相关论文: Associated Legendre Functions and Spherical Harmon…
A ladder structure of operators is presented for the associated Legendre polynomials and the spherical harmonics showing that both belong to the same irreducible representation of so(3,2). As both are also bases of square-integrable…
Associated Legendre functions of fractional degree appear in the solution of boundary value problems in wedges or in toroidal geometries, and elsewhere in applied mathematics. In the classical case when the degree is half an odd integer,…
For the associated Legendre and Ferrers functions of the first and second kind, we obtain new multi-derivative and multi-integral representation formulas. The multi-integral representation formulas that we derive for these functions…
Most of the special functions of mathematical physics are connected with the representation of Lie groups. The action of elements $D$ of the associated Lie algebras as linear differential operators gives relations among the functions in a…
This paper is a contribution to the study of the relations between special functions, Lie algebras and rigged Hilbert spaces. The discrete indices and continuous variables of special functions are in correspondence with the representations…
A correlation function of the classical orthogonal polynomials is defined and determined. The correlation function obeys a second order difference equation in two variables. The correlation function for the Gegenbauer, Chebyshev and…
An algebraic approximation, of order $K$, of a polyhedron correlation function (CF) can be obtained from $\gamma\pp(r)$, its chord-length distribution (CLD), considering first, within the subinterval $[D_{i-1},\, D_i]$ of the full range of…
For integral representations of associated Legendre functions in terms of modified Bessel functions, we establish justification for differentiation under the integral sign with respect to parameters. With this justification, derivatives for…
Clifford-Legendre and Clifford-Gegenbauer polynomials are eigenfunctions of certain differential operators acting on functions defined on $m$-dimensional euclidean space ${\mathbb R}^m$ and taking values in the associated Clifford algebra…
A relationship between partial derivatives of the associated Legendre function of the first kind with respect to its degree, $[\partial P_{\nu}^{m}(z)/\partial\nu]_{\nu=n}$, and to its order, $[\partial P_{n}^{\mu}(z)/\partial\mu]_{\mu=m}$,…
Numerous novel integral and series representations for Ferrers functions of the first kind (associated Legendre functions on the cut) of arbitrary degree and order, various integral, series and differential relations connecting Ferrers…
In a recent paper, Cohl and Costas-Santos derived a number of interesting multi-derivative and multi-integral relations for associated Legendre and Ferrers functions in which the orders of those functions are changed in integral steps.…
Is is shown in this paper that there is a connection between the Riemann zeta-function $\zf$ and the classical Jacobi's polynomials, i.e. the Legendre polynomials, Chebyshev polynomials of the first and the second kind,...
Algebraic hypergeometric functions can be compactly expressed as radical or dihedral functions on pull-back curves where the monodromy group is much simpler. This article considers the classical 3F2-functions with the projective monodromy…
The pseudospherical functions on one-sheet, two-dimensional hyperboloid are discussed. The simplest method of construction of these functions is introduced using the Fock space structure of the representation space of the su(1,1) algebra.…
We present some formulas for the computation of the zeros of the integral-degree associated Legendre functions with respect to the order.
Addition formulas exist in trigonometric functions. Double-angle and half-angle formulas can be derived from these formulas. Moreover, the relation equation between the trigonometric function and the hyperbolic function can be derived using…
One may consider the generalization of Jacobi polynomials and the Jacobi function of the second kind to a general function where the index is allowed to be a complex number instead of a non-negative integer. These functions are referred to…
The use of operational methods of different nature is shown to be a fairly powerful tool to study different problems regarding the theory of Legendre and Legendre-like polynomials. We show how the use of the well known integral…
For a finitary hereditary abelian category $\mathcal{A}$, we define a derived Hall algebra of its root category by counting the triangles and using the octahedral axiom, which is proved to be isomorphic to the Drinfeld double of Hall…