paper we consider the problem y i v + p 2 ( x ) y ' ' + p 1 ( x ) y ' + p 0 ( x ) y = λ y , 0 < x < 1 , y ( s ) ( 1 ) - ( - 1 ) σ y ( s ) ( 0 ) + l = 0 s - 1 α s , l y ( l ) ( 0 ) = 0 , s = 1 , 2 , 3 , y ( 1 ) - ( - 1 ) σ y ( 0 ) = 0 , where λ is a spectral parameter; p j (x) ∈ L 1(0, 1), j = 0, 1, 2, are complex-valued functions; α s;l, s = 1, 2, 3, l = 0 , s - 1 ¯ , are arbitrary complex constants; and σ = 0, 1. The boundary conditions of this problem are regular, but not strongly regular. Asymptotic formulae for eigenvalues and eigenfunctions of the considered boundary value problem are established in the case α 3,2 + α 1,0 ≠ α 2,1. It is proved that the system of root functions of this spectral problem...

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