Comment on the orthogonality of the Macdonald functions of imaginary order

虚阶圆锥函数的正交关系

Orthogonality relations for conical functions of imaginary order

本文推导了虚阶圆锥或Mehler函数的正交关系，并用Dirac delta函数表示，扩展了连带Legendre函数的正交关系。

An orthogonality relation for the Whittaker functions of the second kind of imaginary order

An orthogonality relation for the Whittaker functions of the second kind of imaginary order, $W_{κ,\mathrm{i}μ}(x)$, with $μ\in\mathbb{R}$, is investigated. The integral $\int_{0}^{\infty}\mathrm{d}x\: x^{-2}W_{κ,\mathrm{i}μ}(x)W_{κ,\mathrm{i}μ'}(x)$ is shown to be proportional to the sum $δ(μ-μ')+δ(μ+μ')$, where $δ(μ\pmμ')$ is the Dirac delta distribution. The proportionality factor is found to be $π^{2}/[μ\sinh(2πμ)Γ({1/2}-κ+\mathrm{i}μ) Γ({1/2}-κ-\mathrm{i}μ)]$. For $κ=0$ the derived formula reduces to the orthogonality relation for the Macdonald functions of imaginary order, discussed rece…

带纯虚位移的离散量子力学中多指标正交多项式的离散正交关系

Discrete Orthogonality Relations for the Multi-Indexed Orthogonal Polynomials in Discrete Quantum Mechanics with Pure Imaginary Shifts

该研究探讨带纯虚位移的离散量子力学里多指标正交多项式的离散正交关系，验证了三类情形(1)多项式满足该关系并推测其归一化常数。

Orthogonality of the Ferrers' Associated Legendre Functions of the Second Kind with Imaginary Argument

In this work we study the associated Legendre functions of the second kind with a purely imaginary argument $Q^k_\ell(\mathbb{i}\, x)$. We derive the conditions under which they provide a set of square integrable functions when $x \in \mathbb{R}$ and we prove the relevant orthogonality relation that they satisfy.

A method of composition orthogonality and new sequences of orthogonal polynomials and functions for non-classical weights

A new method of composition orthogonality is introduced. It is applied to generate new sequences of orthogonal polynomials and functions. In particular, classical orthogonal polynomials are interpreted in the sense of composition orthogonality. Finally, new sequences of orthogonal polynomials with respect to the weight function $x^αρ^2_ν(x), ρ_ν(x)= 2 x^{ν/2} K_ν(2\sqrt x),\ x >0, ν\ge 0, α> -1$, where $K_ν(z)$ is the modified Bessel function or Macdonald function, are investigated. Differential properties, recurrence relations, explicit representations, generating functions and Rodrigues-type…

On the generalized Lebedev index transform

An essential generalization of the Lebedev index transform with the square of the Macdonald function is investigated. Namely, we consider a family of integral operators with the positive kernel $|K_{(iτ+α)/2}(x)|^2, α\ge 0,\ x >0, \ τ\in \mathbb{R},$ where $K_μ(z)$ is the Macdonald function and $ i$ is the imaginary unit. Mapping properties such as the boundedness, compactness, invertibility are investigated for these operators and their adjoints in the Lebesgue weighted spaces. Inversion theorems are proved. Important particular cases are exhibited. As an interesting application, a solution o…