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In position dependent mass (PDM) problems, the quantum dynamics of the associated systems have been understood well in the literature for particular orderings. However, no efforts seem to have been made to solve such PDM problems for general orderings to obtain a global picture. In this connection, we here consider the general ordered quantum Hamiltonian of an interesting position dependent mass problem, namely, the Mathews-Lakshmanan oscillator, and try to solve the quantum problem for all possible orderings including Hermitian and non-Hermitian ones. The other interesting point in our study is that for all possible orderings, although the Schrödinger equation of this Mathews-Lakshmanan oscillator is uniquely reduced to the associated Legendre differential equation, their eigenfunctions cannot be represented in terms of the associated Legendre polynomials with integral degree and order. Rather the eigenfunctions are represented in terms of associated Legendre polynomials with non-integral degree and order. We here explore such polynomials and represent the discrete and continuum states of the system. We also exploit the connection between associated Legendre polynomials with non-integral degree with other orthogonal polynomials such as Jacobi and Gegenbauer polynomials. [ABSTRACT FROM AUTHOR]

An explicit solution to the classical electrostatic problem of the electrostatic force between N conducting spheres requires the expansion of the electrostatic potential. The boundary conditions for the potentials are given by Gauss which requires an expansion of the inverse of the distance that encompasses six angles. A sum rule for this expansion is presented in terms of associated Legendre polynomials with complex exponentials. [ABSTRACT FROM AUTHOR]

The following inequality is established: |Pn(cos [variant_greek_theta])|< [[Square_Root]1+(π4/16)(n+1/2)4 sin4 [variant_greek_theta]]-1, 0<[variant_greek_theta]<π, n=1,2,..., where Pn(x) denotes the Legendre polynomial of degree n. The relation P2n(cos [variant_greek_theta]) + (4/π2)× Q2n(cos [variant_greek_theta]) < [[Square_Root]1+(π4/16)(n+1/2)4 sin4 [variant_greek_theta]]-1, n=1,2,..., on [θn1,θn,n+1], is proven where Qn(x) denotes the Legendre function of second kind, cos θn1 the largest zero of Qn(x), and cos θn,n+1=-cos θn1. Similarly we obtain the inequalities |J0(x)| < [[Square_Root]1+(π4/16)x4]-1, x≠0, and J20(x) + Y20(x)< [[Square_Root]1+(π4/16)x4]-1, x≥y1, where y1=0.893577... is the first positive zero of Y0(x), and J0(x), Y0(x) denote the Bessel functions of the first and second kind, respectively. The results of the present paper arise out of some problems of nuclear and particle physics. [ABSTRACT FROM AUTHOR]

The Grassmannian analogues of the Legendre polonomials are obtained from the corresponding Grassmannian Hermite multinomials. The generating function, recurrence relations, and differential equation are given. In contrast to the Hermitian case there is no Berezin weight function that orthogonalizes these functions. Generalizations to the Tchebyscheff and Gegenbauer cases are also possible. [ABSTRACT FROM AUTHOR]

We study the dependence between prime numbers and the real and imaginary parts of the nontrivial zeros of the Riemann zeta function. The Legendre polynomials and the partial derivatives of the Riemann zeta function are used to investigate the above dependence along with the Riemann hypothesis with physical interpretations. A modified zeta function with finite terms is defined as a new implement for the study of the zeta function and its zeros. [ABSTRACT FROM AUTHOR]

Presents correction to the article 'A Sum Rule for Associated Legendre Polynomials With Spherical Triangles,' by Armik V.M. Khachatourian and Anders O. Wistrom published in a 2003 issue of the journal Journal of Mathematical Physics. Correction ot several equations presented in the article.