We provide closed-form expressions for the degree-derivatives $[\partial^{2}P_ν(z)/\partialν^{2}]_{ν=n}$ and $[\partial Q_ν(z)/\partialν]_{ν=n}$, with $z\in\mathbb{C}$ and $n\in\mathbb{N}_{0}$, where $P_ν(z)$ and $Q_ν(z)$ are the Legendre functions of the first and the second kind, respectively. For $[\partial^{2}P_ν(z)/\partialν^{2}]_{ν=n}$, we find that $$\displaystyle\frac{\partial^{2}P_ν(z)}{\partialν^{2}}\bigg|_{ν=n}=-2P_{n}(z)\textrm{Li}_{2}\frac{1-z}{2}+B_{n}(z)\ln\frac{z+1}{2}+C_{n}(z),$$ where $\textrm{Li}_{2}[(1-z)/2]$ is the dilogarithm function, $P_{n}(z)$ is the Legendre polynomial, while $B_{n}(z)$ and $C_{n}(z)$ are certain polynomials in $z$ of degree $n$. For $[\partial Q_ν(z)/\partialν]_{ν=n}$ and $z\in\mathbb{C}\setminus[-1,1]$, we derive $$\displaystyle \frac{\partial Q_ν(z)}{\partialν}\bigg|_{ν=n}=-P_{n}(z)\textrm{Li}_{2}\frac{1-z}{2}-\frac{1}{2}P_{n}(z)\ln\frac{z+1}{2}\ln\frac{z-1}{2} +\frac{1}{4}B_{n}(z)\ln\frac{z+1}{2}-\frac{(-1)^{n}}{4}B_{n}(-z)\ln\frac{z-1}{2}-\frac{π^{2}}{6}P_{n}(z) +\frac{1}{4}C_{n}(z)-\frac{(-1)^{n}}{4}C_{n}(-z).$$ A counterpart expression for $[\partial Q_ν(x)/\partialν]_{ν=n}$, applicable when $x\in(-1,1)$, is also presented. Explicit representations of the polynomials $B_{n}(z)$ and $C_{n}(z)$ as linear combinations of the Legendre polynomials are given.
Expressions for the derivatives of the Legendre polynomials of the first kind with respect to the order of these polynomials are given. An explicit form for the fourth derivative is presented.
In this paper, we study the scaling properties of Legendre polynomials Pn(x). We show that Pn(ax), where a is a constant, can be expanded as a sum of either Legendre polynomials Pn(x) or their multiple derivatives dkPn(x)/dxk, and we derive a general expression for the expansion coefficients. In addition, we demonstrate that the multiple derivative dkPn(x)/dxk can also be expressed as a sum of Legendre polynomials and we obtain a recurrence relation for the coefficients.
On parameter derivatives of the associated Legendre function of the first kind (with applications to the construction of the associated Legendre function of the second kind of integer degree and order)
Index transforms with the product of the associated Legendre functions are introduced. Mapping properties are investigated in the Lebesgue spaces. Inversion formulas are proved. The results are applied to solve a boundary value problem in a wedge for a third order partial differential equation.
The classical Lebedev index transform (1967), involving squares and products of the Legendre functions is generalized on the associated Legendre functions of an arbitrary order. Mapping properties are investigated in the Lebesgue spaces. Inversion formulas are proved. As an interesting application, a solution to the boundary value problem for a third order partial differential equation is obtained.
The parameter derivatives $[\partial^{2}P_ν(z)/\partialν^{2}]_{ν=0}$ and $[\partial^{3}P_ν(z)/\partialν^{3}]_{ν=0}$, where $P_ν(z)$ is the Legendre function of the first kind
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 representations linking Hermite and Legendre like polynomials and of operational technique allow the derivation of new properties regarding the generating functions, repeated derivatives of irrational functions and so on. We finally extend the method of the integral transform to functions with more than one variable and derive new expansion criteria of a given function …
The main objective of this paper is to give a wide study on the conformable fractional Legendre polynomials (CFLPs). This study is assumed to be a generalization and refinement, in an easy way, of the scalar case into the context of the conformable fractional differentiation. We introduce the CFLPs via different generating functions and provide some of their main properties and convergence results. Subsequently, some pure recurrence and differential recurrence relations, Laplace's first integral formula and orthogonal properties are then developed for CFLPs. We append our study with presenting…
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 generalize some classical multi-integration formulas. As a result of the determination of these formulae, we compute some interesting special values and integral representations for certain particular combinations of the degree and order including the case where there is symmetry and antisymmetry for the degree and order parameters. As a consequence of our analysis…
In this paper, we derive new recurrence relations for two-variable orthogonal polynomials for example Jacobi polynomial, Bateman's polynomial and Legendre polynomial via two different differential operators $Ξ=\left(\frac{\partial }{\partial z} +\sqrt{w} \frac{\partial }{\partial w} \right)$ and $Δ=\left(\frac{1}{w} \frac{\partial }{\partial z} +\frac{1}{z} \frac{\partial }{\partial w} \right)$. We also derive some special cases of our main results.
The differentiability of the one parameter family of Okomoto's functions as functions of $x$ has been analyzed extensively since their introduction in 2005. As an analogue to a similar investigation, in this paper, we consider the partial derivative of Okomoto's functions with respect to the parameter $a$. We place a significant focus on $a = 1/3$ to describe the properties of a nowhere differentiable function $K(x)$ for which the set of points of infinite derivative produces an example of a measure zero set with Hausdorff dimension $1$.
The solution in hyperspherical coordinates for $N$ dimensions is given for a general class of partial differential equations of mathematical physics including the Laplace, wave, heat and Helmholtz, Schrödinger, Klein-Gordon and telegraph equations and their combinations. The starting point is the Laplacian operator specified by the scale factors of hyperspherical coordinates. The general equation of mathematical physics is solved by separation of variables leading to the dependencies: (i) on time by the usual exponential function; (ii) on longitude by the usual sinusoidal function; (iii) on ra…
We comment on a Mazur problem from "Scottish Book" concerning second partial derivatives. It is proved that, if a function $f(x,y)$ of real variables defined on a rectangle has continuous derivative with respect to $y$ and for almost all $y$ the function $\,F_y(x):=f'_y(x,y)$ has finite variation, then almost everywhere on the rectangle there exists the partial derivative $f"_{yx}$. We construct a separately twice differentiable function, whose partial derivative $f'_x$ is discontinuous with respect to the second variable on a set of positive measure. This solves in the negative the Mazur prob…
We first prove some weighted inequalities for compositions of functions on time scales which are in turn applied to establish some new dynamic Opial-type inequalities in several variables. Some generalizations and applications to partial differential dynamic equations are also considered.
