# Lebedev's type index transforms with the squares of the associated Legendre functions

## TL;DR

This paper generalizes Lebedev's index transform using associated Legendre functions, investigates its properties in Lebesgue spaces, proves inversion formulas, and applies it to solve a third-order PDE boundary value problem.

## Contribution

It introduces a generalized Lebedev index transform with associated Legendre functions, analyzes its mapping properties, and provides inversion formulas and an application to PDEs.

## Key findings

- Generalized Lebedev transform with associated Legendre functions.
- Established mapping properties in Lebesgue spaces.
- Derived inversion formulas and solved a boundary value problem.

## Abstract

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.