We consider the complex $ν$ plane structure of the associated Legendre function of the second kind $Q^{-1/2-K}_ν(\coshρ)$. We find that for any noninteger value for $K$ $Q^{-1/2-K}_ν(\coshρ)$ has an infinite number of poles in the complex $ν$ plane, but for any negative integer $K$ there are no poles at all. For $K=0$ or any positive integer $K$ there is only a finite number of poles, with there only being one single pole (at $ν=0$) when $K=0$. This pattern is characteristic of the exceptional points that appear in a wide variety of physical contexts. However, unusually for theories with exceptional points, $Q^{-1/2-K}_ν(\coshρ)$ has an infinite number of them. Other than in the $PT$-symmetry Jordan-block case, exceptional points usually occur at complex values of parameters. While not being Jordan-block exceptional points themselves, the exceptional points associated with the $Q^{-1/2-K}_ν(\coshρ)$ nonetheless occur at real values of $K$.
Exceptional orthogonal polynomials are families of orthogonal polynomials that arise as solutions of Sturm-Liouville eigenvalue problems. They generalize the classical families of Hermite, Laguerre, and Jacobi polynomials by allowing for polynomial sequences that miss a finite number of "exceptional" degrees. In this paper we introduce a new construction of multi-parameter exceptional Legendre polynomials by considering the isospectral deformation of the classical Legendre operator. Using confluent Darboux transformations and a technique from inverse scattering theory, we obtain a fully explic…
A relationship between partial derivatives of the associated Legendre function of the first kind with respect to its degree, $[\partial P_ν^{m}(z)/\partialν]_{ν=n}$, and to its order, $[\partial P_{n}^μ(z)/\partialμ]_{μ=m}$, is established for $m,n\in\mathbb{N}$. This relationship is used to deduce four new closed-form representations of $[\partial P_ν^{m}(z)/\partialν]_{ν=n}$ from those found recently for $[\partial P_{n}^μ(z)/\partialμ]_{μ=m}$ by the present author [R. Szmytkowski, J. Math. Chem. 46 (2009) 231]. Several new expressions for the associated Legendre function of the second kind …
The derivative of the associated Legendre function of the first kind of integer degree with respect to its order, $\partial P_{n}^μ(z)/\partialμ$, is studied. After deriving and investigating general formulas for $μ$ arbitrary complex, a detailed discussion of $[\partial P_{n}^μ(z)/\partialμ]_{μ=\pm m}$, where $m$ is a non-negative integer, is carried out. The results are applied to obtain several explicit expressions for the associated Legendre function of the second kind of integer degree and order, $Q_{n}^{\pm m}(z)$. In particular, we arrive at formulas which generalize to the case of $Q_{…
In our recent works [R. Szmytkowski, J. Phys. A 39 (2006) 15147; corrigendum: 40 (2007) 7819; addendum: 40 (2007) 14887], we have investigated the derivative of the Legendre function of the first kind, $P_ν(z)$, with respect to its degree $ν$. In the present work, we extend these studies and construct several representations of the derivative of the associated Legendre function of the first kind, $P_ν^{\pm m}(z)$, with respect to the degree $ν$, for $m\in\mathbb{N}$. At first, we establish several contour-integral representations of $\partial P_ν^{\pm m}(z)/\partialν$. They are then used to de…
We derive explicit expressions for 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. It is found that {displaymath} \frac{\partial^{2}P_ν(z)}{\partialν^{2}}\bigg|_{ν=0} =-2\Li_{2}\frac{1-z}{2}, {displaymath} where $\Li_{2}z$ is the dilogarithm (this formula has been recently arrived at by Schramkowski using \emph{Mathematica}), and that {displaymath} \frac{\partial^{3}P_ν(z)}{\partialν^{3}}\bigg|_{ν=0} =12\Li_{3}\frac{z+1}{2}-6\ln\frac{z+1}{2}\Li_{2}\frac{z+1}{2} -π^{2}…
A new kind of deformed calculus was introduced recently in studying of parabosonic coordinate representation. Based on this deformed calculus, a new deformation of Legendre polynomials is proposed in this paper, some properties and applications of which are also discussed.
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…
Associated Legendre functions arise in many problems of mathematical physics. By using the generalized Abel-Plana formula, in this paper we derive a summation formula for the series over the zeros of the associated Legendre function of the first kind with respect to the degree. The summation formula for the series over the zeros of the Bessel function, previously discussed in the literature, is obtained as a limiting case. The Wightman function for a scalar field with general curvature coupling parameter is considered inside a spherical boundary on background of constant negative curvature spa…
We present new asymptotic series for the Legendre and Jacobi functions of the first and second kinds in terms of Bessel functions with appropriate arguments. The results are useful in the context of scattering problems, improve on known limiting results, and allow the calculation of corrections to the leading Bessel-function approximations for these functions. Our derivations of these series are based on Barnes-type representations of the Legendre, Jacobi, and Bessel functions; our method appears to be new. We use the results, finally, to obtain asymptotic Bessel function expansions for the ro…
A closed-form formula is derived for the generalized Clebsch-Gordan integral $ \int_{-1}^1 {[}P_ν(x){]}^2P_ν(-x)\D x$, with $ P_ν$ being the Legendre function of arbitrary complex degree $ ν\in\mathbb C$. The finite Hilbert transform of $ P_ν(x)P_ν(-x),-1<x<1$ is evaluated. An analytic proof is provided for a recently conjectured identity $\int_0^1[\mathbf K(\sqrt{1-k^2})]^{3}\D k=6\int_0^1[\mathbf K(k)]^2\mathbf K(\sqrt{1-k^2})k\D k=[Γ(1/4)]^{8}/(128π^2) $ involving complete elliptic integrals of the first kind $ \mathbf K(k)$ and Euler's gamma function $ Γ(z)$.
We introduce an extended Kepler-Coulomb quantum model in spherical coordinates. The Schrödinger equation of this Hamiltonian is solved in these coordinates and it is shown that the wave functions of the system can be expressed in terms of Laguerre, Legendre and exceptional Jacobi polynomials (of hypergeometric type). We construct ladder and shift operators based on the corresponding wave functions and obtain their recurrence formulas. These recurrence relations are used to construct higher-order, algebraically independent integrals of motion to prove superintegrability of the Hamiltonian. The …
Trigonometric formulas are derived for certain families of associated Legendre functions of fractional degree and order, for use in approximation theory. These functions are algebraic, and when viewed as Gauss hypergeometric functions, belong to types classified by Schwarz, with dihedral, tetrahedral, or octahedral monodromy. The dihedral Legendre functions are expressed in terms of Jacobi polynomials. For the last two monodromy types, an underlying `octahedral' polynomial, indexed by the degree and order and having a non-classical kind of orthogonality, is identified, and recurrences for it a…
In this paper we generalize an identity first given by Heinrich Eduard Heine in his treatise, {\it Handbuch der Kugelfunctionen, Theorie und Anwendungen (1881), which gives a Fourier series for $1/[z-\cosψ]^{1/2}$, for $z,ψ\in\R$, and $z>1$, in terms of associated Legendre functions of the second kind with odd-half-integer degree and vanishing order. In this paper we give a generalization of this identity as a Fourier series of $1/[z-\cosψ]^μ$, where $z,μ\in\C$, $|z|>1$, and the coefficients of the expansion are given in terms of the same functions with order given by $\frac12-μ$. We are also …
In the study of the amplitudes for many-particle processes, and also for processes involving particles with spin, the use is made of matrix elements of the rotation group d^j_{μν}(z). In this paper the generalization of the functions d^j_{μν}(z) to arbitrary arguments and indices is studied. At the same time the functions of the second kind, analogous to Legendre functions of the second kind, are investigated. The results obtained play an important part in the introduction of complex angular momenta in many-particle processes. [Added in 2013: The generalized Legendre functions considered here …
We study impact of quantum phase transitions (QPTs) on the distribution of exceptional points (EPs) of the Hamiltonian in complex-extended parameter domain. Analyzing first- and second-order QPTs in the Lipkin model, we find an exponentially and polynomially close approach of EPs to the respective critical point with an increasing size of the system. If the critical Hamiltonian is subject to random perturbations of various kinds, the averaged distribution of EPs close to the critical point still carries decisive information on the QPT type. We therefore claim that properties of the EP distribu…
It is conjectured that the exceptional-point (EP) singularity of a one-parametric quasi-Hermitian $N$ by $N$ matrix Hamiltonian $H(t)$ can play the role of a quantum phase-transition interface connecting different dynamical regimes of a unitary quantum system. Six realizations of the EP-mediated quantum phase transitions "of the third kind" are described in detail. Fairly realistic Bose-Hubbard (BH) and discrete anharmonic oscillator (AO) models of any matrix dimension $N$ are considered in the initial, intermediate, or final phase. In such a linear algebraic illustration of the changes of pha…
We construct a new family of quasi-solvable and N-fold supersymmetric quantum systems where each Hamiltonian preserves an exceptional polynomial subspace of codimension 2. We show that the family includes as a particular case the recently reported rational radial oscillator potential whose eigenfunctions are expressed in terms of the X_2-Laguerre polynomials of the second kind. In addition, we find that the two kinds of the X_2-Laguerre polynomials are ingeniously connected with each other by the N-fold supercharge.
